NCERT Solutions Class 8 Maths Chapter 04 Quadrilaterals

Page 82

Question. Observe the following figures.
Answer: Figures (i), (ii) and (iii) are quadrilaterals while the others are not. This is because these three have four angles and four sides.
In simple words: A quadrilateral is a shape with four corners and four straight sides. Figures (i), (ii), and (iii) have this, but (iv) and (v) do not.

Exam Tip: When asked to identify quadrilaterals, always count the number of sides and angles - a true quadrilateral must have exactly four of each.

Question. Are there other ways to define a rectangle?
Answer: Yes, there are several other ways to define a rectangle, though all definitions carry the same meaning. A rectangle can be described as: (a) a parallelogram in which every angle is a right angle; (b) a parallelogram where even a single right angle is present, because then all other angles automatically become 90 degrees as well; (c) a quadrilateral whose diagonals are equal in length and split each other in half; (d) a quadrilateral with opposite sides equal and one right angle.
In simple words: A rectangle can be described in many ways. The simplest way is: all angles are 90 degrees, or the diagonals are equal and cut each other in half.

Exam Tip: Learn all four definitions - examiners may ask you to prove that a shape is a rectangle using a different definition than the main one you learned.

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Question. She already has one 8 cm long strip. What should be the length of the other strip? Where should they both be joined?
Answer: The second strip must also be 8 cm long, since a rectangle's diagonals are always equal in length. The two strips should be joined at their midpoints, which means they should be joined at 4 cm from each end.
In simple words: Both strips must be the same length. They must be joined at the middle point of each strip.

Exam Tip: Remember that in a rectangle, the diagonals are always equal and always cut each other exactly in the middle - this is a key property to use.

Question 1. What is the length of the other diagonal?
Answer: The other diagonal is 8 cm long, because in a rectangle the two diagonals are always equal in length.
In simple words: Both diagonals of a rectangle have the same length.

Exam Tip: This is one of the most important properties of rectangles - if you know one diagonal, you know the other one too.

Question 2. What is the point of intersection of the two diagonals?
Answer: The two diagonals of a rectangle always cut each other in half. So the point where they meet is the midpoint of each diagonal.
In simple words: The diagonals meet at their midpoints.

Exam Tip: In any parallelogram (including rectangles), the diagonals always bisect each other - this is a property you must know.

Question 3. What should the angle be between the diagonals?
Answer: When the two diagonals meet, they form two different angles. One angle will be acute (less than 90 degrees) and the other will be obtuse (more than 90 degrees).
In simple words: The diagonals make one sharp angle and one wide angle where they meet.

Exam Tip: In a rectangle, the diagonals never meet at right angles - they form one acute and one obtuse angle.

Page 85

Question. Can the following equalities be used to establish that ΔAOD ≅ ΔCOB?
Answer: No, these equalities cannot be used to prove that ΔAOD and ΔCOB are congruent. To use the SAS (Side-Angle-Side) test for congruence, you need the angle to be between the two equal sides. Here, we have AO = CO and ∠AOB = ∠COD, but this is SSA (Side-Side-Angle), which is not a valid congruence rule. The angle is not between the two matching sides, so we cannot conclude that the triangles are congruent.
In simple words: For congruence, the angle must be in the middle of the two sides. Here it is not, so the triangles are not congruent.

Exam Tip: Always check that the equal angle sits between the two equal sides for SAS - if it does not, the rule does not work.

Question. Can you find all the remaining angles?
Answer: Since ∠AOB and ∠COD are vertically opposite angles, ∠AOB = ∠COD = 60 degrees. Because ∠AOB and ∠BOC form a linear pair (they sit on a straight line), they add up to 180 degrees. So ∠BOC = 180 degrees - 60 degrees = 120 degrees. Since ∠BOC and ∠AOD are also vertically opposite, ∠AOD = ∠COB = 120 degrees.
In simple words: Opposite angles are equal. Angles on a line add up to 180 degrees. Use these rules to find: ∠AOB = 60°, ∠BOC = 120°, ∠COD = 60°, ∠AOD = 120°.

Exam Tip: When two lines cross, they make four angles - two pairs of equal angles. Angles next to each other add up to 180 degrees.

Question. Can you find the value of a?
Answer: In triangle AOB, the sides OA and OB are equal (since the diagonals of a rectangle bisect each other). When two sides of a triangle are equal, the angles opposite to them are also equal. So ∠OAB = ∠OBA = a. Using the angle sum property of triangles: a + a + 60 degrees = 180 degrees. This gives us 2a = 120 degrees, so a = 60 degrees.
In simple words: When two sides are equal, the angles across from them are equal too. Add all three angles to get 180 degrees, then solve for a.

Exam Tip: In an isosceles triangle (two equal sides), the angles opposite those sides are equal - use this fact first before applying the angle sum property.

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Question. Can we now identify what type of quadrilateral ABCD is?
Answer: ABCD is a rectangle. We know this because all four angles of the quadrilateral are 90 degrees, and its diagonals do not meet each other at right angles. These are the defining properties of a rectangle.
In simple words: All angles are 90 degrees, so it is a rectangle. In a rectangle, the diagonals are not perpendicular.

Exam Tip: If all angles are 90 degrees, the shape is a rectangle. If all angles are 90 degrees AND the diagonals are perpendicular, then it is a square.

Question. What can we say about its sides?
Answer: In a rectangle, sides that are across from each other are always equal. So the opposite sides must be equal in length.
In simple words: The opposite sides of a rectangle are equal.

Exam Tip: This is a key property of rectangles - opposite sides are equal, and opposite sides are also parallel to each other.

Question. Will ABCD remain a rectangle if the angles between the diagonals are changed? Can we generalise this?
Answer: ABCD will continue to be a rectangle as long as the angles between the diagonals stay the same. It will remain a rectangle until the angle between the diagonals becomes exactly 90 degrees. Once the diagonals meet at a right angle, the shape becomes a square instead of just a rectangle.
In simple words: A rectangle stays a rectangle if you change the diagonal angle - but if the angle becomes 90 degrees, it becomes a square.

Exam Tip: The key difference is the angle between diagonals: if it is 90 degrees, you have a square; if not, you have a rectangle.

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Question. Can you find the other angles?
Answer: You can find each angle by using the angle sum property in each of the four triangles formed by the diagonals - triangles COD, AOB, BOC, and AOD.
In simple words: The sum of angles in any triangle is 180 degrees. Use this for each of the four triangles to find all angles.

Exam Tip: When a quadrilateral has diagonals drawn, it splits into four triangles - use the angle sum property for each one.

Question. What is the value of a (in degrees) in terms of x?
Answer: In triangle AOB, the sides OA and OB are equal. The angles opposite to equal sides are also equal, so ∠OAB = ∠OBA = a. Using the angle sum property: a + a + x = 180 degrees. Simplifying: 2a = 180 - x, so a = (180 - x)/2.
In simple words: The two base angles are equal. Add them with the top angle to get 180 degrees, then solve for a.

Exam Tip: When you see equal sides in a triangle, write down the equal angles first - this makes the calculation much easier.

Question. What can we say about AB and CD, and AD and BC?
Answer: Sides that sit across from each other in a rectangle are always equal in length. Therefore, AB = CD and AD = BC.
In simple words: Opposite sides of a rectangle are equal.

Exam Tip: This property - that opposite sides are equal - is true for all rectangles and is one of the main ways to identify them.

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Question. In the earlier definition, we stated that a rectangle has (a) opposite sides of equal length, and (b) all angles equal to 90°. Would we be wrong if we just define a rectangle as a quadrilateral in which all the angles are 90°?
Answer: No, we would not be wrong, though the definition might seem incomplete. If you say "a rectangle is a quadrilateral where all angles are 90 degrees," this automatically makes the opposite sides equal as well. This happens because any quadrilateral with all angles equal to 90 degrees must be a parallelogram. In every parallelogram, opposite sides are always equal. So even without mentioning that opposite sides are equal, the fact that all angles are 90 degrees forces this property to happen. Therefore, the definition using only the 90-degree angle condition is actually sufficient and complete.
In simple words: If all angles are 90 degrees, then opposite sides must be equal - this happens automatically. So you only need one definition, not both.

Exam Tip: Understand that some definitions are complete even though they seem short - all angles = 90° is enough to define a rectangle fully.

Question. If you think that this definition is incomplete, try constructing a quadrilateral in which the angles are all 90° but the opposite sides are not equal. Are you able to construct such a quadrilateral?
Answer: If you try to draw a quadrilateral with all angles equal to 90 degrees, this is what happens: First, draw one side AB. At points A and B, draw right angles and extend the next two sides AD and BC. These two new sides automatically become parallel to each other. Now, to close the figure, join points D and C. The line segment DC becomes parallel to AB all by itself. As a result, both pairs of opposite sides end up parallel, which means the shape is now a parallelogram with all right angles - that is, exactly a rectangle. In a parallelogram, opposite sides are always equal. So it is impossible to create a quadrilateral where all angles are 90 degrees but opposite sides are not equal. Therefore, the definition "a quadrilateral where all angles are 90 degrees" is complete enough to define a rectangle.
In simple words: Try to draw such a shape - you will find that opposite sides automatically become equal. You cannot make a different shape.

