## 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 = ……

(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

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°?

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.

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.

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.
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.

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)

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)
CLUE parallelogram (opposite angles are equal)

Angle sum property of triangles
= 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.

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

Given
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?)