RBSE Solutions Class 8 Maths Chapter 6 Polygons Important Questions

I. Objective Type Questions

Question 1. Which of the following closed curve has four sides
(a) diagonal
(b) quadrilaterals
(c) triangle
(d) circle
Answer: (b) quadrilaterals
In simple words: A quadrilateral is a closed shape that has exactly four sides. Other options are shapes with different numbers of sides or properties.

🎯 Exam Tip: Remember basic geometric definitions. A quadrilateral is any polygon with four sides and four vertices.

Question 2. Which of the quadrilaterals has all angles has right angles, opposite side equal and diagonals bisect each other,
(a) rectangle
(b) rhombus
(c) square
(d) none of the options
Answer: (a) rectangle
In simple words: A rectangle is a special type of quadrilateral where all angles are 90 degrees, opposite sides are the same length, and the diagonals cut each other in half.

🎯 Exam Tip: Distinguish between definitions carefully. A square also has these properties but additionally has all sides equal.

Question 3. Which of the following quadrilaterals is a regular quadrilateral?
(a) rhombus
(b) square
(c) kite
(d) rectangle
Answer: (b) square
In simple words: A regular quadrilateral is a shape where all sides are equal and all angles are equal. Only a square fits this description.

🎯 Exam Tip: A regular polygon has all sides equal and all interior angles equal. For a quadrilateral, only the square meets this condition.

Question 4. The angle of a square are
(a) 90°, 120°, 120°, 30°
(b) All angles are 90°
Answer: (b) All angles are 90°
In simple words: A square is a type of rectangle, and all angles inside a rectangle are 90 degrees. So, a square always has four 90-degree angles.

🎯 Exam Tip: Remember that a square is a regular polygon, meaning all its interior angles are equal and each measures 90 degrees.

Question 6. The sum of the measures of the external angles of any polygon is
(a) 90°
(b) 180°
(c) 240°
(d) 360°
Answer: (d) 360°
In simple words: No matter how many sides a polygon has, if you add up all its outside (exterior) angles, the total will always be 360 degrees. Imagine walking around the polygon, turning at each corner. You will always turn a full circle.

🎯 Exam Tip: This is a fundamental property of all convex polygons; the sum of their exterior angles (one at each vertex) is always 360 degrees.

Question 7. A quadrilateral whose diagonals bisect each other at right angles is ....
(a) Rectangle
(b) Square
(c) Parallelogram
(d) Trapezium
Answer: (b) Square
In simple words: In a square, the diagonals cut each other exactly in half, and they cross at a perfect 90-degree angle. This property is also true for a rhombus, but a square is a specific type of rhombus.

🎯 Exam Tip: Both a square and a rhombus have diagonals that bisect each other at right angles. However, only a square has equal diagonals and 90-degree angles. The question specifically asks for "a quadrilateral" and square is the most complete answer among the choices that satisfy the property.

Question 8. Sum of all exterior angles of n sided regular polygon is .......
(a) 360°
(b) 720°
(c) 540°
(d) 900°
Answer: (a) 360°
In simple words: For any polygon, whether it has many sides or few, the sum of its exterior angles will always add up to 360 degrees. This is a fixed rule in geometry.

🎯 Exam Tip: This property holds true for all convex polygons, not just regular ones, and is important for solving problems involving polygon angles.

Question 10. Sum of all the exterior angles formed by increasing the sides of a convex polygon in same order is
(a) 360°
(b) 270°
(c) 180°
(d) 90°
Answer: (a) 360°
In simple words: When you extend each side of a convex polygon in the same direction, the angles formed outside (exterior angles) always add up to 360 degrees, regardless of how many sides the polygon has.

🎯 Exam Tip: This is a constant for any convex polygon. It's helpful to remember that "exterior angles" means one angle taken at each vertex by extending a side.

II. Fill in the blanks

Fill in the blanks choosing the right option.

Question 1. A quadrilateral which has two pairs of equal adjacent sides is called ........(parallelogram/kite)
Answer: kite
In simple words: A kite is a special kind of quadrilateral where two pairs of its touching (adjacent) sides are equal in length.

