In a cyclic quadrilateral ABCD ∠A = (2x + 4)°∠B = (y + 3)°∠C

1.	In a cyclic quadrilateral ABCD ∠A = (2x + 4)°∠B = (y + 3)°∠C = (2y + 10)°∠D = (4x - 5)°. Find the four angles.
Answer» In a cyclic quadrilateral sum of opposite angles is 180°. Given, in a cyclic quadrilateral ABCD , ∠A = (2x + 4)° ∠B = (y + 3)° ∠C = (2y + 10)° ∠D = (4x - 5)° ∴ ∠A + ∠C = 180° and ∠B + ∠D = 180° ⇒ 2x + 4 + 2y + 10 = 180 and y + 3 + 4x – 5 = 180 ⇒ x + y = 83 --------- (1) and y + 4x = 182 ---------- (2) Subtracting eq1 from eq2. ⇒ y + 4x – x – y = 182 – 83 ⇒ 3x = 99 ⇒ x = 33 Substituting in eq1. ⇒ y = 50 ∠A = 2 × 33 + 4 = 70° ∠B = 50 + 3 = 53° ∠C = 2 × 50 + 10 = 110° ∠D = 4 × 33 – 5 = 127°

In a cyclic quadrilateral ABCD ∠A = (2x + 4)°∠B = (y + 3)°∠C = (2y + 10)°∠D = (4x - 5)°. Find the four angles.

Answer»

In a cyclic quadrilateral sum of opposite angles is 180°.

Given, in a cyclic quadrilateral ABCD ,

∠A = (2x + 4)°

∠B = (y + 3)°

∠C = (2y + 10)°

∠D = (4x - 5)°

∴ ∠A + ∠C = 180° and ∠B + ∠D = 180°

⇒ 2x + 4 + 2y + 10 = 180 and y + 3 + 4x – 5 = 180

⇒ x + y = 83 --------- (1) and y + 4x = 182 ---------- (2)

Subtracting eq1 from eq2.

⇒ y + 4x – x – y = 182 – 83

⇒ 3x = 99

⇒ x = 33

Substituting in eq1.

⇒ y = 50

∠A = 2 × 33 + 4 = 70°

∠B = 50 + 3 = 53°

∠C = 2 × 50 + 10 = 110°

∠D = 4 × 33 – 5 = 127°