Question

In the given figure, AB = AD = DC = PB and ∠DBC = x°. Determine, in terms of x :

1. ∠ABD,
2. ∠APB.

Hence or otherwise, prove that AP is parallel to DB.

Answer 1

Given – In the figure, AB = AD = DC = PB and DBC = X°

Join AC and BD

To find : The measure of ∠ABD and ∠APB

Proof : ∠DAC = ∠DBC = X  ...[Angels in the same segment]

But ∠DCA = ∠DAC = X    ...[∵ AD = DC]

Also, we have, ∠ABD = ∠DAC  ...[Angles in the same segment]

In ∆ABP, ext ∠ABC = ∠BAP + ∠APB

But, ∠BAP = ∠APB   ...[∵ AB = BP]

  2 × X = ∠APB + ∠APB = 2∠APB

∴ 2∠APB = 2X

`=>` ∠APB = X

∴ ∠APB = ∠DBC = X,

But these are corresponding angles

∴ AP || DB