Answer:
m∠A = 91°
m∠B = 146°
m∠C = 89°
m∠D = 34°
Step-by-step explanation:
- If the four vertices of a quadrilateral lie on the edge of a circle, then this quadrilateral is called cyclic quadrilateral
- In the cyclic quadrilateral each two opposite angles are supplementary (means the sum of their measures is 180°)
- The sum of the measures of the interior angles of any quadrilateral is 360°
In quadrilateral ABCD
∵ A, B, C, And D lie on the circumference of the circle
∴ ABCD is a cyclic quadrilateral
∴ The sum of the measures of each opposite angles is 180°
∵ ∠A and ∠C are opposite angle in the cyclic quadrilateral ABCD
∴ m∠A + m∠C = 180°
∵ m∠A = (2x + 3)°
∵ m∠C = (2x + 1)°
- Add them and equate the answer by 180
∴ (2x + 3) + (2x + 1) = 180
- Add the like terms in the left hand side
∴ 4x + 4 = 180
- Subtract 4 from both sides
∴ 4x = 176
- Divide both sides by 4
∴ x = 44
Substitute the value of x in the expressions of angle A, C, D
∵ m∠A = 2(44) + 3 = 88 + 3
∴ m∠A = 91°
∵ m∠C = 2(44) + 1 = 88 + 1
∴ m∠C = 89°
∵ m∠D = x - 10
∴ m∠D = 44 - 10
∴ m∠D = 34°
- ∠B and ∠D are opposite angles in the cyclic quadrilateral ABCD
∴ m∠B + m∠D = 180°
∴ m∠B + 34 = 180
- Subtract 34 from both sides
∴ m∠B = 146°
