CBSE Class 8 Mathematics Understanding Quadrilaterals Worksheet Set 04

Understanding Quadrilaterals

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Question 1. A simple closed curve made up of only line segments is called a
(A) Circle
(B) Polygon
(C) Line segment
(D) None of the options
Answer: (B) Polygon
A polygon is defined as a simple closed curve made up entirely of line segments.
In simple words: A polygon is a closed shape formed by straight line segments.

Exam Tip: Remember that curves like circles are not made of line segments, and open curves are not polygons.

Question 2. In the following figure, which of the polygon is octagon?
(A) Circle
(B) Rectangle
(C) Quadrilateral
(D) Octagon
Answer: (D)
An octagon is a polygon with 8 sides. Figure (D) represents an 8-sided polygon.
In simple words: Count the number of sides; an octagon has 8 sides.

Exam Tip: Memorize polygon names by their side counts (triangle = 3, quadrilateral = 4, pentagon = 5, hexagon = 6, heptagon = 7, octagon = 8).

Question 3. In the following polygon, which of the polygon is regular polygon?
(A) Rectangle
(B) Right-angled triangle
(C) Square
(D) Trapezium
Answer: (C) Square
A regular polygon has all sides of equal length and all interior angles of equal measure. Among the choices, a square satisfies both conditions.
In simple words: A regular polygon has equal sides and equal angles, like a square.

Exam Tip: A rectangle has equal angles but unequal adjacent sides, so it is not regular.

Question 4. Find the angle measure x in the following figure:
(A) 60°
(B) 50°
(C) 45°
(D) 90°
Answer: (B) 50°
Using the exterior angle property or angle sum property of the polygon, the exterior angle sum is 360°. Solving for x gives 50°.
In simple words: The sum of exterior angles of any polygon is always 360 degrees.

Exam Tip: Always check if interior or exterior angles are being summed up.

Question 5. The sum of the measures of the external angles of any polygon is
(A) 180°
(B) 360°
(C) 270°
(D) 540°
Answer: (B) 360°
The sum of the exterior angles taken in order for any convex polygon is always 360°.
In simple words: Going around any polygon once totals 360 degrees of turning.

Question 6. Find x in the following figure:
(A) 110°
(B) 108°
(C) 105°
(D) 100°
Answer: (B) 108°
For a regular pentagon, the sum of interior angles is \((5 - 2) \times 180° = 540°\). Each interior angle is \(\frac{540°}{5} = 108°\).
In simple words: Divide the total interior angle sum of a pentagon by 5 to find each angle.

Exam Tip: Formula for each interior angle of a regular polygon of n sides is \(\frac{(n-2) \times 180°}{n}\).

Question 7. One angle of a quadrilateral is 150° and the other three angles are equal. Find the measure of other three angles
(A) 75°
(B) 85°
(C) 95°
(D) 70°
Answer: (A) 75°
Let each of the three equal angles be x.
Sum of angles in a quadrilateral = 360°
\(150° + x + x + x = 360°\)
\(150° + 3x = 360°\)
\(3x = 360° - 150° = 210°\)
\(x = \frac{210°}{3} = 75°\)
In simple words: Subtract the given angle from 360 and divide the remainder by 3.

Exam Tip: The sum of all interior angles of a quadrilateral is always 360 degrees.

Question 8. Which of the following statement is false in the following statements?
(A) The opposite sides of a parallelogram are of equal length.
(B) A kite has 4 sides.
(C) The opposite angles of a parallelogram are of equal measure.
(D) The adjacent angles in a parallelogram are complementary
Answer: (D) The adjacent angles in a parallelogram are complementary
Adjacent angles in a parallelogram are supplementary (their sum is 180°), not complementary (90°).
In simple words: Adjacent angles add up to 180 degrees, making them supplementary.

Exam Tip: Supplementary means sum to 180 degrees; complementary means sum to 90 degrees.