🎯 Exam Tip: Remember the properties of a kite: two distinct pairs of equal-length adjacent sides. This distinguishes it from a rhombus (all sides equal) or a parallelogram (opposite sides equal).

Question 2. Those polygons whose all diagonals are in the interior, are called .......
Answer: convex
In simple words: A convex polygon is a shape where all its diagonals stay completely inside the shape. This means it doesn't have any "dents" or inward-pointing corners.

🎯 Exam Tip: This is the key definition of a convex polygon. If even one diagonal lies partly or wholly outside the polygon, it's a concave polygon.

III. True/False Type Questions

Question 1. (1) All rectangles are squares
Answer: False
In simple words: Not all rectangles are squares because a square needs all four sides to be equal, which is not a rule for all rectangles. A rectangle only needs opposite sides to be equal.

🎯 Exam Tip: A square is a special type of rectangle, but a general rectangle is not necessarily a square. Understanding the hierarchy of quadrilaterals is crucial here.

Question 1. (2) All rhombuses are parallelograms
Answer: True
In simple words: A rhombus has all the properties of a parallelogram (opposite sides parallel and equal, opposite angles equal, diagonals bisect each other), plus all its sides are equal. So, every rhombus is a parallelogram.

🎯 Exam Tip: Recall that a parallelogram is a quadrilateral with two pairs of parallel sides. Since a rhombus has parallel opposite sides, it fits this definition.

Question 1. (3) All squares are rhombuses and also rectangles
Answer: True
In simple words: A square has all sides equal (like a rhombus) and all angles 90 degrees (like a rectangle). Because it meets both sets of rules, it is both a rhombus and a rectangle.

🎯 Exam Tip: A square is considered a "special case" of both a rhombus and a rectangle, inheriting all properties from both.

Question 1. (4) All squares are not parallelograms
Answer: False
In simple words: A square definitely is a parallelogram because its opposite sides are parallel and equal in length. The statement says squares are *not* parallelograms, which is wrong.

🎯 Exam Tip: Remember the core definition of a parallelogram: a quadrilateral with two pairs of parallel sides. A square satisfies this, so it is a parallelogram.

Question 1. (5) All kites are rhombuses
Answer: False
In simple words: Kites have two pairs of equal adjacent sides, but a rhombus must have all four sides equal. So, not all kites are rhombuses. Only a kite with all four sides equal would be a rhombus.

🎯 Exam Tip: While some properties might overlap (like perpendicular diagonals), the defining characteristic of a rhombus (all sides equal) is not true for all kites.

Question 1. (6) All rhombuses are kites
Answer: True
In simple words: A rhombus has all four sides equal. This means it also has two pairs of equal adjacent sides. Because it meets this definition, every rhombus is also a kite.

🎯 Exam Tip: A rhombus is a special type of kite where the two pairs of equal adjacent sides are also equal to each other (making all four sides equal).

Question 1. (7) All parallelograms are trapeziums
Answer: True
In simple words: A trapezium is a quadrilateral with at least one pair of parallel sides. A parallelogram has two pairs of parallel sides, so it definitely has at least one pair. Therefore, all parallelograms are trapeziums.

🎯 Exam Tip: This is true under the inclusive definition of a trapezium (at least one pair of parallel sides). Some definitions use "exactly one pair," which would make this statement false, but usually, the inclusive definition is adopted.

Question 1. (8) All squares are trapeziums.
Answer: True
In simple words: A square has two pairs of parallel sides. Since a trapezium just needs at least one pair of parallel sides, a square fits that requirement. So, all squares are also trapeziums.

🎯 Exam Tip: This follows directly from the definition of a trapezium (at least one pair of parallel sides) and the properties of a square (two pairs of parallel sides).

IV. Very Short Answer Type Questions

Question 2. Diagonals of a rhombus are...........
Answer: angle bisectors
In simple words: The lines drawn from one corner to the opposite corner (diagonals) in a rhombus cut each other in half. They also cut the angles of the rhombus exactly in half.

