If ∠PAC : ∠PCA = 4 : 5, ∠P is :

1. 40°

2. 60°

3. 105°

4. 45°

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

$\Rightarrow \dfrac{∠\text{PAC}}{∠\text{PCA}} = \dfrac{4}{5} \\[1em] \Rightarrow \dfrac{60°}{∠\text{PCA}} = \dfrac{4}{5} \\[1em] \Rightarrow ∠\text{PCA} = \dfrac{60° \times 5}{4} \\[1em] \Rightarrow ∠\text{PCA} = \dfrac{300°}{4} = 75°.$

In △PCA,

By angle sum property of triangle,

⇒ ∠APC + ∠PAC + ∠PCA = 180°

⇒ ∠APC + 60° + 75° = 180°

⇒ ∠APC + 135° = 180°

⇒ ∠APC = 180° - 135° = 45°.

Hence, Option 4 is the correct option.