Question 9. Two adjacent sides AB and BC of a parallelogram ABCD are in the ratio of 5:3. If the perimeter is 200 cm, find the lengths of AB and BC.
(A) 25 cm, 50 cm
(B) 40 cm, 37.50 cm
(C) 62.50 cm, 37.50 cm
(D) 60 cm, 62.50 cm
Answer: (C) 62.50 cm, 37.50 cm
Let the sides be \(5x\) and \(3x\).
Perimeter = \(2(AB + BC) = 200\)
\(2(5x + 3x) = 200\)
\(16x = 200\)
\(x = \frac{200}{16} = 12.5 \text{ cm}\)
\(AB = 5 \times 12.5 = 62.50 \text{ cm}\)
\(BC = 3 \times 12.5 = 37.50 \text{ cm}\)
In simple words: Use the perimeter formula for a rectangle or parallelogram: twice the sum of adjacent sides.

Exam Tip: Perimeter of a parallelogram is \(2 \times (\text{sum of adjacent sides})\).

Question 10. The measures of two adjacent angles of a parallelogram are in the ratio 3:2. Find the measure of each of the angles of the parallelogram.
(A) \(120°, 60°, 90°, 90°\)
(B) \(110°, 70°, 90°, 90°\)
(C) \(100°, 70°, 90°, 90°\)
(D) \(108°, 108°, 72° \text{ and } 72°\)
Answer: (D) \(108°, 108°, 72° \text{ and } 72°\)
Let the adjacent angles be \(3x\) and \(2x\).
Adjacent angles are supplementary: \(3x + 2x = 180°\)
\(5x = 180° \implies x = 36°\)
Angles are \(3 \times 36° = 108°\) and \(2 \times 36° = 72°\).
Since opposite angles are equal, the four angles are \(108°, 108°, 72°, 72°\).
In simple words: Adjacent angles sum to 180 degrees, and opposite angles are equal.

Question 11. In the given figure, ABCD is a parallelogram. Find the angle measures x, y and z.
(A) \(110°, 20° \text{ and } 30°\)
(B) \(130°, 20° \text{ and } 30°\)
(C) \(120°, 20° \text{ and } 40°\)
(D) \(120°, 30° \text{ and } 40°\)
Answer: (B) \(130°, 20° \text{ and } 30°\)
Using parallel lines properties and angle sum in triangles, z = 30°, y = 20°, and x = 130°.
In simple words: Apply alternate interior angles and triangle angle sum rules.

Exam Tip: Look for Z-shapes (alternate interior angles) when parallel lines are cut by a transversal.

Question 12. In a parallelogram ABCD. The bisector of angle A and angle B meet at O. Measurement of angle AOB.
(A) 45°
(B) 90°
(C) 75°
(D) 120°
Answer: (B) 90°
Adjacent angles A and B are supplementary (\(\angle A + \angle B = 180°\)).
Bisectors halve them: \(\frac{1}{2}\angle A + \frac{1}{2}\angle B = 90°\).
In triangle AOB, \(\angle AOB = 180° - (\frac{1}{2}\angle A + \frac{1}{2}\angle B) = 180° - 90° = 90°\).
In simple words: Angle bisectors of adjacent angles in a parallelogram always meet at a right angle.

Exam Tip: This is a standard geometric result: angle between adjacent angle bisectors in a parallelogram is always 90°.

Question 13. In the adjoining figure, ABCD is a parallelogram. Side DC of the parallelogram is produced to E. If exterior angle \(\angle C = 110°\), then find \(\angle ABC\).
(A) 60°
(B) 110°
(C) 120°
(D) 90°
Answer: (B) 110°
Since DC is produced to E, the exterior angle at C is 110°. Opposite angles in a parallelogram are equal, or by corresponding angles, \(\angle ABC = \angle BCE = 110°\).
In simple words: Opposite angles in a parallelogram are equal.

Exam Tip: Exterior angle of a side produced equals the opposite interior angle.

Question 14. ABCD is a parallelogram, in which \(\angle A = 4x - 15\) and \(\angle B = 5x - 3\). Find the measure of \(\angle A\) and \(\angle B\).
Answer:
Adjacent angles of a parallelogram are supplementary.
\(\angle A + \angle B = 180°\)
\((4x - 15) + (5x - 3) = 180°\)
\(9x - 18 = 180°\)
\(9x = 198°\)
\(x = 22°\)
\(\angle A = 4(22) - 15 = 88 - 15 = 73°\)
\(\angle B = 5(22) - 3 = 110 - 3 = 107°\)
In simple words: Set the sum of adjacent angles to 180, solve for x, then substitute back.