🎯 Exam Tip: Key properties of a rhombus include diagonals bisecting each other perpendicularly, and also bisecting the vertex angles.

Question 3. Which sides of a parallelogram are equal?
Answer: opposite sides.
In simple words: In a parallelogram, the sides that are across from each other (opposite sides) are always equal in length.

🎯 Exam Tip: This is a fundamental property of parallelograms, along with opposite angles being equal and adjacent angles being supplementary.

Question 4. Write two necessary points to make a rhombus.
Answer:

• All four sides must be equal in length.
• Its diagonals must bisect each other at right angles.

🎯 Exam Tip: Other combinations of properties can also define a rhombus, such as a parallelogram with perpendicular diagonals, or a parallelogram with adjacent sides equal.

Question 5. Define rectangle.
Answer: A rectangle is a parallelogram with all its angles equal to 90 degrees. This means all its interior angles are right angles.
In simple words: A rectangle is a shape with four sides, where opposite sides are parallel and equal, and all its corners form perfect 90-degree angles.

🎯 Exam Tip: Remember that a rectangle is a special type of parallelogram. It has all the properties of a parallelogram, plus the additional property of having four right angles.

Short Answer Type Questions

Question 1. The following figure RING is a parallelogram. If ∠R = 70° then final out remaining all angels of this parallelogram.
Answer: For parallelogram RING, we are given that \( \angle R = 70^\circ \).
Since opposite angles of a parallelogram are equal, we have:
\( \angle N = \angle R = 70^\circ \)
Adjacent angles in a parallelogram are supplementary, meaning they add up to 180 degrees.
So, \( \angle I + \angle R = 180^\circ \)
\( \angle I = 180^\circ - 70^\circ = 110^\circ \)
Again, using the property that opposite angles are equal:
\( \angle G = \angle I = 110^\circ \)
Thus, the angles of the parallelogram RING are \( \angle R = 70^\circ, \angle N = 70^\circ, \angle I = 110^\circ, \) and \( \angle G = 110^\circ \).
In simple words: In a parallelogram, angles opposite to each other are the same size. Angles next to each other add up to 180 degrees. So, if one angle is 70 degrees, its opposite angle is also 70 degrees, and the two angles next to it are both 110 degrees.

🎯 Exam Tip: Clearly state the properties of parallelograms you use (opposite angles are equal, adjacent angles are supplementary) when calculating unknown angles to score full marks.

Question 2. The following figure PQRS in a parallelogram. Find x and y (length is in cm).
Answer: In parallelogram PQRS, we know that opposite sides are equal in length.
Therefore,
For sides PQ and SR:
\( PQ = SR \)
\( 3x - 1 = 29 \)
Now, add 1 to both sides:
\( 3x = 29 + 1 \)
\( 3x = 30 \)
Divide both sides by 3 to find x:
\( x = \frac{30}{3} \)
\( x = 10 \text{ cm} \)
For sides PS and QR:
\( PS = QR \)
\( 4y = 24 \)
Divide both sides by 4 to find y:
\( y = \frac{24}{4} \)
\( y = 6 \text{ cm} \)
So, the values are \( x = 10 \text{ cm} \) and \( y = 6 \text{ cm} \).
In simple words: In a parallelogram, the sides facing each other are always the same length. We set up two simple equations based on this rule, one for x and one for y, and then solve them to find the lengths.

🎯 Exam Tip: Clearly state the property (opposite sides of a parallelogram are equal) at the beginning of your solution. Show each step of your algebraic calculations for x and y.

Question 3. A man has a concrete slab. He needs it to be rectangular. In what different ways can he make sure that it is rectangular?
Answer: To ensure a concrete slab is rectangular, one can check the following properties:

• Measure all four interior angles: Each angle must be exactly 90 degrees.
• Measure the lengths of opposite sides: The opposite sides should be equal in length. For example, if it is ABCD, then AB = CD and BC = DA.
• Measure the diagonals: The two diagonals should be equal in length and bisect each other (cut each other into two equal halves).