Exam Tip: This shows why certain definitions are complete - the properties they state force other properties to happen too.

Question. Consider a quadrilateral ABCD with all angles measuring 90°. What can we say about the opposite sides of such a quadrilateral?
Answer: If a quadrilateral ABCD has all angles measuring 90 degrees, then the opposite sides must be parallel to each other. This occurs because when all angles are right angles, the sides automatically become parallel in pairs, so AB is parallel to CD. Additionally, opposite sides must be equal in length. Since the opposite sides are parallel and the angles are 90 degrees, the quadrilateral becomes a parallelogram with all right angles - that is, a rectangle. In every rectangle ABCD, AB = CD and BC = AD. Thus, when all angles in a quadrilateral equal 90 degrees, the opposite sides are both equal and parallel.
In simple words: All angles are 90 degrees means: opposite sides are equal, and opposite sides are parallel.

Exam Tip: A 90-degree angle at every corner forces the opposite sides to be parallel and equal - this is a direct consequence of how parallel lines work.

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Question. Is it wrong to write ΔBAD ≅ ΔCDB? Why?
Answer: Yes, it is wrong to write ΔBAD ≅ ΔCDB in a rectangle ABCD. Although triangles BAD and CDB may appear to share some equal sides, they do not match in size. In a rectangle, AB = CD and BC = AD (opposite sides are equal). Triangle BAD uses sides BA, AD, and diagonal BD. Triangle CDB uses sides CD, CB, and the same diagonal BD. While diagonal BD is common to both triangles, the other sides are: BA compared to CD (equal) and AD compared to CB (equal). However, the angles between these sides are different. In triangle BAD, the angle between BA and AD is ∠BAD. In triangle CDB, the angle between CD and CB is ∠DCB. These two angles are not equal. Even though two pairs of sides match, the included angles are different, so the SAS (Side-Angle-Side) condition for congruence fails. Therefore, the triangles cannot be congruent.
In simple words: The two triangles have equal sides, but the angles between those sides are different. So they are not congruent.

Exam Tip: For SAS congruence, the equal angle MUST sit between the two equal sides. If the angle is not in the middle, SAS does not work.

Question. Are the opposite sides of a rectangle parallel?
Answer: Yes, opposite sides of a rectangle are parallel. They are not only parallel but also equal in length.
In simple words: In a rectangle, opposite sides are parallel and equal.

Exam Tip: A rectangle is a special type of parallelogram - so it has all the properties of a parallelogram, including parallel opposite sides.

Page 92

Question. Let us consider the Carpenter's Problem again. If the wooden strips have to be placed such that the thread passing through their endpoints forms a square, what must be done?
Answer: The two equal threads must cross each other at their midpoints and form a 90-degree angle.
In simple words: The two sticks (representing diagonals) must be equal, cross in the middle, and meet at 90 degrees.

Exam Tip: A square has very specific properties for its diagonals: they are equal, bisect each other, and are perpendicular.

Question. What more needs to be done to get equal sidelengths as well? Can this be achieved by properly choosing the angle between the diagonals?
Answer: If the two equal threads cross at their midpoints and at a 90-degree angle, the side lengths of the quadrilateral automatically become equal. Yes, this can be achieved by setting the angle between the diagonals to 90 degrees. When diagonals are equal, bisect each other, and meet at 90 degrees, the resulting quadrilateral is a square with all sides equal.
In simple words: When the diagonals cross at 90 degrees and bisect each other, all four sides automatically become equal.

Exam Tip: For a square, focus on three things: diagonals are equal, diagonals bisect each other, and diagonals are perpendicular - these three conditions guarantee a square.

Question. By the SSS condition for congruence, ΔBOA ≅ ΔBOC. Can this be used to find the angles ∠BOA and ∠BOC formed by the diagonals?
Answer: Yes, this can be used to find the angles. In congruent triangles, matching parts are equal. Since ΔBOA and ΔBOC are congruent by SSS (all three corresponding sides are equal), their matching angles are also equal. You can use this equality to calculate the specific angles ∠BOA and ∠BOC that the diagonals make when they intersect.
In simple words: When two triangles are congruent, their angles are also equal. Use this to find the angles the diagonals make.

Exam Tip: Remember that in congruent triangles, not just sides but also angles match up - use this fact to find unknown angles.

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Question. Using this fact, construct a square with a diagonal of length 8 cm.
Answer: Follow these steps: (1) Use a ruler to draw a straight line segment AC that is 8 cm long. (2) Find the exact midpoint O of segment AC. To do this, take a compass and open it to more than half the distance of AC. Draw curved arcs above and below the line AC, starting from point A. Without changing the compass width, draw arcs from point C as well. These arcs will cross at two locations. Join these crossing points with a straight line. This new line meets AC at point O, which is the midpoint. (3) The line that passes through O and is perpendicular to AC becomes the second diagonal BD of the square. (4) In a square, both diagonals are equal in length. So BD must also be 8 cm long. Place the compass point at O and set the radius to half of AC, which is 4 cm. Mark two points on the perpendicular line - these are B and D, each at 4 cm away from O. This ensures BO = OD = 4 cm, making BD = 8 cm. (5) Connect A to B, B to C, C to D, and D to A. This completed figure is your square ABCD.
In simple words: Draw the diagonal. Find its middle. Draw another diagonal through that middle at 90 degrees, also 8 cm long. Connect the four endpoints.

Exam Tip: The key idea: in a square, both diagonals are equal and cross at the middle at 90 degrees - use this to construct the shape accurately.

Question. Verify if this is true by going through geometric reasoning in Deduction 1 and Deduction 2, and see if they apply to a square as well.
Answer: Yes, in a square, both deductions are completely true based on geometric reasoning. First, the diagonals of a square are always equal in length. This is because a square is a special type of parallelogram where all angles are 90 degrees and all sides are equal. Since the diagonals meet at right angles and split the square into four matching right triangles, each half of both diagonals is the same length. This makes the full lengths of AC and BD equal, so if one diagonal is 8 cm, the other must also be 8 cm. Second, the diagonals of a square always cut each other into two equal parts. A square has opposite sides that are parallel, so it works like a parallelogram. In every parallelogram, diagonals cut each other into two equal pieces. Therefore, where the diagonals meet is the midpoint of both AC and BD. Thus, both Deduction 1 and Deduction 2 hold true for every square.
In simple words: In a square, diagonals are equal and bisect each other because a square is a special parallelogram. Both facts are always true.

Exam Tip: A square is a special case of both a rectangle and a rhombus - it inherits and confirms all the diagonal properties of both shapes.

Question. What are the measures of ∠1, ∠2, ∠3, and ∠4? See if you can reason and/or experiment to figure this out!
Answer: In triangle ADC, sides AD and DC are equal. Therefore, ∠1 = ∠3 (angles opposite equal sides are equal). Using the angle sum property for triangle ADC: ∠1 + ∠3 + 90 degrees = 180 degrees. Substituting ∠1 for ∠3: ∠1 + ∠1 + 90 degrees = 180 degrees, which gives 2∠1 = 90 degrees, so ∠1 = 45 degrees. Thus, ∠1 = ∠3 = 45 degrees. Similarly, in triangle ABC, sides AB and BC are equal. Therefore, ∠4 = ∠2 (angles opposite equal sides are equal). Using the angle sum property for triangle ABC: ∠2 + ∠4 + 90 degrees = 180 degrees. Substituting ∠2 for ∠4: ∠2 + ∠2 + 90 degrees = 180 degrees, which gives 2∠2 = 90 degrees, so ∠2 = 45 degrees. Thus, ∠2 = ∠4 = 45 degrees.
In simple words: In each triangle, two sides are equal, so two angles are equal. Use the angle sum property to find that all four angles are 45 degrees.

Exam Tip: When a square is divided by its diagonals, each angle at a corner is split into two 45-degree angles - this is a key pattern to remember.

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Question 1. Find all the other angles inside the following rectangles.
Answer:
(i) In triangle OAB, since the diagonals of a rectangle cut each other in half, AO = BO. The angles opposite equal sides are equal, so ∠OBA = ∠OAB = 30 degrees. Similarly, ∠OCD = ∠ODC = 30 degrees. In triangle OAB, using the angle sum property: 30 + 30 + ∠AOB = 180 degrees, so ∠AOB = 120 degrees. Since vertically opposite angles are equal, ∠DOC = ∠AOB = 120 degrees. Angles ∠BOC and ∠AOB form a linear pair: ∠BOC = 180 - 120 = 60 degrees. Since vertically opposite angles are equal, ∠AOD = ∠BOC = 60 degrees. In triangle OBC, since BO = CO, we have ∠OBC = ∠OCB. Using the angle sum property: ∠OBC + ∠OCB + 60 = 180, so 2∠OBC = 120, giving ∠OBC = ∠OCB = 60 degrees. Similarly, ∠OAD = ∠ODA = 60 degrees.
In simple words: Use the fact that equal sides make equal angles. The diagonals create four triangles. Find angles in each triangle using the angle sum property of 180 degrees.