Exam Tip: Verify your answer by checking if \(73° + 107° = 180°\).

Question 15. Which of the following statement is true?
(A) Opposite angles are supplementary in parallelogram.
(B) Diagonals bisect each other at right angle in rectangle.
(C) Diagonals are not equal in square.
(D) Diagonals bisect each other at right angle in rhombus.
Answer: (D) Diagonals bisect each other at right angle in rhombus.
A rhombus has perpendicular diagonals that bisect each other.
In simple words: The diagonals of a rhombus intersect at 90 degrees.

Question 16. Find the number of sides of a regular polygon whose each exterior angle has a measure of 60°.
(A) 6
(B) 8
(C) 12
(D) 5
Answer: (A) 6
Number of sides \(n = \frac{360°}{\text{Exterior Angle}} = \frac{360°}{60°} = 6\).
In simple words: Divide 360 by the measure of one exterior angle to get the number of sides.

Exam Tip: Sum of exterior angles is always 360° for any convex polygon.

Question 17. Find the measure of each exterior angle of a regular polygon of 9 sides.
(A) 45°
(B) 60°
(C) 40°
(D) 35°
Answer: (C) 40°
Measure of each exterior angle = \(\frac{360°}{n} = \frac{360°}{9} = 40°\).
In simple words: Divide 360 by the number of sides.

Exam Tip: Exterior angle formula is \(\frac{360°}{n}\).

Question 18. Two adjacent sides of a parallelogram are 5 cm and 4 cm respectively. Find its perimeter.
(A) 18 cm
(B) 14 cm
(C) 10 cm
(D) 16 cm
Answer: (A) 18 cm
Perimeter = \(2(\text{side}_1 + \text{side}_2) = 2(5 + 4) = 2(9) = 18 \text{ cm}\).
In simple words: Add the two adjacent sides and multiply the sum by 2.

Exam Tip: Opposite sides are equal, so perimeter is the sum of all 4 outer sides.

Question 19. In the given figure, ABCD is parallelogram in which \(\angle DAB = 75°\) and \(\angle DBC = 60°\), calculate \(\angle CDB\) and \(\angle ADB\).
(A) 30°, 60°
(B) 90°, 30°
(C) 45°, 60°
(D) 60°, 35°
Answer: (C) 45°, 60°
Since ABCD is a parallelogram, \(\angle DCB = \angle DAB = 75°\).
In triangle BCD, sum of angles = 180°
\(\angle CDB = 180° - (75° + 60°) = 180° - 135° = 45°\).
Since AB || CD, alternate interior angle \(\angle ADB = \angle DBC = 60°\).
In simple words: Use triangle angle sum and alternate interior angles for parallel lines.

Exam Tip: Parallel lines create alternate interior angles that are equal.

Question 20. The diagonal of a rhombus are 6 cm and 8 cm, find the length of all sides of rhombus.
(A) 3 cm
(B) 6 cm
(C) 4 cm
(D) 5 cm
Answer: (D) 5 cm
Diagonals of a rhombus bisect each other at right angles.
Half diagonals are \(d_1 = \frac{6}{2} = 3 \text{ cm}\) and \(d_2 = \frac{8}{2} = 4 \text{ cm}\).
Side \(s = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm}\).
In simple words: Use Pythagoras theorem on the right triangle formed by half-diagonals.

Question 21. The point of intersection of the diagonal of a quadrilateral divides one diagonal in the ratio 1:2. Can it be a parallelogram?
(A) No
(B) Yes
(C) May be
(D) None of the options
Answer: (A) No
Diagonals of a parallelogram always bisect each other, meaning they divide each other in the ratio 1:1, not 1:2.
In simple words: Parallelogram diagonals cut each other exactly in half.

Exam Tip: Bisecting means dividing into two equal halves (ratio 1:1).

Question 22. In the following polygons, which of the polygon is concave polygon?
(A) Regular hexagon
(B) Rectangle
(C) Arrowhead / reflex polygon
(D) Parallelogram
Answer: (D)
A concave polygon has at least one interior angle greater than 180° and at least one diagonal lies outside the polygon.
In simple words: A concave polygon has a "cave-in" vertex pointing inwards.

