Question

The sides AB and BC of a square ABCD are produced to P and Q respectively so that BP= CQ. Prove that PD and AQ are equal and perpendicular to each other.

Answer 1

ABCD is a square and AP = DQ

we will consider ΔDAQ = ΔABP = 90°

DQ = AP this is given

AD = AB ( this is because the sides of the square are equal)

We will use the RHS congruency rule here

ΔDAQ

≅

ΔABP

∠1 = ∠3

∠1 + ∠4 = 90°

∠3 + ∠4 = 90°

In ΔAOD,

By ASP,

∠3 + ∠4 + ∠AOΔ = 180°

90° = ∠AOΔ = 180°

∠AOΔ = 90°

So AP and DQ are perpendicular.