Exam Tip: When diagonals are drawn in a rectangle, you get four triangles - solve for angles in each triangle separately using properties of isosceles triangles.

Question 2. Draw a quadrilateral whose diagonals have equal lengths of 8 cm that bisect each other, and intersect at an angle of (i) 30° (ii) 40° (iii) 90° (iv) 140°
Answer: For each angle, follow this method: Draw two straight lines that cross at a point, forming the given angle. Mark the intersection point as O. Measure 4 cm on each line from O in both directions - this creates eight points total (four pairs, two per line). These eight points represent the four vertices of your quadrilateral. Connect the four endpoints to complete the shape. (i) When the angle is 30 degrees, you get a quadrilateral that is elongated and narrow. (ii) At 40 degrees, the shape is still elongated but slightly wider. (iii) At 90 degrees, the diagonals are perpendicular and the shape becomes a square. (iv) At 140 degrees, the quadrilateral is again elongated. In every case, the diagonals remain 8 cm long (4 cm on each side of the center) and bisect each other at the center point O.
In simple words: Draw two lines crossing at the right angle. Mark points 4 cm away on all sides. Connect the four outer points. The shape changes with different angles.

Exam Tip: This exercise shows that rectangles and squares are just special cases - any quadrilateral with equal diagonals that bisect each other exists for any angle between them.

Question 3. Consider a circle with centre O. Line segments PL and AM are two perpendicular diameters of the circle. What is the figure APML? Reason and/or experiment to figure this out.
Answer: Figure APML is a square. Here is why: PL and AM are diameters of the circle and meet each other at right angles. Therefore, they divide the circle into four equal parts, creating four arcs of 90 degrees each. The points A, P, M, and L are located on the circle at the ends of these two diameters. Because the arcs AP, PM, ML, and LA are all equal, the chords connecting them are also equal. In a circle, equal arcs produce equal chords, so AP = PM = ML = LA. This means all four sides of quadrilateral APML are equal. Now look at the angle at point P, which is ∠APM. This angle is inscribed in a semicircle (created by diameter AM), and a well-known fact states that any angle in a semicircle is 90 degrees. So ∠APM = 90 degrees. Similarly, all other angles - at points A, M, and L - are also 90 degrees because each is formed using a diameter. Since all four sides are equal and all four angles are 90 degrees, APML is a square.
In simple words: Two perpendicular diameters create four equal sides and four right angles. This makes a square inscribed in the circle.

Exam Tip: Any angle inscribed in a semicircle is 90 degrees - this is a fundamental circle theorem to remember and apply.

Question 4. We have seen how to get 90° using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact 90° using these?
Answer: You can create an exact 90-degree angle using two equal sticks and a thread by applying the idea of a right-angled triangle. First, take the thread and tie it so that its total length exactly matches the diagonal of the square you want to build. Next, position the two sticks so they meet at one end, much like two sides of a square meeting at a corner. Now stretch the thread between the two free ends of the sticks - the ends that are not connected to each other. Adjust the two sticks until the stretched thread becomes tight and forms the longest distance between these two ends. When this happens, the angle between the two sticks is exactly 90 degrees. This works because of a property of right triangles: in a right-angled triangle, the side opposite the right angle (the diagonal) is always the longest side. When you have two equal sticks representing two equal sides and the thread representing the diagonal, adjusting until the thread is taut and longest forces the angle between the sticks to be 90 degrees.
In simple words: Tie the thread to the diagonal length. Hold the two sticks together at one end and stretch the thread between their other ends. Adjust until the thread is tight and longest. The angle is now 90 degrees.

Exam Tip: This demonstrates the converse of the Pythagorean theorem - if the diagonal is longest, the angle must be 90 degrees.

Question 5. Can opposite sides that are parallel and equal be used as a definition of a rectangle? In other words, is every quadrilateral with opposite sides parallel and equal also a rectangle?
Answer: No, this cannot serve as a rectangle's definition. When a quadrilateral has opposite sides that are parallel and equal, it becomes a parallelogram, but not every parallelogram is a rectangle. A rectangle requires an additional special condition - all four angles must measure 90 degrees. Many parallelograms have opposite sides that are equal and parallel, yet their angles are not right angles (they appear slanted instead). This means a parallelogram might resemble a tilted shape rather than a rectangle.
In simple words: A shape with opposite sides parallel and equal is called a parallelogram, but it is not always a rectangle unless all its angles are 90 degrees.

Exam Tip: Remember that a rectangle is a special parallelogram - it has all the features of a parallelogram plus the extra requirement that every angle must be 90 degrees.

Question 6. Is it possible to construct a quadrilateral with three angles equal to 90° and the fourth angle not equal to 90°?
Answer: No, this is not possible. The total of all angles in any quadrilateral is always 360 degrees. If three angles each measure 90 degrees, their sum equals 270 degrees. The fourth angle must then equal 360 minus 270, which gives 90 degrees. This forces the fourth angle to also be 90 degrees.
In simple words: If three corners of a four-sided shape are right angles, the last corner must also be a right angle because all four angles must add to 360 degrees.

Exam Tip: This is a key property - use the angle sum formula (360 degrees for quadrilaterals) to prove why the fourth angle cannot differ from the others when three are already fixed at 90 degrees.

Question 7. Are there quadrilaterals that have parallel opposite sides that are not rectangles?
Answer: Yes, many quadrilaterals have parallel opposite sides but are not rectangles. Any quadrilateral whose opposite sides are parallel is called a parallelogram. However, not every parallelogram is a rectangle. Examples include:
- Parallelogram: opposite sides are parallel and equal, but angles are not necessarily 90 degrees.
- Rhombus: all four sides are equal and opposite sides are parallel, but angles are not all 90 degrees (they are only 90 degrees in a square).
A square is a special case of a parallelogram, but since you asked for shapes that are not rectangles, both rhombus and general parallelograms qualify as answers.
In simple words: A shape can have opposite sides that are parallel and equal without being a rectangle. If the angles are not all 90 degrees, it is a different shape like a parallelogram or rhombus.

Exam Tip: Distinguish between parallelogram (opposite sides parallel), rhombus (all sides equal), and rectangle (opposite sides parallel AND all angles 90 degrees) - these are three different types of quadrilaterals.

Question 8. Construct a parallelogram that is not a rectangle by recalling how parallel lines can be constructed using a ruler and a set-square or a compass and a ruler.
Answer:
Steps of Construction:
Step 1: Draw a line segment AB.
Step 2: Construct a line parallel to AB through a point C.
- Choose a point C anywhere above AB (but not directly above it).
- Using a set-square, slide it along the ruler to draw a line through C that runs parallel to AB.
- Mark a point D on this new line. Now CD is parallel to AB.
Step 3: Construct a line through B parallel to CD.
- Again use the set-square to draw a line through B that runs parallel to CD.
- This line will meet the line through C at point D automatically.
Step 4: Join A to D and B to C to create quadrilateral ABCD where AB is parallel to CD and BC is parallel to AD, but the angles are not 90 degrees. This is a parallelogram, not a rectangle.
In simple words: Draw two sets of parallel lines using a set-square. Where they meet, you get a four-sided shape with opposite sides parallel - this is a parallelogram. If you choose a point C that is not directly above AB, the angles will not be 90 degrees.

Exam Tip: The key to constructing a non-rectangular parallelogram is choosing point C at an angle rather than directly above or below line AB - this ensures the angles will not all be right angles.

Question 9. Is a rectangle a parallelogram?
Answer: Yes, a rectangle is a parallelogram. A parallelogram is defined as a quadrilateral in which opposite sides are parallel. A rectangle has all these features: opposite sides are parallel, opposite sides are equal, and all angles measure 90 degrees. Since a rectangle satisfies all the requirements of a parallelogram and additionally has the special property that all angles are right angles, it is a special type of parallelogram.
In simple words: A rectangle is a parallelogram with the added rule that all four corners must be right angles.

Exam Tip: Recognize that a rectangle is a special case of a parallelogram - it has all the properties of a parallelogram, plus the extra condition of having 90-degree angles.

Question 10. Draw a parallelogram with adjacent sides of lengths 4 cm and 5 cm and an angle of 30° between them.
Answer:
Steps of Construction:
Step 1: Draw line segments AB = 4 cm and AD = 5 cm with a 30-degree angle between them.
Step 2: Draw a line parallel to AB through point D and a line parallel to AD through point B.
Step 3: Mark the point where these two lines meet as C.
ABCD is the required parallelogram.
In simple words: Draw two lines at a 30-degree angle to each other with lengths 4 cm and 5 cm. Then draw parallel lines from the ends to complete the parallelogram shape.

Exam Tip: Use a protractor to measure the 30-degree angle accurately, and ensure the parallel lines are drawn correctly using a set-square to get a proper parallelogram.