Exam Tip: Convex polygons have all interior angles less than 180°.

Question 23. Diagonals of a square are equal and bisect each other at .......
(A) 90°
(B) 60°
(C) 75°
(D) None of the options
Answer: (A) 90°
The diagonals of a square are perpendicular bisectors of each other.
In simple words: Square diagonals cross at right angles.

Exam Tip: This property is shared by both squares and rhombuses.

Question 24. The angle between the altitudes of a parallelogram, through the same vertex of an obtuse angle of the parallelogram is 60 degree. Find the angles ofamboat the parallelogram.
(A) 150°, 120°, 60° and 30°
(B) 150°, 150°, 30° and 30°
(C) 120°, 60°, 90° and 45°
(D) None of the options
Answer: (D) None of the options
The angle between the altitudes drawn from an obtuse-angled vertex equals the acute angle of the parallelogram. If the angle between altitudes is 60°, the acute angle is 60° and the obtuse angle is \(180° - 60° = 120°\). Thus, the angles are 120°, 60°, 120°, 60°.
In simple words: The angle between altitudes equals the acute angle of the parallelogram.

Exam Tip: Parallelogram angles come in equal opposite pairs: two acute and two obtuse.

Question 25. How many diagonals have a regular hexagon?
(A) 9
(B) 5
(C) 7
(D) 8
Answer: (A) 9
Number of diagonals = \(\frac{n(n-3)}{2}\). For a hexagon (\(n = 6\)): \(\frac{6(6-3)}{2} = \frac{6 \times 3}{2} = 9\).
In simple words: Use the diagonal formula \(n(n-3)/2\).

Exam Tip: Memorize the formula for diagonals of an n-sided polygon: \(\frac{n(n-3)}{2}\).

Question 26. The angles of a quadrilateral are in the ratio 3:4:5:6. Find the measure of each angle.
(A) 60°, 90°, 100° and 110°
(B) 75°, 85°, 90° and 110°
(C) 60°, 80°, 100° and 120°
(D) 80°, 85°, 95° and 100°
Answer: (C) 60°, 80°, 100° and 120°
Let the angles be \(3x, 4x, 5x, 6x\).
Sum of angles = 360°
\(3x + 4x + 5x + 6x = 360°\)
\(18x = 360° \implies x = 20°\)
Angles are: \(3(20) = 60°\), \(4(20) = 80°\), \(5(20) = 100°\), \(6(20) = 120°\).
In simple words: Sum the ratio parts to equal 360, solve for x, then multiply.

Exam Tip: Always verify that \(60° + 80° + 100° + 120° = 360°\).

Question 27. In the given figure, find the measure of the \(\angle BAD\).
(A) 80°
(B) 70°
(C) 90°
(D) 105°
Answer: (A) 80°
Using polygon exterior and interior angle relationships, \(\angle BAD = 80°\).
In simple words: Sum of angles around the vertex gives 80 degrees.

Exam Tip: Linear pairs sum up to 180°.

Question 28. If an angle of a parallelogram is two-third of its adjacent angle; find these two angles of the parallelogram.
(A) 108°, 72°
(B) 110°, 70°
(C) 100°, 80°
(D) 120°, 60°
Answer: (A) 108°, 72°
Let adjacent angles be \(x\) and \(\frac{2}{3}x\).
\(x + \frac{2}{3}x = 180°\)
\(\frac{5}{3}x = 180°\)
\(x = \frac{180 \times 3}{5} = 108°\)
Other angle = \(\frac{2}{3} \times 108° = 72°\).
In simple words: Adjacent angles add to 180; set up the equation and solve.

Exam Tip: Adjacent angles of a parallelogram are always supplementary.

Question 29. In the given figure, ABCD is a parallelogram and angle \(D = 115°\). Determine the measure of angle A and angle B.
(A) 60°, 45°
(B) 65°, 115°
(C) 45°, 65°
(D) 30°, 75°
Answer: (B) 65°, 115°
Opposite angles are equal, so \(\angle B = \angle D = 115°\).
Adjacent angles are supplementary, so \(\angle A = 180° - 115° = 65°\).
In simple words: Opposite angles are equal, and adjacent angles add up to 180.

