# Mathematics Questions and Answers – Angle Sum Property of a Quadrilateral

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Angle Sum Property of a Quadrilateral”.

1. Which of the following does not form a quadrilateral?
a) b) c) d) Explanation: Sum of the four angles of the quadrilateral is always 360°. So, to check whether a given figure is quadrilateral or not, we need to check the angle sum property of the given figure.
Since the sum of angles of the figure containing angles 100°, 60°, 80°, 70° is equal to 310°, it will no form a quadrilateral.

2. Find the value of x. a) 105°
b) 50°
c) 60°
d) 90°

∠A + ∠B + ∠C + ∠D = 360°  (Angle sum property of quadrilateral)
⇒ x + 100° + 80° + 75° = 360°
⇒ x = 360° – 255°
⇒ x = 105°.

3. Find the value of (x + z) if w : x : y : z = 6 : 1 : 3 : 2. a) 30°
b) 180°
c) 120°
d) 90°

Explanation: As w : x : y : z = 6 : 1 : 3 : 2, w = 6k, x = k, y = 3k and z = 2k  ——–(i)
From Figure, x + y + z + w = 360°  (Angle sum property of quadrilateral)
⇒ k + 3k + 2k + 6k = 360°  (from equation i)
⇒ 12k = 360°
⇒ k = 30°
Therefore, x + z = k + 2k = 3k = 90°.

4. Find the value of x and y. a) 30°, 50°
b) 50°, 30°
c) 80°, 40°
d) 40°, 80°

Explanation: From Figure, ∠PQT + ∠PQR = 180°  (Linear Pair)
⇒ 130° + 20° + x = 180°
⇒ x = 30° ———- (i)
From Figure, (x + y) + (80 + y) + (20 + x) + 100° = 360°  (Angle sum property of quadrilateral)
⇒ 200° + 2x + 2y = 360°
⇒ 60° + 2y = 160°
⇒ y = 50°.

5. What is the value of ∠BCD if BC || AD and ∠1 : ∠2 = 1 : 2? a) 30°
b) 80°
c) 95°
d) 67°

Explanation: Since BC || AD, ∠CBA + ∠DAB = 180°  (Interior angles on the same side of transversal)
⇒ 110° + 50° + ∠ABD = 180°
⇒ ∠ABD = 30°  ————- (i)
Also, ∠CBD = ∠BDA = 50°  —————– (ii) (Alternate interior angles)
Now, ∠1 : ∠2 = 1 : 2
⇒ ∠1 = ½ x 50°
⇒ ∠1 = 25°  ——————- (iii)
In quadrilateral ABCD, ∠A + ∠B + ∠C + ∠D = 360°  (Angle sum property of quadrilateral)
⇒ 110° + (50° + 30°) + (25° + 50°) + ∠C = 360°
⇒ ∠C = 95°.

6. What is the value of ∠POQ if OP and OQ are angle bisectors? a) 30°
b) 85°
c) 105°
d) 60°

Explanation: As it is given that OP and OQ are angle bisectors, ∠SPO = ∠OPQ and ∠RQO = ∠OQP
Also, In quadrilateral PQRS, ∠P + ∠Q + ∠R + ∠S = 360°  (Angle sum property of quadrilateral)
⇒ 2∠OPQ + 2∠OQP + (110° + 60°) = 360°  (Angle bisectors)
⇒ 2(∠OPQ + ∠OQP) = 190°
⇒ (∠OPQ + ∠OQP) = 95°  ————— (i)
In ΔPOQ, ∠OPQ + ∠OQP + ∠POQ = 180°  (Angle sum property of triangle)
⇒ 95° + ∠POQ = 180°  (From equation i)
⇒ ∠POQ = 105°.

7. Which among the following relation is correct? a) x1 + y1 > x2 + y2
b) x1 + y1 < x2 + y2
c) x1 + y1 = ½ (x2 + y2)
d) x1 + y1 = x2 + y2

Explanation: From Figure,
∠QRS = 180° – y1  —————- (i) (Linear Pair)
And ∠QPS = 180° – x1  —————- (ii) (Linear Pair)
In quadrilateral PQRS, ∠P + ∠Q + ∠R + ∠S = 360°  (Angle sum property of quadrilateral)
⇒ (180° – x1) + x2 + (180° – y1) + y2 = 360°
⇒ 360° – x1 + x2 – y1 + y2 = 360°
⇒ x2 + y2 = x1 + y1.

Sanfoundry Global Education & Learning Series – Mathematics – Class 9.

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