Question 11. What are the remaining angles of the parallelogram with adjacent sides of 4 cm and 5 cm and an angle of 30°?
Answer:
In parallelogram ABCD, the sum of adjacent angles equals 180 degrees:
\( \angle A + \angle B = 180° \) [Adjacent angles]
\( 30° + \angle B = 180° \) [Given that \( \angle A = 30° \)]
\( \angle B = 180° - 30° = 150° \)

Opposite angles in a parallelogram are equal:
\( \angle B = \angle D = 150° \)
\( \angle A = \angle C = 30° \)

The opposite sides of a parallelogram are equal in length:
\( BC = AD = 5 \text{ cm} \) [Opposite sides are equal]
\( DC = AB = 4 \text{ cm} \) [Opposite sides are equal]
In simple words: When one angle is 30 degrees, the next angle must be 150 degrees because adjacent angles add to 180 degrees. The opposite angles are equal to these, so the other 30-degree angle is across from the first 30-degree angle, and the other 150-degree angle is across from the first 150-degree angle. The opposite sides have the same lengths as the sides you drew.

Exam Tip: Remember two key rules for parallelograms: adjacent angles sum to 180 degrees, and opposite angles are equal. Use these to find all missing angles quickly.

Question 12. What can we say about the angles of a parallelogram?
Answer: In a parallelogram, adjacent pairs of angles (angles next to each other) add up to 180 degrees. Also, opposite pairs of angles (angles across from each other) are always equal to one another.
In simple words: In a parallelogram, two angles that sit next to each other always add to 180 degrees. Two angles across from each other are always the same size.

Exam Tip: These two angle properties - adjacent angles supplementary and opposite angles equal - are defining features of all parallelograms and help identify them.

Question 13. Will the opposite angles be equal in all parallelograms? If yes, how can we be sure?
Answer: Yes, the opposite angles of a parallelogram are always equal. We can prove this as follows:

Let \( \angle P = x \)

Since \( \angle P + \angle R = 180° \) [Adjacent angles of parallelogram]
\( \implies \angle R = 180° - \angle P = 180° - x \)

Similarly, since \( \angle A + \angle R = 180° \),
\( \angle A = 180° - \angle R = 180° - (180° - x) = 180° - 180° + x = x \)

Thus, \( \angle P = \angle A = x \)

By the same method, we can show that \( \angle R = \angle E = 180° - x \)

This proves that opposite angles of a parallelogram are always equal.
In simple words: If one angle is x degrees, the angle next to it must be (180 - x) degrees. The angle across from the first angle is also (180 - x) degrees minus 180 degrees plus x, which equals x. So opposite angles are always the same.

Exam Tip: Use the property that adjacent angles sum to 180 degrees to prove opposite angles are equal - this is a common exam question requiring algebraic reasoning.

Question 14. What can we say about the sides of a parallelogram?
Answer: The opposite sides of a parallelogram are equal in length.
In simple words: In a parallelogram, the side across from one side has the same length as that side.

Exam Tip: This is a fundamental property - opposite sides of any parallelogram must be equal, which is why if you know two adjacent side lengths, you can determine all four side lengths.

Question 15. Is it wrong to write \( \triangle ABD \cong \triangle CBD \)? Why?
Answer: Yes, it is wrong to write \( \triangle ABD \cong \triangle CBD \) based on the figure shown. In the figure:

- The two triangles ABD and CBD share the common side BD.
- They also have some equal angles marked in the figure.
- However, for congruence, we need one of these conditions:
- SSS (all three sides equal)
- SAS (two sides and the included angle equal)
- ASA or AAS
- RHS (for right triangles)

In this picture:
1. Only one side (BD) is common. There is no information showing that AB equals BC or AD equals CD. So the three-side equality (SSS) is missing.
2. The equal angle markings are not the included angle between known equal sides. Even if some angles appear equal, we do not know that the sides around those angles are equal. So SAS congruence does not apply.
3. The shape appears to be a parallelogram or general quadrilateral. In such shapes, the diagonals (like BD) divide the quadrilateral into two triangles, but these triangles are usually not congruent because the sides differ.
In simple words: These two triangles share one side, but we do not know if their other sides are equal. Without equal sides in the right places, we cannot say the triangles are congruent.

Exam Tip: Always check that you have enough information (SSS, SAS, ASA, AAS, or RHS) before claiming two triangles are congruent - sharing one common side and having some equal angles is NOT enough.

Question 16. Are the diagonals of a parallelogram always equal? Check with the parallelogram that you have constructed.
Answer: No, the diagonals of a parallelogram are not always equal. In any parallelogram:
- Opposite sides are parallel
- Opposite sides are equal
- Diagonals bisect each other (they cut each other into two equal halves)
- But they do not have to be equal in length

In the parallelogram that was constructed, one diagonal measures 8 cm while the other measures 4.6 cm, showing that the diagonals can have different lengths.
In simple words: A parallelogram's two diagonals can be different lengths. They cross each other at their midpoints, but one does not have to be as long as the other.

Exam Tip: Remember that diagonals of a parallelogram bisect each other (cut in half at their meeting point), but they are NOT equal in length unless the parallelogram is a rectangle or square.

Question 17. Do the diagonals of a parallelogram bisect each other (intersect at their midpoints)? Reason and/or experiment to figure this out.
Answer: Yes, the diagonals of a parallelogram always bisect each other. In a parallelogram:
- Opposite sides are parallel.
- When the diagonals cross, they form pairs of alternate interior angles that are equal.
- Because of these equal angles and the parallel sides, the triangles created by the diagonals are congruent, which shows that AO equals OC and BO equals OD (where O is the intersection point).
- So the diagonals meet at point O, which is the midpoint of both diagonals.
In simple words: When the two diagonals of a parallelogram cross, they cut each other into two equal parts. The crossing point is the exact middle of each diagonal.

Exam Tip: Use the properties of parallel sides and alternate interior angles to prove congruence of the triangles formed by the diagonals - this is a standard proof for showing that diagonals bisect each other.

Question 18. Is it wrong to write \( \triangle AOE \cong \triangle SOY \)? Why?
Answer: Yes, it is wrong to write \( \triangle AOE \cong \triangle SOY \). To prove two triangles are congruent, we need one of these rules:
- SSS (all three sides equal)
- SAS (two sides and included angle equal)
- ASA (two angles and included side equal)
- RHS (right angle, hypotenuse, side)

In the figure:
- Triangle AOE is on the left.
- Triangle SOY is on the right.
Even though the diagram shows some equal sides and angles, the positions of angles and sides do not match in the same order, so the triangles cannot be paired correctly under any congruency rule.
In simple words: Even though both triangles might have some equal sides and angles, the arrangement of these equal parts does not follow any congruence rule, so the triangles are not congruent.

Exam Tip: When proving triangle congruence, the order and position of equal sides and angles matter - you must match them to one of the five standard rules (SSS, SAS, ASA, AAS, RHS).

Question 19. Do the diagonals of a parallelogram intersect at a particular angle?
Answer: No, the diagonals of a parallelogram do not intersect at any fixed or particular angle. In a parallelogram, the diagonals always bisect each other (they cut each other into two equal halves), but the angle at which they intersect can vary. This angle depends on the shape of the parallelogram:
- Square: diagonals intersect at 90 degrees
- Rectangle: diagonals intersect at not 90 degrees (usually acute or obtuse)
- Rhombus: diagonals intersect at 90 degrees
- General parallelogram: diagonals may intersect at any acute or obtuse angle
In simple words: The two diagonals of a parallelogram always cross at their midpoints, but they can cross at many different angles depending on what shape the parallelogram is.

Exam Tip: Different types of parallelograms have different diagonal properties - remember that rectangles don't have perpendicular diagonals, but rhombuses and squares do.

Question 20. Are squares the only quadrilaterals that have all sides equal in length?
Answer: No, squares are not the only quadrilaterals that have all sides equal. A square has all four sides equal and all angles equal (90 degrees). However, there is another quadrilateral that also has all sides equal - the rhombus. A rhombus has all four sides equal and opposite sides are parallel. So a rhombus is a quadrilateral with all sides equal, but it is not a square unless its angles become 90 degrees.
In simple words: Both a square and a rhombus have all four sides the same length. The difference is that a square has all right angles, while a rhombus does not.

Exam Tip: Distinguish between a square (all sides equal AND all angles 90 degrees) and a rhombus (all sides equal but angles not necessarily 90 degrees).

Question 21. Can we complete this quadrilateral so that all its sides are of the same length?
Answer: Yes. Measure AB using a compass and keep this length as the radius. Mark a point C whose distance from both B and D equals AB (or AD). From B and D, cut arcs using this radius as the measurement. The intersection of these arcs gives point C. Now you have a quadrilateral with all sides equal in length, with one of its angles being 50 degrees.
In simple words: Use a compass to measure the length of side AB. Then, from points B and D, draw arcs with this same length. Where the two arcs meet, you have found point C. Connect all points to get a four-sided shape with all equal sides.