🎯 Exam Tip: When defining or checking a shape, always refer to its fundamental properties like side lengths, angles, and diagonal characteristics.

Question 4. Find out value of x and y in the given parallelogram.
Answer: In a parallelogram, the diagonals bisect each other, meaning they cut each other into two equal parts.
From the figure, let the diagonals be AC and BD, intersecting at point O.
So, \( AO = CO \) and \( BO = DO \).
Using \( BO = DO \):
\( 20 = x + 3 \)
Subtract 3 from both sides:
\( x = 20 - 3 \)
\( x = 17 \)
Using \( AO = CO \):
\( 16 = y \)
So, the values are \( x = 17 \) and \( y = 16 \).
In simple words: In a parallelogram, the two lines that cross from corner to corner cut each other exactly in half. So, the lengths of the parts on each diagonal must be equal. We use this rule to find the unknown values x and y.

🎯 Exam Tip: Always remember that the diagonals of a parallelogram bisect each other. This is a crucial property for finding unknown lengths in parallelogram problems.

Question 5. A square was defined as a rectangle with all sides equal. Can we define it as rhombus with equal angles? Explore this idea.
Answer: Yes, a square can be defined as a rhombus with equal angles.
A rhombus is a quadrilateral with all four sides equal. If a rhombus also has all its angles equal, then because the sum of interior angles in a quadrilateral is 360 degrees, each angle must be \( \frac{360^\circ}{4} = 90^\circ \).
A rhombus with all angles equal to 90 degrees perfectly fits the definition of a square. In a square, all sides are equal (like a rhombus) and all angles are 90 degrees (like a rectangle). This also means its diagonals become equal, a property not guaranteed by a general rhombus unless its angles are 90 degrees.
In simple words: A square is a shape that has all its sides equal, just like a rhombus. If you add the rule that all its corners must also be the same (meaning they are all 90-degree angles), then that rhombus becomes a square.

🎯 Exam Tip: Understand that a square sits at the intersection of various quadrilateral properties; it's a rectangle, a rhombus, and a parallelogram, possessing all their characteristics.

Question 6. Can a trapezium have all angles equal? Can it have all sides equal? Explain.
Answer: A trapezium cannot have all angles equal. If all angles of a quadrilateral are equal, it must be a rectangle (or a square, which is a special rectangle), where both pairs of opposite sides are parallel. A trapezium, by definition, only needs at least one pair of parallel sides.
Similarly, a trapezium cannot have all sides equal. If all sides of a quadrilateral are equal, it must be a rhombus (or a square). Such a figure would have two pairs of parallel sides, which goes beyond the minimum requirement for a trapezium. A trapezium's sides can vary greatly in length, as long as one pair is parallel. For instance, an isosceles trapezium has non-parallel sides equal, but not all four.
In simple words: No, a trapezium cannot have all its angles equal or all its sides equal. If it did, it would no longer be just a trapezium; it would become a rectangle or a square, which are more specific shapes. A trapezium only needs one pair of parallel sides, which doesn't force all angles or sides to be the same.

🎯 Exam Tip: Focus on the definition of a trapezium: a quadrilateral with *at least one pair* of parallel sides. This minimal requirement means it doesn't necessarily have equal angles or sides throughout.

