Question

If angle PAC : angle PCA = 5: 4 ; angle P is ______.

Options

• 40°

• 60°

• 105°

• 45°

Answer 1

If angle PAC : angle PCA = 5: 4 ; angle P is 45°. 

Explanation:

We know that,

Sum of co-interior angles in a trapezium is 180°.

In trapezium ABDC,

⇒ ∠B + ∠BAC = 180°

⇒ 60° + ∠BAC = 180°

⇒ ∠BAC = 180° − 60° = 120°

From figure,

⇒ ∠PAC + ∠BAC = 180°

⇒ ∠PAC + 120° = 180°

⇒ ∠PAC = 180° − 120° = 60°

Given,

⇒ ∠PAC : ∠PCA = 4 : 5

⇒ `(∠PAC​)/(∠PCA) = 4/5`

⇒ `(60°)/(∠PCA) ​= 4/5​`

⇒ `∠PCA = (60°×5)/4​`

⇒ ∠PCA = `(300°)/4`​ = 75°

In △PCA,

By angle sum property of triangle,

⇒ ∠APC + ∠PAC + ∠PCA = 180°

⇒ ∠APC + 60° + 75° = 180°

⇒ ∠APC + 135° = 180°

⇒ ∠APC = 180° − 135° = 45°