Exam Tip: When constructing a quadrilateral with equal sides, use a compass to maintain consistent measurements - the intersection of arcs from two different points will locate the fourth vertex accurately.

Question 22. What are the other angles of the rhombus ABCD that we have constructed?
Answer:
\( \angle A + \angle B = 180° \) [Adjacent angles]
\( 50° + \angle B = 180° \) [Given \( \angle A = 50° \)]
\( \angle B = 180° - 50° = 130° \)

Now, using the property that opposite angles of a parallelogram are equal:
\( \angle A = \angle C = 50° \) [Opposite angles of parallelogram]
\( \angle B = \angle D = 130° \) [Opposite angles of parallelogram]
In simple words: When one angle is 50 degrees, the next angle is 180 minus 50, which is 130 degrees. The opposite angles are equal, so there is another 50-degree angle and another 130-degree angle.

Exam Tip: In any parallelogram (including a rhombus), adjacent angles are supplementary (add to 180 degrees) and opposite angles are equal - use these two properties to find all angles quickly.

Question 23. In rhombus GAME, it can be seen that \( \triangle GAE \cong \triangle MAE \). How can we verify this?
Answer: In triangles GAE and MAE:

\( GE = ME \) [Sides of rhombus]
\( GA = MA \) [Sides of rhombus]
\( AE = EA \) [Common side]

By the SSS (Side-Side-Side) rule, \( \triangle GAE \cong \triangle MAE \)
In simple words: Both triangles have all three sides equal. The two sides GE and ME are equal because they are sides of the rhombus. The two sides GA and MA are equal because they are also sides of the rhombus. The side AE is the same in both triangles. Since all three sides are equal, the triangles must be congruent.

Exam Tip: When using the SSS rule, check all three pairs of sides carefully - in this case, two pairs are equal because they are sides of the rhombus, and one pair is the same side shared by both triangles.

Question 24. So a rhombus is a parallelogram and a rectangle is also a parallelogram. How can this be represented using a Venn diagram? Where will the set of squares occur in this diagram?
Answer: A Venn diagram shows the relationships between these shapes as follows:

The largest set is "Parallelogram" - this is the outer circle. Inside this circle are two overlapping circles: "Rectangle" and "Rhombus." The overlapping region (intersection) where both circles meet represents the "Square," because a square is both a rectangle (all angles are 90 degrees) and a rhombus (all sides are equal).

This arrangement shows that:
- Every rectangle and every rhombus is a parallelogram
- A square is the only shape that is both a rectangle and a rhombus
- A rectangle has right angles but not necessarily equal sides
- A rhombus has equal sides but not necessarily right angles
In simple words: Draw one big circle for parallelograms. Inside, draw two overlapping circles - one for rectangles and one for rhombuses. Where they overlap is where squares go, because squares have both properties (right angles like rectangles, and equal sides like rhombuses).

Exam Tip: A Venn diagram for quadrilaterals helps visualize that a square is a special case - it satisfies both the rectangle condition (all angles 90 degrees) and the rhombus condition (all sides equal).

Question 25. Are the diagonals of a rhombus equal?
Answer: No, the diagonals of a rhombus are not equal.
In simple words: The two diagonals of a rhombus can be different lengths from each other.

Exam Tip: This distinguishes a rhombus from a square - a square has equal diagonals, but a rhombus does not.

Question 26. Do the diagonals of a rhombus intersect at any particular angle?
Answer: Yes, the diagonals of a rhombus always intersect at a particular angle - they meet at a right angle (90 degrees). In a rhombus:
- The diagonals are not equal in length, but
- They always cut each other exactly in half (bisect each other), and
- They always intersect at 90 degrees, forming four right angles at their intersection point

This property is true for every rhombus.
In simple words: The two diagonals of a rhombus always cross at 90-degree angles, even though one diagonal might be longer than the other.

Exam Tip: Remember that rhombus diagonals are perpendicular (intersect at 90 degrees) and bisect each other - these are defining properties that distinguish a rhombus from a general parallelogram.

Question 27. In the rhombus GAME, we have \( \triangle GEO \cong \triangle MEO \). Why?
Answer: In triangles GEO and MEO:

\( EG = EM \) [Sides of rhombus]
\( GO = MO \) [Diagonals of rhombus bisect each other]
\( EO = EO \) [Common side]

By the SSS (Side-Side-Side) rule, \( \triangle GEO \cong \triangle MEO \)
In simple words: Both triangles share the side EO. The sides EG and EM are equal because they are sides of the rhombus. The sides GO and MO are equal because the diagonals of a rhombus cut each other in half. Since all three sides match, the triangles are congruent.

Exam Tip: Use properties specific to rhombuses - equal sides and bisecting diagonals - to establish which parts of triangles are equal, then apply the SSS rule.

Question 28. Find the remaining angles in the following quadrilaterals.
(i) In parallelogram PEAR with \( \angle P = 40° \):
Answer:
\( \angle P + \angle E = 180° \) [Adjacent angles]
\( 40° + \angle E = 180° \) [Given \( \angle P = 40° \)]
\( \angle E = 180° - 40° = 140° \)
\( \angle R = \angle E = 140° \) [Opposite angles of parallelogram]
\( \angle A = \angle P = 40° \) [Opposite angles of parallelogram]

(ii) In parallelogram PQRS with \( \angle P = 110° \):
Answer:
\( \angle P + \angle Q = 180° \) [Adjacent angles]
\( 110° + \angle Q = 180° \) [Given \( \angle P = 110° \)]
\( \angle Q = 180° - 110° = 70° \)
\( \angle S = \angle Q = 70° \) [Opposite angles of parallelogram]
\( \angle R = \angle P = 110° \) [Opposite angles of parallelogram]

(iii) In rhombus XWVU with a 30-degree angle shown in triangle XUV:
Answer:
In triangle XUV:
\( UX = UV \) [Sides of rhombus]
\( \angle UXV = \angle UVX = 30° \) [Angles opposite to equal sides]
\( \angle UXV + \angle UVX + \angle U = 180° \) [Angle sum property of triangle]
\( 30° + 30° + \angle U = 180° \)
\( \angle U = 180° - 60° = 120° \)
\( \angle W = \angle U = 120° \) [Opposite angles of rhombus]

In triangle XWV:
\( WX = WV \) [Sides of rhombus]
\( \angle WXV = \angle WVX \) [Angles opposite to equal sides]
\( \angle WXV + \angle WVX + \angle W = 180° \) [Angle sum property of triangle]
\( \angle WXV + \angle WXV + 120° = 180° \) [Since \( \angle WXV = \angle WVX \)]
\( 2\angle WXV + 120° = 180° \)
\( 2\angle WXV = 60° \)
\( \angle WXV = 30° \)

(iv) In rhombus OIAE with a 20-degree angle shown in triangle EAO:
Answer:
In triangle EAO:
\( AE = AO \) [Sides of rhombus]
\( \angle AOE = \angle AEO = 20° \) [Angles opposite to equal sides]
\( \angle AOE + \angle AEO + \angle A = 180° \) [Angle sum property of triangle]
\( 20° + 20° + \angle A = 180° \)
\( \angle A = 180° - 40° = 140° \)
\( \angle I = \angle A = 140° \) [Opposite angles of rhombus]

In triangle OIE:
\( OI = IE \) [Sides of rhombus]
\( \angle IEO = \angle IOE \) [Angles opposite to equal sides]
\( \angle IEO + \angle IOE + \angle I = 180° \) [Angle sum property of triangle]
\( \angle IEO + \angle IEO + 140° = 180° \) [Since \( \angle IEO = \angle IOE \)]
\( 2\angle IEO + 140° = 180° \)
\( 2\angle IEO = 40° \)
\( \angle IEO = 20° \)
In simple words: Use two main rules: (1) adjacent angles in a parallelogram add to 180 degrees, and (2) opposite angles are equal. For rhombuses, also use the fact that equal sides create equal angles opposite to them in triangles.

Exam Tip: When finding missing angles, first apply the parallelogram angle rules, then break the shape into triangles and use the triangle angle sum property (180 degrees) along with the property that equal sides create equal opposite angles.

Question 29. Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect at an angle of 140°.
Answer:
Steps of Construction:
- Draw a straight line segment AC = 7 cm.
- To find the midpoint of AC, mark the midpoint O of AC by measuring 3.5 cm from A or C.
- At point O, use a protractor to construct an angle of 140 degrees.
- Draw a line through O along this angle. This will be the direction of the second diagonal.
- Mark half of the shorter diagonal (2.5 cm) on both sides of O. Since the diagonals of a parallelogram bisect each other, the half-length of the shorter diagonal = 5 divided by 2 = 2.5 cm.
- From O, measure 2.5 cm on each side along the 140-degree line. Mark these points as B and D.
- Join the vertices to get ABCD, the required parallelogram.
In simple words: Draw one diagonal 7 cm long and mark its middle point O. From O, draw a line at 140 degrees and mark points 2.5 cm on each side of O on this line. Connect the four points to make a parallelogram.

