Quadrilaterals — Practice

Question 1

A quadrilateral has three angles of 80°, 95°, and 110°. Find the fourth angle.

Accepted Answer

✦ Answer: Sum = 360°. Fourth angle = 360° − (80 + 95 + 110) = 360° − 285° = 75°.

Question 2

What is the relationship between adjacent angles of a parallelogram?

Accepted Answer

✦ Answer: Adjacent (consecutive) angles of a parallelogram are SUPPLEMENTARY (their sum = 180°). So if one angle is 65°, the adjacent angle is 115°.

Question 3

Find the diagonal of a rectangle with length 12 cm and breadth 5 cm.

Accepted Answer

Step 1 — Use Pythagoras (rectangle has 90° angles).
  d² = L² + B² = 12² + 5² = 144 + 25 = 169

Step 2 — Take square root.
  d = √169 = 13 cm

Step 3 — Verify.
  Use the Pythagorean triple (5, 12, 13) — common in problems.

✦ Answer: The diagonal is 13 cm.

Question 4

A rhombus has diagonals 24 cm and 10 cm. Find: (a) its side length, (b) its area.

Accepted Answer

Step 1 — Half-diagonals.
  Diagonals of rhombus bisect at 90°.
  Half-diagonals: 12 cm and 5 cm.

Step 2 — Side using Pythagoras.
  s² = 12² + 5² = 144 + 25 = 169
  s = 13 cm

Step 3 — Area formula.
  Area = (1/2) × d₁ × d₂
        = (1/2) × 24 × 10
        = 120 cm²

✦ Answer: (a) Side = 13 cm. (b) Area = 120 cm².

Question 5

In parallelogram ABCD, ∠A = (2x + 25)° and ∠B = (3x − 5)°. Find x and all four angles.

Accepted Answer

Step 1 — Use 'consecutive angles supplementary' (parallelogram property).
  ∠A + ∠B = 180°
  (2x + 25) + (3x − 5) = 180
  5x + 20 = 180
  5x = 160
  x = 32

Step 2 — Find all four angles.
  ∠A = 2(32) + 25 = 89°
  ∠B = 3(32) − 5 = 91°
  ∠C = ∠A = 89° (opposite angles equal in parallelogram)
  ∠D = ∠B = 91° (opposite angles equal)

Step 3 — Verify.
  Sum = 89 + 91 + 89 + 91 = 360° ✓
  Adjacent pairs: 89 + 91 = 180° ✓
  Opposite pairs: equal ✓

✦ Answer: x = 32. Angles: ∠A = ∠C = 89°; ∠B = ∠D = 91°.

Practice — Quadrilaterals

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