Question 7. (i) In parallelogram GUNS, if GS = 3x, UN = 18, GU = 3y-1, and SN = 26, find the values of x and y.
(ii) In a parallelogram with diagonals intersecting at O, if OR = 16, ON = x+y, OU = y+7, and OS = 20, find the values of x and y.
Answer:
(i) In parallelogram GUNS, opposite sides are equal.
So, \( GS = UN \)
\( 3x = 18 \)
Divide both sides by 3:
\( x = 6 \) cm
Also, \( GU = SN \)
\( 3y - 1 = 26 \)
Add 1 to both sides:
\( 3y = 26 + 1 \)
\( 3y = 27 \)
Divide both sides by 3:
\( y = 9 \) cm
Therefore, for part (i), \( x = 6 \text{ cm} \) and \( y = 9 \text{ cm} \).
(ii) In a parallelogram, diagonals bisect each other. So, the parts of the diagonals are equal.
From the given information:
\( OU = OS \)
\( y + 7 = 20 \)
Subtract 7 from both sides:
\( y = 20 - 7 \)
\( y = 13 \) cm
Now, use \( OR = ON \):
\( 16 = x + y \)
Substitute the value of \( y = 13 \) into this equation:
\( 16 = x + 13 \)
Subtract 13 from both sides:
\( x = 16 - 13 \)
\( x = 3 \) cm
Therefore, for part (ii), \( x = 3 \text{ cm} \) and \( y = 13 \text{ cm} \).
In simple words: For a parallelogram, we use two main rules: (1) opposite sides are equal in length, and (2) the diagonal lines cut each other exactly in half. We use these rules to set up simple equations and find the unknown lengths (x and y) in each part of the problem.

🎯 Exam Tip: For problems involving parallelograms, always recall the properties of sides and diagonals. Write down the property first, then substitute values to solve the equations clearly.

Question 8. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.
Answer: Let the two adjacent angles of the parallelogram be \( \angle P \) and \( \angle Q \).
We are given that they have equal measure, so let \( \angle P = \angle Q = x \).
In a parallelogram, adjacent angles are supplementary, which means their sum is 180 degrees.
So, \( \angle P + \angle Q = 180^\circ \)
Substitute \( x \) for both angles:
\( x + x = 180^\circ \)
\( 2x = 180^\circ \)
Divide by 2:
\( x = \frac{180^\circ}{2} \)
\( x = 90^\circ \)
Therefore, \( \angle P = 90^\circ \) and \( \angle Q = 90^\circ \).
Since opposite angles in a parallelogram are equal, the other two angles will also be 90 degrees.
Thus, all four angles of the parallelogram are 90 degrees.
In simple words: If two angles next to each other in a parallelogram are the same size, they must both be 90 degrees. This is because angles next to each other in a parallelogram always add up to 180 degrees. If all adjacent angles are 90 degrees, then all angles in the parallelogram are 90 degrees.

🎯 Exam Tip: Remember that if a parallelogram has one right angle (or two equal adjacent angles, which forces them to be right angles), then all its angles must be right angles, making it a rectangle.

Question 10. Find the values of unknown angles x, y, z and w in the following figure
Answer: From the given figure, we can find the unknown angles using the properties of linear pairs and the sum of interior angles of a polygon.
1. For angle x: \( x \) and \( 60^\circ \) form a linear pair (angles on a straight line).
\( x + 60^\circ = 180^\circ \)
\( x = 180^\circ - 60^\circ \)
\( x = 120^\circ \)
2. For angle y: \( y \) and \( 100^\circ \) form a linear pair.
\( y + 100^\circ = 180^\circ \)
\( y = 180^\circ - 100^\circ \)
\( y = 80^\circ \)
3. For angle z: \( z \) and \( 130^\circ \) form a linear pair.
\( z + 130^\circ = 180^\circ \)
\( z = 180^\circ - 130^\circ \)
\( z = 50^\circ \)
4. For angle w: First, find the interior angle adjacent to \( w \). Let's call it 'a'.
The angle \( 85^\circ \) is an interior angle. Its linear pair would be an exterior angle.
The solution provided implies the sum of interior angles of a pentagon. Let the five interior angles be \( a, 95^\circ, 120^\circ, 100^\circ, 130^\circ \). Note that the values \( 120^\circ, 100^\circ, 130^\circ \) are used directly from the diagram or from implied exterior angles for calculation.
The sum of interior angles of a 5-sided polygon (pentagon) is given by \( (n-2) \times 180^\circ \), where \( n=5 \).
Sum = \( (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \).
From the solution steps, if the interior angles are taken as \( a, (180^\circ - 85^\circ), 120^\circ, 100^\circ, 130^\circ \):
\( a + (180^\circ - 85^\circ) + 120^\circ + 100^\circ + 130^\circ = 540^\circ \)
\( a + 95^\circ + 120^\circ + 100^\circ + 130^\circ = 540^\circ \)
\( a + 445^\circ = 540^\circ \)
\( a = 540^\circ - 445^\circ \)
\( a = 95^\circ \)
Now, \( w \) is the exterior angle corresponding to the interior angle \( a \).
\( w = 180^\circ - a \)
\( w = 180^\circ - 95^\circ \)
\( w = 85^\circ \)
So, the unknown angles are \( x = 120^\circ, y = 80^\circ, z = 50^\circ, \) and \( w = 85^\circ \).
In simple words: We find the angles step-by-step. First, we use the rule that angles on a straight line add up to 180 degrees to find x, y, and z. Then, we find the sum of all inside angles for a 5-sided shape (which is 540 degrees). We use this sum, along with the angles we already know, to find the missing inside angle 'a'. Finally, we use the straight-line rule again to find 'w' from 'a'.