Exam Tip: The key to this construction is remembering that diagonals of a parallelogram bisect each other - the midpoint of both diagonals is the same point O.

Question 30. Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.
Answer:
Steps of Construction:
- Draw a point O. This will be the point where the diagonals intersect.
- Draw a line through O and mark 2.5 cm on both sides of O.
- Mark point A 2.5 cm to the left of O. Mark point C 2.5 cm to the right of O. Thus, AC = 5 cm (full diagonal).
- Construct a perpendicular line to AC at point O. Use a set-square or compass to draw a line at 90 degrees to AC.
- On the perpendicular line, mark 2 cm on both sides of O. Mark point B 2 cm above O.
- Mark point D 2 cm below O. Thus, BD = 4 cm (full diagonal).
- Join the points A-B, B-C, C-D, and D-A. The figure formed is the required rhombus ABCD.
In simple words: Draw a point O in the middle. From O, mark points 2.5 cm away on a horizontal line (this is half of the 5 cm diagonal). From O, mark points 2 cm away on a vertical line at 90 degrees (this is half of the 4 cm diagonal). Connect the four points to make a rhombus.

Exam Tip: For a rhombus, the diagonals must be perpendicular (at 90 degrees) - use a set-square to ensure the angle is exactly 90 degrees for an accurate construction.

Question 31. Join the ends. What is the quadrilateral that you get? Justify your answer.
Answer: You will get a square because the diagonals are equal and bisecting each other at right angles. A quadrilateral with equal diagonals that bisect each other at 90 degrees must be a square, since this combination of properties - equal diagonals, bisection, and perpendicularity - uniquely defines a square among all quadrilaterals.
In simple words: When the diagonals are equal in length and cross at 90-degree angles at their midpoints, the quadrilateral formed is a square.

Exam Tip: A square is the only quadrilateral where the diagonals are equal, bisect each other, AND meet at 90 degrees - these three properties together identify a square uniquely.

Question 32. Extend one of the diagonals on both sides by 2 cm. What quadrilateral will you get now? Justify your answer.
Answer: You will get a rhombus because the diagonals are unequal and bisecting each other at a right angle. When one diagonal is extended to become longer while the other remains unchanged, the diagonals are no longer equal. However, they still bisect each other at 90 degrees, which is the defining property of a rhombus - unequal diagonals that are perpendicular and bisect each other.
In simple words: When the diagonals have different lengths and cross at 90-degree angles at their midpoints, the quadrilateral is a rhombus.

Exam Tip: A rhombus has unequal diagonals (unlike a square) but still has perpendicular diagonals that bisect each other - remembering this distinction helps identify rhombuses correctly.

Question 33. Can you join them to get a quadrilateral? What type of quadrilateral is this? Justify your answer.
Answer: Yes, you can join them taking one side as common to get a quadrilateral. Since all sides of this quadrilateral are equal and its diagonals are not equal, it is a rhombus. When two isosceles triangles are joined along one of their equal sides, the resulting quadrilateral has all four sides equal (the two equal sides from each triangle), making it a rhombus with unequal diagonals.
In simple words: Two isosceles triangles joined along a shared side create a four-sided shape with all sides equal. This is a rhombus if the diagonals are not equal.

Exam Tip: When joining two congruent isosceles triangles, remember that the type of quadrilateral depends on which side you use as the common edge - a different edge produces a different quadrilateral type.

Question 34. Take two cardboard cutouts of an isosceles triangle with side lengths 8 cm, 8 cm and 6 cm. What are the different ways they can be joined to get a quadrilateral? What quadrilaterals are these? Justify your answers.
Answer:
Method 1: Joining along the 6 cm side
If you join the two triangles taking the 6 cm side as common, you get a quadrilateral where all sides are equal (8 cm, 8 cm, 8 cm, 8 cm). Since all sides of the quadrilateral are equal and its diagonals are not equal, this quadrilateral is a rhombus.

Method 2: Joining along one 8 cm side
If you join the two triangles taking one 8 cm side as common, you get a quadrilateral where opposite sides are equal (6 cm opposite to 6 cm, and 8 cm opposite to 8 cm). Since opposite sides are equal, this quadrilateral is a parallelogram. The angles in this parallelogram are not all 90 degrees, so it is not a rectangle.
In simple words: Depending on which side you choose to join the two triangles, you get either a rhombus (all sides 8 cm, from joining along the base) or a parallelogram (sides 6 cm and 8 cm alternating, from joining along the slant side).

Exam Tip: When constructing quadrilaterals from triangles, the choice of which side serves as the common edge determines the final shape and its properties - always check which edges are equal in the resulting quadrilateral to identify its type.

Question. Take two cardboard cutouts of a scalene triangle with sides 6 cm, 9 cm and 12 cm. What are the different ways they can be joined to get a quadrilateral? Are you able to identify the different quadrilaterals that are obtained by joining the triangles? Justify your answer whenever you identify a quadrilateral.
Answer: You can join the two triangles in three different ways by placing a common side between them. When you join them using the 12 cm side as the shared edge, you get a shape with two pairs of neighbouring sides that are equal in length - this is a kite. If you join them along the 6 cm side, you get a shape where opposite sides match in length - this forms a parallelogram. Similarly, joining them with the 9 cm side as the common edge also produces a parallelogram because the opposite sides remain equal.
In simple words: Two identical triangles can be put together in three ways. Depending on which side you pick to join them, you get either a kite or a parallelogram.

Exam Tip: Always identify which sides become equal when triangles are joined, and use this to name the resulting quadrilateral - kites have two pairs of equal adjacent sides, while parallelograms have equal opposite sides.

Question. In the kite ABCD, show that the diagonal BD (i) bisects angles ABC and ADC, (ii) bisects the diagonal AC (that is, AO = OC), and is perpendicular to it.
Answer: Let the diagonals of kite ABCD meet at point O.
Given: AB = BC and AD = DC (the two pairs of equal adjacent sides that define the kite).
(i) To show BD bisects angles ABC and ADC:
When you compare triangle ABD with triangle CBD, you find:
AB = CB (sides of the kite)
AD = CD (sides of the kite)
BD = BD (same side shared by both triangles)
By the SSS rule (all three sides match), triangle ABD is congruent to triangle CBD. Using CPCTC, this means angle ABD = angle DBC and angle ADB = angle CDB. Therefore, BD splits both angle ABC and angle ADC into two equal parts.
(ii) To show BD bisects AC and is perpendicular to it:
Now compare triangle AOB with triangle COB:
AB = CB (given sides of the kite)
angle ABO = angle CBO (since BD bisects angle ABC, from part (i))
OB = OB (same side)
By the SAS rule, triangle AOB is congruent to triangle COB. By CPCTC, AO = OC, so BD cuts AC into two equal segments. Also, angle AOB = angle COB. Since these two angles share the line AC, they must add up to 180°. Because they are equal and sum to 180°, each one must be 90°. Therefore, BD is perpendicular to AC.
In simple words: In a kite, one diagonal always splits the other diagonal in half and meets it at a right angle. It also divides the angles at the ends of that diagonal into two equal parts.

Exam Tip: Use congruence of triangles formed by the diagonals to establish angle bisection and perpendicularity - this is a standard proof structure for kite properties.

Question. Construct a trapezium. Measure the base angles (marked in the figure). Can you find the remaining angles without measuring them?
Answer: When you measure the base angles of trapezium PQRS (where PQ is parallel to SR), you get angle S = 75° and angle R = 65°. Since PQ and SR are parallel lines, angles on the same side of a cutting line sum to 180°.
For angle P: angle S + angle P = 180°, so 75° + angle P = 180°, giving angle P = 105°.
For angle Q: angle R + angle Q = 180°, so 65° + angle Q = 180°, giving angle Q = 115°.
In simple words: When two sides of a trapezium are parallel, any two angles next to each other on the same leg add up to 180°. So you can find the missing angles without measuring them.

Exam Tip: Remember the co-interior angle property - when a line cuts two parallel lines, angles on the same side of the cutting line sum to 180°.

Question. How do we construct an isosceles trapezium? Construct an isosceles trapezium UVWX, with UV parallel to XW. Measure angle U.
Answer: To build an isosceles trapezium UVWX where UV is parallel to XW:
- Draw a straight line segment UV of any comfortable length.
- Sketch a line that runs parallel to UV.
- At point U, draw an arc that hits the parallel line at point X.
- At point V, using the same arc distance, draw another arc hitting the parallel line at point W.
- Connect X to U and W to V.
Using a protractor, angle U = 80°.
Quadrilateral UVWX is now an isosceles trapezium because UV is parallel to XW and the non-parallel sides UX and VW are equal in length.
In simple words: An isosceles trapezium is made by drawing two parallel lines and then connecting them with two equal-length slanted sides.

Exam Tip: In an isosceles trapezium, the base angles (angles along each parallel side) are equal to each other - this is a defining feature.