Question 30. Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle.
(A) 60°, 60°, 120° and 120°
(B) 45°, 45°, 120° and 120°
(C) 65°, 65°, 125° and 125°
(D) 68°, 68°, 112° and 112°
Answer: (D) 68°, 68°, 112° and 112°
Let the smallest angle be \(x\). The adjacent angle is \(2x - 24°\).
\(x + (2x - 24°) = 180°\)
\(3x = 204° \implies x = 68°\)
Other angle = \(2(68) - 24 = 136 - 24 = 112°\).
The four angles are 68°, 68°, 112°, 112°.
In simple words: Set the sum of adjacent angles to 180 and solve for x.

Exam Tip: Smallest angle in a parallelogram is always acute unless it's a rectangle.

Question 31. In the given figure, ABCD is a parallelogram. Determine the measure of x.
(A) 135°
(B) 60°
(C) 120°
(D) 90°
Answer: (C) 120°
Using interior and exterior angle relationships in a parallelogram, \(x = 120°\).
In simple words: Opposite angles of a parallelogram are equal, and exterior angle matches interior alternate.

Exam Tip: Linear pair with interior angle gives the exterior angle.

Question 32. In the given figure, ABCD is a square. Find the value of x.
(A) 120°
(B) 130°
(C) 110°
(D) 100°
Answer: (B) 130°
Using triangle angle properties around the intersection O of diagonals in a square, where diagonals intersect at 90°, \(x = 130°\).
In simple words: Use angle sum properties in triangles formed by square diagonals.

Exam Tip: Diagonals of a square bisect the vertex angles (each is 45°).

Question 33. In the given figure, ABCD is a rectangle. Find the value of x.
(A) 48°
(B) 68°
(C) 45°
(D) 58°
Answer: (B) 68°
In rectangle ABCD, diagonals are equal and bisect each other, forming isosceles triangles. Solving gives \(x = 68°\).
In simple words: Diagonals of a rectangle form isosceles triangles with the sides.

Exam Tip: Base angles of isosceles triangles are equal.

Question 34. The sides BA and DC of a quadrilateral ABCD are produced as shown in the given figure. Find the sum of angle EAD and angle BCF.
(A) 200°
(B) 180°
(C) 150°
(D) 210°
Answer: (B) 180°
Exterior angles formed by producing sides of a quadrilateral relate to interior angles summing to 360°, giving a sum of 180° for these specific exterior angles.
In simple words: Sum of specific exterior angle pairs in a quadrilateral equals 180 degrees.

Exam Tip: Use linear pairs with interior angles to find exterior angles.

Question 35. In the given figure, ABCD is a rhombus. Find the value of x.
(A) 45°
(B) 30°
(C) 60°
(D) 40°
Answer: (B) 30°
Using properties of a rhombus where diagonals bisect vertex angles and opposite angles are equal, \(x = 30°\).
In simple words: Diagonals of a rhombus bisect the corner angles.

Exam Tip: Rhombus diagonals bisect interior angles.

Question 36. State the name of a regular polygon of 8 sides
Answer:
Octagon.
In simple words: An 8-sided regular polygon is called a regular octagon.

Exam Tip: Prefix "octo-" means eight.

Question 37. Find the angle measure of x in fig 1.
Answer:
Sum of interior angles of a quadrilateral = 360°
\(60° + 130° + 120° + x = 360°\)
\(310° + x = 360°\)
\(x = 360° - 310° = 50°\)
In simple words: Subtract the sum of the three given angles from 360 degrees.

Exam Tip: Quadrilateral interior angles always add up to 360°.

Question 38. Find the sum of the angles in the given figure.
Answer:
The given figure is a pentagon (5 sides).
Sum of interior angles = \((n - 2) \times 180° = (5 - 2) \times 180° = 3 \times 180° = 540°\).
In simple words: Use the formula \((n-2) \times 180\) where n is the number of sides.

Exam Tip: A pentagon can be split into 3 triangles, giving \(3 \times 180°\).

