## Chapter 3 Understanding Quadrilaterals Exercise 3.3

**Question 1: Given a parallelogram ABCD. Complete each statement along with the definition or property used.**

(ii) ∠DCB = ……

(iii) OC = ……

(iv) m ∠DAB + m ∠CDA = ……

Answer:

**(i) AD = ……**(ii) ∠DCB = ……

(iii) OC = ……

(iv) m ∠DAB + m ∠CDA = ……

Answer:

(i) AD = BC (opposite sides are equal and parallel)

(ii) ∠DCB = ∠DAB (Opposite angles are equal)

(iii) OC = OA (Diagonals bisect each other)

(iv) m ∠DAB + m ∠CDA = 180° (Interior opposite angles, as AB||DC)

**Question 2: Consider the following parallelograms. Find the values of the unknown x, y, z**

**Answer:**i)

y = 100° (opposite angles are equal in parallelogram)

y + x = 180° (adjacent angles are supplementary)

100 + x = 180°

x = 80°

x = z = 80° (opposite angles are equal in parallelogram)

ii)

50° + y = 180° (adjacent angles in a parallelogram are supplementary)

50° + x = 180° (adjacent angles in a parallelogram are supplementary)

y = 130°

x = 130°

y = z = 130° (interior alternate angles)

iii)

x = 90° (Vertically Opposite Angles)

y + x + 30° = 180° (angle sum property of a triangle)

y + 90° + 30° = 180°

y + 120° = 180°

y = 180 - 120

y = 60°

y = z = 60° (interior alternate angles)

iv)

y = 80° (opposite angles are equal)

x + 80° = 180° (adjacent angles are supplementary)

x = 100°

z = 80° (corresponding angles)

v)

y = 112° (opposite angles are equal)

y + 40° + x = 180° (angle sum property of a triangle)

112° + 40° + x = 180°

152° + x = 180°

x = z = 28° (interior alternate angles)

**Question 3: Can a quadrilateral ABCD be a parallelogram if**

(i) ∠D + ∠B = 180°?

(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?

(iii)∠A = 70° and ∠C = 65°?

Answer:

(i) ∠D + ∠B = 180°?

(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?

(iii)∠A = 70° and ∠C = 65°?

Answer:

i) Yes, a quadrilateral ABCD be a parallelogram if ∠D + ∠B = 180°.

ii) No, in a parallelogram, opposite sides are equal, but AD ≠ BC.

iii) No, in a parallelogram, opposite angles are equal, but ∠A ≠ ∠C).

**Question 4: Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.**

Answer:

Answer:

**Question 5: 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.**

Answer:

Answer:

Ratio of adjacent angles = 3:2

Find: Measure of each angle

(Adjacent angles of the parallelogram are supplementary )

3x + 2x = 180°

5x = 180°

x = 36°

3x = 108°

2x = 72°

Therefore, the angles of parallelogram are 72°, 108°, 72°, 108°.

**Question 6: Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.**

Answer:

Adjacent angles are equal.

Answer:

Find: Measure of each angle

(Adjacent angles of parallelogram are supplementary)

x + x = 180°

2x = 180°

x = 90°

Therefore, the measure of each of the angle of the parallelogram is 90°.

**Question 7: The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.**

**Answer:**HOPE is a parallelogram.

y = 40° (alternate interior angles)

In triangle HPO,

70° = y + z (exterior angle property of a triangle)

z = 30°

x = 110° (opposite angles are equal in parallelogram)

Therefore, x = 110°, y = 40°, z = 30°.

**Question 8: The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)**

**Answer:**i)

GUNS is a parallelogram.

Opposite sides are equal.

3x = 18

x = 18/3

x = 6

3y - 1 = 26

3y = 26 + 1

3y = 27

y = 27/3

y = 9

Therefore, x = 6, y = 9.

ii)

RUNS is a parallelogram.

Opposite diagonals are equal.

20 = y + 7

y = 20 - 7

y = 13

16 = x + y

16 = x + 13

x = 16 - 13

x = 3

Therefore, y = 13, x = 3.

**Question 9: In the above figure both RISK and CLUE are parallelograms. Find the value of x.**

**RISK parallelogram (adjacent angles are supplementary)**

**Answer:**CLUE parallelogram (opposite angles are equal)

= 70° + 60° + x = 180°

= 130° + x = 180°

= x = 180 - 130°

= x = 50°

Therefore, x = 50°.

**Question 10: Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig 3.32)**

**The co-interior angles ∠M + ∠L = 180°. The lines NM and KL are parallel. KLMN is a trapezium as one pair of opposite sides are parallel.**

**Answer:****Question 11: Find m∠C in Fig 3.33 if AB || DC?**

**Given**

**Answer:**AB || DC

BC is a transversal.

Co-interior angles are supplementary.

m∠A + m∠C = 180°

120° + m∠C = 180°

∠C = 60°

Therefore, measure of ∠C is 60°.

**Question 12: Find the measure of ∠P and ∠S if SP || RQ ? in Fig 3.34. (If you find m∠R, is there more than one method to find m∠P?)**

**Answer:**Given

SP || RQ

PQ and SR are transversal.

Co-interior angles are supplementary.

∠P + ∠Q = 180°

∠S + ∠R = 180°

∠P + 130° = 180°

∠P = 50°

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