Question. What type of quadrilateral is XWZY?
Answer: Since XW is parallel to UV, when perpendiculars are drawn from X and W to line UV, they create right angles. Using the property that angles on the same side of a cutting line sum to 180°:
a = 180° - angle XYZ = 90° and
b = 180° - angle WZY = 90°
All four angles of quadrilateral XWZY are right angles, which means it is a rectangle.
In simple words: When you drop straight lines from the ends of one parallel side to the other, you create four right angles, making a rectangle.

Exam Tip: A quadrilateral with all four angles being 90° is always a rectangle - check this when parallel lines create perpendicular segments.

Question. In triangles UXY and VWZ, show that triangle UXY is congruent to triangle VWZ.
Answer: When comparing triangle UXY with triangle VWZ:
angle XYU = angle WZV (both are 90°, since XWZY is a rectangle)
UX = VW (the non-parallel sides of the isosceles trapezium are equal)
XY = WZ (opposite sides of the rectangle are equal)
By the RHS (Right angle - Hypotenuse - Side) rule, triangle UXY is congruent to triangle VWZ.
In simple words: Two right triangles with the same hypotenuse length and the same other side length must be identical shapes.

Exam Tip: The RHS rule is used only for right triangles - make sure the right angle, hypotenuse, and one other side all match before claiming congruence.

Question 1. Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.
Answer: When you join two equilateral triangles (each with all sides 4 cm and all angles 60°) along one shared side, all four sides of the resulting quadrilateral remain 4 cm. The angles you get are 60°, 120°, 60°, and 120° in order around the shape. The angle becomes 120° at the vertex where the two triangles meet (since two 60° angles join together), while the other two angles stay at 60°.
In simple words: When two identical equilateral triangles are put together, you get a diamond-shaped quadrilateral with all sides the same length and with alternating angles of 60° and 120°.

Exam Tip: When triangles are joined edge-to-edge, angles at the joining vertex combine - add them together to find the resulting angle in the new shape.

Question 2. Construct a kite whose diagonals are of lengths 6 cm and 8 cm.
Answer: To build a kite with diagonals 6 cm and 8 cm:
- Draw a line segment AC (the first diagonal) of 6 cm length.
- Mark point O at the centre of AC.
- At point O, sketch a perpendicular line.
- On this perpendicular, mark points D and B such that the distance from D to O is one portion of 8 cm (for example, 3 cm) and the distance from O to B is the remaining portion (5 cm). This makes DB total 8 cm.
- Connect A to D, D to C, C to B, and B back to A.
The figure ADCB (or ABCD) is now a kite because one diagonal (DB) bisects the other (AC) at right angles, and two pairs of neighbouring sides are equal.
In simple words: A kite can be built by drawing two straight lines that cross at right angles, then connecting the four end points in the right way.

Exam Tip: In a kite, the main diagonal always bisects the other diagonal at 90° - use this property to place the four vertices correctly.

Question 3. Find the remaining angles in the following trapeziums.
Answer: For the first trapezium ABCD with angle D = 135° and angle C = 105°:
Since AD is parallel to BC (or the parallel sides are AB and DC), angles next to each other sum to 180°.
angle A + angle D = 180° → angle A + 135° = 180° → angle A = 45°
angle B + angle C = 180° → angle B + 105° = 180° → angle B = 75°
For the second trapezium PQRS with angle P = 100°:
angle S + angle P = 180° → angle S + 100° = 180° → angle S = 80°
In an isosceles trapezium, angles at each parallel side are equal, so angle P = angle Q = 100°.
angle R + angle Q = 180° → angle R + 100° = 180° → angle R = 80°
In simple words: In any trapezium, angles that are next to each other on the slanted sides always add up to 180°. Use this to find missing angles.

Exam Tip: For isosceles trapeziums, remember that the angles along each parallel side are equal - this cuts down the calculation work.

Question 4. Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the following questions: (i) What is the quadrilateral that is both a kite and a parallelogram? (ii) Can there be a quadrilateral that is both a kite and a rectangle? (iii) Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?
Answer: (i) A rhombus is the shape that has the features of both a kite and a parallelogram. It qualifies as a kite because it has two pairs of equal adjacent sides, and it qualifies as a parallelogram because opposite sides are parallel. Therefore: rhombus = kite + parallelogram.
(ii) Yes, but only in one special situation. A square is a quadrilateral that is both a kite (all sides are equal, so it contains two pairs of equal adjacent sides) and a rectangle (all angles measure 90°). So a square is both a kite and a rectangle.
(iii) No, not every kite is a rhombus. The correct relationship is: Every rhombus must be a kite because it always has two pairs of equal adjacent sides. However, a kite is not always a rhombus. The reason is that in a rhombus all four sides are identical in length, but in a kite only the neighbouring side pairs match - the non-adjacent sides can be different lengths. So the relationship only goes one direction: rhombus → kite, but kite ↛ rhombus.
In simple words: A rhombus is a special kite where all four sides are the same length. A square is the most special quadrilateral - it is both a kite and a rectangle at the same time.

Exam Tip: When comparing quadrilateral types, always think about which properties they must have - this helps you identify relationships like "every rhombus is a kite, but not every kite is a rhombus".

Question 5. If PAIR and RODS are two rectangles, find angle IOD.
Answer: In triangle RIO:
angle ORI + angle IOR + angle I = 180° (angle sum property of triangles)
angle ORI + angle IOR + 90° = 180° (since angle I = 90°, being a corner of rectangle PAIR)
30° + angle IOR + 90° = 180° (given that angle ORI = 30°)
angle IOR = 180° - 120° = 60°
Since angle DOR = 90° (a corner angle of rectangle RODS):
angle IOD + angle IOR = 90°
angle IOD + 60° = 90°
angle IOD = 30°
In simple words: You can find angle IOD by first working out angle IOR using the angle sum property in triangle RIO, then using the right angle of rectangle RODS to find the remaining angle.

Exam Tip: When shapes overlap, find angles inside the triangles they create, then use those results to calculate angles between the larger shapes.

Question 6. Construct a square with diagonal 6 cm without using a protractor.
Answer: To make a square having a diagonal of 6 cm without a protractor:
- Draw a straight line segment AC of length 6 cm - this will be one diagonal.
- Find and mark the midpoint O of this line segment.
- At point O, use a compass to draw a circle. The circle should pass through both A and C.
- The perpendicular line through O (drawn using a ruler and set square, or by folding) will hit this circle at two points. Call these points B and D.
- Connect A to B, B to C, C to D, and D back to A.
- The quadrilateral ABCD is a square because its diagonals are equal (both 6 cm), they bisect each other at O, they cross at 90°, and all sides turn out equal.
In a square, the diagonals always bisect each other at right angles, so constructing perpendicular lines through the midpoint gives you the other diagonal, and joining the ends of both diagonals builds the square.
In simple words: Draw one diagonal, find its middle point, then sketch a perpendicular line through that point. The points where this line meets a circle centred at the middle point give you the other two corners of the square.

Exam Tip: Remember that square diagonals are equal in length, bisect each other, and meet at 90° - use these properties for constructions without a protractor.

Question 7. CASE is a square. The points U, V, W and X are the midpoints of the sides of the square. What type of quadrilateral is UVWX? Find this by using geometric reasoning, as well as by construction and measurement. Find other ways of constructing a square within a square such that the vertices of the inner square lie on the sides of the outer square.
Answer: UVWX is a square.
Geometric Reasoning:
Let CASE be a square with sides of equal length. Points U, V, W, X are placed at the midpoints of sides CA, AS, SE, EC respectively.
(a) First, UVWX forms a parallelogram. Using the midpoint theorem:
In triangle CAS, since U and V are midpoints → UV is parallel to CS
In triangle ASE, since V and W are midpoints → VW is parallel to AE
In triangle SEC, since W and X are midpoints → WX is parallel to CS
In triangle CEA, since X and U are midpoints → XU is parallel to AE
Therefore, UVWX is a parallelogram (opposite sides are parallel).
(b) Next, all four sides of UVWX are equal. By the midpoint theorem, each side of UVWX equals half the length of a diagonal of CASE. Since the diagonals of a square are equal, all sides of UVWX are equal.
(c) The angles of UVWX are all 90°. In square CASE, the diagonals are perpendicular to each other (they meet at a right angle). Since UV is parallel to one diagonal and VW is parallel to the other, UV and VW meet at 90°. Therefore, all angles in UVWX are 90°.
Conclusion: UVWX has all sides equal and all angles 90°, making it a square.
Other ways to build a square inside a square: You can place the inner square's corners at different positions along the outer square's sides (not just at midpoints). For any choice of how far along the first side you place the first corner, if you then place each new corner the same distance along each subsequent side, you will create an inner square. The inner square will be rotated at some angle inside the outer square, but it will still be a true square because of the symmetry of the construction.
In simple words: When you mark the middle points of a square's four sides and connect them, you get another square inside it, rotated 45° and tilted. You can also create other squares by picking different positions on the sides, as long as you space them equally around the outer square.

Exam Tip: The midpoint theorem is key here - remember that a line joining two midpoints of a triangle's sides is parallel to the third side and half its length.