Question 39. Find the sum of the angles in the following figure.
Answer:
For a pentagon, the sum of interior angles is 540°.
In simple words: Sum of angles in a 5-sided polygon is 540 degrees.

Exam Tip: Always subtract 2 from the number of sides before multiplying by 180.

Question 40. How many diagonals can be there in fig.
Answer:
The figure is a pentagon (\(n = 5\)).
Number of diagonals = \(\frac{n(n-3)}{2} = \frac{5(5-3)}{2} = \frac{5 \times 2}{2} = 5\).
In simple words: A pentagon has 5 diagonals.

Question 41. Find X in the following figure.
Answer:
The figure is a quadrilateral with given interior angles. One angle is 90°, another is 90° (from perpendiculars), another is 110°.
Sum = 360°
\(90° + 90° + 110° + X = 360°\)
\(290° + X = 360°\)
\(X = 360° - 290° = 70°\)
In simple words: Subtract the sum of known angles from 360.

Exam Tip: Perpendicular symbols indicate 90-degree angles.

Question 42. Find the number of sides of a regular Polygon. Whose each exterior angle has a measure of 60°.
Answer:
Number of sides \(n = \frac{360°}{\text{Exterior Angle}} = \frac{360°}{60°} = 6\).
In simple words: Divide 360 by 60 to get 6 sides (a hexagon).

Exam Tip: Exterior angle sum is always 360°.

Question 43. Find the measure of each exterior angle of a regular polygon of 9 sides.
Answer:
Measure of each exterior angle = \(\frac{360°}{n} = \frac{360°}{9} = 40°\).
In simple words: Divide 360 by 9 to get 40 degrees.

Exam Tip: Regular polygons have all exterior angles equal.

<2M>

Question 44. ABCD is a Trapezium in which AB || DC. If \(\angle A = \angle B = 40°\), what are the measures of the other two angles.
Answer:
Since AB || DC, the consecutive interior angles on the same side of transversal are supplementary.
\(\angle A + \angle D = 180° \implies 40° + \angle D = 180° \implies \angle D = 140°\)
Similarly, \(\angle B + \angle C = 180° \implies 40° + \angle C = 180° \implies \angle C = 140°\)
The other two angles are 140° and 140°.
In simple words: Consecutive interior angles between parallel lines add up to 180 degrees.

Exam Tip: Co-interior angles sum to 180° when lines are parallel.

Question 45. Length of two adjacent sides of a Parallelogram are 4cm and 3cm. Find its perimeter.
Answer:
Perimeter of parallelogram = \(2(\text{side}_1 + \text{side}_2)\)
Perimeter = \(2(4 \text{ cm} + 3 \text{ cm}) = 2(7 \text{ cm}) = 14 \text{ cm}\).
In simple words: Add the two adjacent side lengths and multiply by 2.

Exam Tip: Perimeter is the total distance around the outside of the shape.

Question 46. In the following figure, given a parallelogram ABCD. Find x and y.
Answer:
Opposite sides of a parallelogram are equal in length.
\(3x = 26 \implies x = \frac{26}{3} \text{ cm}\)
\(3y - 1 = 18 \implies 3y = 19 \implies y = \frac{19}{3} \text{ cm}\)
In simple words: Set opposite sides equal to each other and solve for x and y.

Exam Tip: Opposite sides of a parallelogram are parallel and equal.

Question 47. Two adjacent angles of a Parallelogram are in the ratio of 2 : 3. Find all the 4 angles of the Parallelogram.
Answer:
Let the adjacent angles be \(2x\) and \(3x\).
Adjacent angles are supplementary: \(2x + 3x = 180°\)
\(5x = 180° \implies x = 36°\)
Angles are \(2(36°) = 72°\) and \(3(36°) = 108°\).
Since opposite angles are equal, all four angles are 72°, 108°, 72°, 108°.
In simple words: Adjacent angles add to 180, and opposite angles are equal.

Exam Tip: Always list all four angles when asked for "all the 4 angles".

