Question

ABCD is a square. A is joined to point P on BC and D is joined to a point Q on AB. If AP=DQ, prove that AP⊥DQ
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Answer 1

In ΔADQ and ΔAPB

AD = AB ( Sides of square are equal)

∠A = ∠B ( Each angle of square is right angle)

DQ = AP (Given)

So, By SAS congruency criterion, ΔADQ ≅ ΔAPB

⇒ ∠1 = ∠2 (Corresponding parts of the congruent triangles are equal)

Now, using angles sum property of triangle in ΔADM

∠1 + ∠3 + ∠4 = 180

⇒ ∠2 + ∠3 + ∠4 = 180 ( Because ∠1 = ∠2 )

But, ∠2 + ∠3 = 90

⇒ 90 + ∠4 = 180

⇒ ∠4 = 90°

Therefore, AP ⊥ DQ