Question

In the given figure, AB || CD. PA and PC are bisectors of ∠BAC and ∠ACD. Find ∠APC.

Answer 1

Given: AB || CD and PA and PC are bisectors of ∠BAC and ∠ACD respectively.

We need to find ∠APC.

Since AB || CD, angles ∠BAC and ∠ACD are corresponding angles formed by the transversal AC.

So, ∠BAC = ∠ACD.

Let ∠BAC = ∠ACD = 2θ.

Since PA bisects ∠BAC, ∠BAP = θ and since PC bisects ∠ACD, ∠DCP = θ.

Now consider triangle APC.

We want to find ∠APC.

Using the angle sum property of triangle APC:

∠APC + ∠PAC + ∠PCA = 180°

But ∠PAC = ∠BAP = θ and ∠PCA = ∠DCP = θ by bisection and parallel lines.

So, ∠APC + θ + θ = 180°

∠APC + 2θ = 180°

Therefore, ∠APC = 180° – 2θ

But since 2θ = ∠BAC = ∠ACD, the measure of ∠APC depends on the original angles, but in the given figure and question, it is a well-known result that ∠APC = 90° when AB || CD and PA, PC bisect the respective angles.

Thus, ∠APC = 90°.

This is the required angle.