Question 48. Diagonal AC and BD of a rectangle ABCD intersect each other at a point O in the following fig. If OA = 5cm. Find AC and BD.
Answer:
Diagonals of a rectangle are equal and bisect each other.
Since \(OA = 5 \text{ cm}\), \(OC = 5 \text{ cm}\), so \(AC = OA + OC = 5 + 5 = 10 \text{ cm}\).
Since diagonals are equal, \(BD = AC = 10 \text{ cm}\).
In simple words: Diagonals bisect each other into equal halves and are equal in length.

Question 49. In a Parallelogram PQRS, given that OQ = 4cm, and PR is 5 more than SQ. Find OP.
Answer:
Diagonals of a parallelogram bisect each other.
\(SQ = 2 \times OQ = 2 \times 4 = 8 \text{ cm}\).
Given \(PR = SQ + 5 = 8 + 5 = 13 \text{ cm}\).
\(OP = \frac{1}{2} \times PR = \frac{13}{2} = 6.5 \text{ cm}\).
In simple words: Find diagonal SQ first, add 5 to get PR, then take half to find OP.

Exam Tip: Bisecting means dividing into two equal parts.

Question 50. Find \(\angle C\) in the following figure ABDC.
Answer:
Using polygon angle properties or parallel line properties given in the figure, \(\angle C\) is supplementary to the adjacent interior angle. For standard trapezium or parallelogram configurations, \(\angle C = 180° - 40° = 140°\) (if consecutive) or equal to opposite angle.
In simple words: Use co-interior or opposite angle rules depending on the quadrilateral type.

Exam Tip: Identify the specific type of quadrilateral first before applying formulas.

Question 51. Find the measure of x in the following figure.
Answer:
Using parallel lines and transversal angle properties (corresponding/alternate/interior angles), \(x = 110°\).
In simple words: Corresponding angles or alternate exterior angles are equal.

Exam Tip: Parallel lines cut by a transversal form equal corresponding angles.

<4M>

Question 52. Two adjacent angles of a parallelogram have equal measure. Find the measures of each of the angles of the parallelogram.
Answer:
Let the two adjacent angles be \(x\) and \(x\).
Adjacent angles of a parallelogram are supplementary:
\(x + x = 180°\)
\(2x = 180° \implies x = 90°\)
Since opposite angles are equal, all four angles are 90°, 90°, 90°, 90°.
Therefore, the parallelogram is a rectangle or square, and each angle measures 90°.
In simple words: If adjacent angles are equal and add up to 180, each must be 90 degrees.

Exam Tip: A parallelogram with all 90° angles is a rectangle.

Question 53. EFGH is a Parallelogram. Find the angle measure x, y, and z. Also give the properties used to find the angles.
Answer:
1. Property used: Alternate interior angles / Opposite angles of a parallelogram.
2. Calculation: Using triangle angle sum and parallel line properties, \(z = 40°\) (alternate interior angle), \(y = 70°\), and \(x = 180° - (40° + 70°) = 70°\) (or via angle sum in triangle).
In simple words: Use alternate interior angles for parallel lines and angle sum of a triangle.

Exam Tip: Always state the geometric property used (e.g., alternate interior angles, opposite angles equal).

Question 54. Diagonal AC of a rhombus ABCD is equal to one of its sides BC. Find all the angles of Rhombus.
Answer:
In rhombus ABCD, all sides are equal: \(AB = BC = CD = DA\).
Given diagonal \(AC = BC\), so \(AC = AB = BC\).
Thus, triangle ABC is an equilateral triangle, which means \(\angle BAC = \angle BCA = \angle ABC = 60°\).
Similarly, triangle ADC is equilateral, so \(\angle DAC = \angle DCA = \angle ADC = 60°\).
Angles of the rhombus are:
\(\angle A = \angle DAB = \angle DAC + \angle BAC = 60° + 60° = 120°\)
\(\angle C = \angle DCB = 120°\)
\(\angle B = 60°\)
\(\angle D = 60°\)
The angles are 120°, 60°, 120°, 60°.
In simple words: The diagonal splits the rhombus into two equilateral triangles, giving angles of 60 and 120 degrees.

Exam Tip: An equilateral triangle has all three interior angles equal to 60°.

Question 55. Name the Quadrilaterals whose diagonals
(i) Bisect each other
(ii) Are perpendicular bisector of each other
(iii) Are equal
Answer:
(i) Parallelogram, Rhombus, Rectangle, Square.
(ii) Rhombus, Square.
(iii) Rectangle, Square.
In simple words: Parallelograms share bisecting diagonals, while squares, rectangles, and rhombuses have special diagonal properties.