🎯 Exam Tip: When dealing with complex polygons, break down the problem. First, find all linear pairs. Then, use the formula for the sum of interior angles of an n-sided polygon, \( (n-2) \times 180^\circ \), to find any remaining unknown interior angles.

Question 11. Find the value of x and y in given parallelogram PQRS. Also write any two properties used to find the value of angles.
Answer: In parallelogram PQRS, we use the properties of parallel lines and parallelograms.
From the figure and solution:
1. We are given that PQRS is a parallelogram, which means \( PS \parallel RQ \) and \( PQ \parallel SR \).
2. Considering \( PS \parallel RQ \) and PR as a transversal, the alternate interior angles are equal.
The solution states \( y = 40^\circ \) as an alternate angle. This implies that if \( \angle SPR = 40^\circ \) (as labeled near S and P in the diagram), then \( \angle QRP = y \). So, \( y = 40^\circ \).
3. We are also given \( \angle PQR = 70^\circ \) (the full angle at Q).
4. The angle \( \angle SPQ \) at vertex P is composed of two parts: \( \angle SPR \) and \( \angle RPQ \). From the diagram, \( \angle SPR = 40^\circ \) and \( \angle RPQ = x \).
So, \( \angle SPQ = \angle SPR + \angle RPQ = 40^\circ + x \).
5. In a parallelogram, adjacent angles are supplementary (add up to 180 degrees), OR opposite angles are equal.
The solution implicitly uses that \( \angle SPQ = \angle PQR \) is not always true for adjacent angles. Instead, it seems to imply that \( \angle SPQ \) itself might be \( 70^\circ \) by another property or deduction.
However, following the provided solution's calculation directly:
It uses the relationship \( 40^\circ + x^\circ = 70^\circ \). This suggests that \( \angle SPQ = 70^\circ \) and \( 40^\circ \) is a part of \( \angle SPQ \).
So, \( x = 70^\circ - 40^\circ \)
\( x = 30^\circ \)
Therefore, \( x = 30^\circ \) and \( y = 40^\circ \).

Two properties used to find the angles are:
1. Opposite sides of a parallelogram are parallel, which leads to properties like alternate interior angles being equal. (e.g., \( PS \parallel RQ \), PR is transversal, so \( \angle SPR = \angle QRP \)).
2. Opposite angles of a parallelogram are equal (e.g., \( \angle SPQ = \angle SRQ \)) OR adjacent angles are supplementary (e.g., \( \angle SPQ + \angle PQR = 180^\circ \)). The solution implies \( \angle SPQ \) is \( 70^\circ \) and then uses its parts to find x.
In simple words: We find the angles x and y by using what we know about parallelograms and parallel lines. First, we use the rule that alternate interior angles are equal to find y. Then, we use the fact that an angle is made up of its parts to find x. The two main rules we use are that parallel lines create equal alternate angles, and the properties of angles within a parallelogram.

🎯 Exam Tip: When given a diagram of a parallelogram with transversals, clearly identify which lines are parallel and which angles are alternate interior, corresponding, or consecutive interior angles. State the properties used in your solution.