Question

Abcd is a square a is joined to a point p on bc and d is joined to a point q on ab.If ap=dq.Prove that ap and dq are perpendicular to each other

Answer 1

Proving the congruency of triangles PAB &DAQ

PA=DQ

angle B=angle A( since they both are 90°)

AB=AD( since both of them are sides of the square)

therefore the triangles are congruent by RHS axiom.

angle DAP+angle PAB=90°

and angle PAB = angle QDA (by CPCT)

therefore angle DAP+ angle QDA=90°

therefore angle DMA=90° ( considering triangle DMA and applying the angle sum property of triangles).