Exam Tip: A square satisfies all three conditions (bisecting, perpendicular, and equal diagonals).

Question 56. Two adjacent angles of a Parallelogram are (3x - 4) and (3x + 10). Find the angles of the Parallelogram.
Answer:
Adjacent angles of a parallelogram are supplementary.
\((3x - 4) + (3x + 10) = 180°\)
\(6x + 6 = 180°\)
\(6x = 174° \implies x = 29°\)
First angle = \(3(29) - 4 = 87 - 4 = 83°\)
Second angle = \(3(29) + 10 = 87 + 10 = 97°\)
Since opposite angles are equal, the four angles of the parallelogram are 83°, 97°, 83° and 97°.
In simple words: Set the sum of the two adjacent expressions to 180, solve for x, and find the angles.

Exam Tip: Check that \(83° + 97° = 180°\).

Question 57. Given, a rectangle ABCD in the following figure and its diagonals meet at O. Find x, if \(OA = 2x + 4\) and \(OD = 3x + 1\). And also find BD.
Answer:
Diagonals of a rectangle are equal and bisect each other, which means \(OA = OC = OB = OD\).
Therefore, \(OA = OD\).
\(2x + 4 = 3x + 1\)
\(4 - 1 = 3x - 2x \implies x = 3\)
Substitute x to find OD:
\(OD = 3(3) + 1 = 9 + 1 = 10 \text{ cm}\)
Since O is the midpoint of diagonal BD, \(BD = 2 \times OD = 2 \times 10 = 20 \text{ units}\).
In simple words: Equate OA and OD since all half-diagonals in a rectangle are equal, find x, then double OD for BD.

Exam Tip: All four half-diagonals from the center O to the vertices of a rectangle are equal in length.

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Question 58. Find the length of the diagonal of a rectangle whose sides are 12 cm and 5 cm.
Answer:
In rectangle ABCD, let length \(l = 12 \text{ cm}\) and breadth \(b = 5 \text{ cm}\).
The diagonal d forms a right-angled triangle with the two adjacent sides.
Using Pythagoras theorem:
\(d^2 = l^2 + b^2\)
\(d^2 = 12^2 + 5^2\)
\(d^2 = 144 + 25 = 169\)
\(d = \sqrt{169} = 13 \text{ cm}\)
In simple words: Use Pythagoras theorem (\(\text{diagonal} = \sqrt{\text{length}^2 + \text{breadth}^2}\)).

Exam Tip: Memorize Pythagorean triples like 5-12-13 to solve instantly.

Question 59. Length of the diagonals AC and BD of a rhombus are 6 cm and 8 cm respectively. Find the length of each side of the rhombus.
Answer:
Diagonals of a rhombus bisect each other at right angles (90°).
Let diagonals be \(d_1 = 6 \text{ cm}\) and \(d_2 = 8 \text{ cm}\).
Half-diagonals are \(\frac{6}{2} = 3 \text{ cm}\) and \(\frac{8}{2} = 4 \text{ cm}\).
Each side s of the rhombus is the hypotenuse of a right-angled triangle formed by the half-diagonals:
\(s = \sqrt{3^2 + 4^2}\)
\(s = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm}\)
The length of each side of the rhombus is 5 cm.
In simple words: Halve both diagonals and apply Pythagoras theorem to find the side length.

Exam Tip: All four sides of a rhombus are equal in length.

Question 60. Fill in the blanks.
a) A Parallelogram having all sides equal is called a ------------
b) A rectangle with a pair of adjacent sides equal is a --------
c) A quadrilateral having exactly one pair of parallel sides is called ----
d) A Parallelogram with each angle a right angle is called ------
e) A quadrilateral with each pair of opposite sides parallel is called a -----
Answer:
a) Rhombus
b) Square
c) Trapezium
d) Rectangle
e) Parallelogram
In simple words: These are the standard definitions of fundamental quadrilaterals.

Exam Tip: Learn these definitions thoroughly as they are frequently tested in objective questions.