Question 8. If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.
Answer: Yes, a quadrilateral with four equal sides and even just one angle of 90° must be a square.
Geometric Reasoning:
Let quadrilateral ABCD have these properties:
AB = BC = CD = DA (all four sides are equal)
angle D = 90°
A quadrilateral where all four sides are equal is always a rhombus. In any rhombus, opposite sides are parallel (because it is also a parallelogram). Now, in a parallelogram, if one angle measures 90°, then all four angles automatically become 90° - this is because opposite angles in a parallelogram are equal, and adjacent angles sum to 180°. So if one angle is 90°, the opposite angle is also 90°, and each adjacent angle must be 180° - 90° = 90°.
Therefore, we get a figure that is both a rhombus (all sides equal) and a rectangle (all angles 90°). A quadrilateral that has these two properties at the same time is, by definition, a square.
Conclusion: Any four-sided shape with all sides equal and at least one right angle must be a square.
In simple words: If all four sides of a quadrilateral are the same length and one corner angle is a right angle, then all four corners must be right angles, making it a square.

Exam Tip: Remember that in a parallelogram, if you know one angle is 90°, you can immediately conclude all four angles are 90° - this is a powerful shortcut for identifying rectangles and squares.

Question 9. What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer.
Answer: A quadrilateral where both pairs of opposite sides are equal is a parallelogram.
Justification:
If in quadrilateral ABCD:
AB = CD (one pair of opposite sides are equal), and
BC = AD (the other pair of opposite sides are equal),
then this satisfies a known property of parallelograms. When both pairs of opposite sides in a quadrilateral are equal in length, the quadrilateral must be a parallelogram - this is the converse of the property "in a parallelogram, opposite sides are equal".
Here is why: Draw one diagonal to split the quadrilateral into two triangles. The equal opposite sides ensure these two triangles are congruent by the SSS (side-side-side) rule. When the triangles are congruent, their matching angles are equal. In particular, the alternate interior angles formed by the diagonal (acting as a transversal cutting the two opposite sides) become equal. Equal alternate interior angles mean the opposite sides must be parallel. A quadrilateral with both pairs of opposite sides parallel is, by definition, a parallelogram.
In simple words: If you draw a quadrilateral where opposite sides are equal in length, you are drawing a parallelogram. The opposite sides will automatically be parallel to each other.

Exam Tip: The converse of the parallelogram property is just as useful as the original - if opposite sides are equal, you have a parallelogram, even if the figure doesn't look like a typical one.

Question 10. Will the sum of the angles in a quadrilateral such as the following one also be 360°? Find the answer using geometric reasoning as well as by constructing this figure and measuring.
Answer: Yes, the sum of the angles in a concave quadrilateral (where one vertex is pushed inward) is also 360°.
After measuring the angles in such a quadrilateral ABCD:
angle A = 40°
angle B = 50°
angle C = 40°
angle D = 230° (this is a reflex angle, measured on the inside of the concave vertex)
Sum = 40° + 50° + 40° + 230° = 360°
Geometric Reasoning:
When you measure the interior angles of any quadrilateral - whether it is convex (all corners pointing outward) or concave (one corner pointing inward) - and you add them together, you always get 360°. This is true because any quadrilateral can be divided into two triangles by drawing a diagonal. Each triangle has angles summing to 180°, so two triangles give 180° + 180° = 360°. Even when a quadrilateral is concave, if you carefully measure the reflex angle at the pushed-in corner, this property still holds.
In simple words: No matter what shape a four-sided figure is - even if one corner is pushed inward - its four angles will always add up to 360°.

Exam Tip: When a quadrilateral is concave, the angle at the pushed-in vertex becomes a reflex angle (bigger than 180°) - make sure to measure it correctly on the interior side of the shape.

Question 11. State whether the following statements are true or false. Justify your answers.

Question 11. (i) A quadrilateral whose diagonals are equal and bisect each other must be a square.
Answer: False. A quadrilateral where the diagonals are equal in length and bisect each other is a rectangle. A square is a special type of rectangle that also has all four sides equal in length. The statement only tells us the diagonals are equal and bisect each other, but says nothing about whether all sides are equal. So the shape might be a rectangle that is not a square - for example, a rectangle with length 6 cm and width 4 cm satisfies the diagonal condition but is not a square.
In simple words: Equal diagonals that split each other in half make a rectangle, not necessarily a square. A square needs the extra condition that all four sides must be equal.

Exam Tip: Do not mix up the properties of rectangles and squares - a rectangle's diagonals are equal and bisect, but a square adds the requirement that all sides be equal too.

Question 11. (ii) A quadrilateral having three right angles must be a rectangle.
Answer: True. The sum of all interior angles in any quadrilateral is always 360°. If three angles are each 90°, then:
90° + 90° + 90° = 270°
Fourth angle = 360° - 270° = 90°
So all four angles are right angles. A quadrilateral with all four angles measuring 90° is a rectangle (or possibly a square, which is a special type of rectangle). Therefore, the statement is true.
In simple words: If three corners of a quadrilateral are right angles, the fourth corner must also be a right angle, making it a rectangle.

Exam Tip: The angle sum property of quadrilaterals (360°) is your tool for this kind of problem - use it to find missing angles.

Question 11. (iii) A quadrilateral whose diagonals bisect each other must be a parallelogram.
Answer: True. If the diagonals of a quadrilateral bisect each other, each diagonal is split into two equal sections by the meeting point. This is a defining property of parallelograms. More specifically, when you draw a line that splits both diagonals in half, the triangles formed by these halves are congruent. From this congruence, you can prove that opposite sides are parallel. A quadrilateral with both pairs of opposite sides parallel is, by definition, a parallelogram. So the statement is true - it is the converse of the parallelogram property "in a parallelogram, diagonals bisect each other".
In simple words: If the two diagonals of a quadrilateral split each other exactly in half, then the quadrilateral must be a parallelogram.

Exam Tip: The property "diagonals bisect each other" is a unique marker of parallelograms among quadrilaterals - if you see this property, you can confidently identify a parallelogram.

Question 11. (iv) A quadrilateral whose diagonals are perpendicular to each other must be a rhombus.
Answer: False. Perpendicular diagonals alone are not enough to guarantee a rhombus. A quadrilateral could have perpendicular diagonals and still be a kite, or even some other shape. In a true rhombus, the diagonals are perpendicular AND they bisect each other. Just having perpendicularity without the bisection property does not force all four sides to be equal, which is the defining feature of a rhombus. For example, a kite has perpendicular diagonals but is not necessarily a rhombus.
In simple words: Diagonals that cross at right angles can belong to a kite or other shapes, not just rhombuses. A rhombus needs both perpendicular diagonals AND diagonals that bisect each other.

Exam Tip: When identifying quadrilaterals by diagonal properties, check for BOTH perpendicularity and bisection if you want to prove a rhombus - one property alone is not enough.

Question 11. (v) A quadrilateral in which the opposite angles are equal must be a parallelogram.
Answer: True. This is the converse of a key parallelogram property. The original property states "in a parallelogram, opposite angles are equal". The converse - which is also true for quadrilaterals - states "if a quadrilateral has opposite angles equal, then it is a parallelogram". If opposite angles are equal, you can use the fact that the angle sum is 360° to show that both pairs of opposite sides must be parallel, which defines a parallelogram.
In simple words: If the angles across from each other in a quadrilateral are the same size, the quadrilateral must be a parallelogram.

Exam Tip: Many properties of parallelograms have true converses - use this to identify parallelograms from their angle or side properties.

Question 11. (vi) A quadrilateral in which all the angles are equal is a rectangle.
Answer: True. If all four angles in a quadrilateral are equal, and the sum of angles in any quadrilateral is 360°, then each angle must measure 360° ÷ 4 = 90°. A quadrilateral where all angles measure 90° is, by definition, a rectangle. (A square is a special case of a rectangle where all sides are also equal, but any quadrilateral with all 90° angles qualifies as a rectangle.)
In simple words: If all four corners of a quadrilateral are the same angle size, that angle must be 90°, making the shape a rectangle.

Exam Tip: Use the angle sum property (360°) combined with the "equal angles" condition to find that each angle must be 90°.

Question 11. (vii) Isosceles trapeziums are parallelograms.
Answer: False. An isosceles trapezium has exactly one pair of opposite sides that are parallel (the two bases), while the other pair of opposite sides are equal in length but NOT parallel (the two legs). By contrast, a parallelogram must have BOTH pairs of opposite sides parallel. Because isosceles trapeziums do not have two pairs of parallel sides, they are not parallelograms. In general, isosceles trapeziums and parallelograms are different categories of quadrilaterals. Only in an extremely special case - which would no longer truly be just a trapezium - could an isosceles trapezium become a parallelogram (and at that point it would actually be a rectangle).
In simple words: A trapezium has only one pair of parallel sides, while a parallelogram must have two pairs of parallel sides. So trapeziums and parallelograms are different types of shapes.

Exam Tip: Always check how many pairs of parallel sides a quadrilateral has - one pair = trapezium, two pairs = parallelogram.