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Quadrilaterals — MATHS STD 9 — Question

Rajasthan BoardEnglish MediumSTD 9MATHSQuadrilaterals3 Marks
Question
$ \text{ABCD}$ is a parallelogram and $AP$ and $CQ$ are perpendiculars from vertices $A$ and $C$ on diagonal $BD$ respectively.

Show that :
  1. $\triangle APB  \cong \triangle CQD$
  2. $AP = CQ.$
✓

Answer

Given: $ABCD$ is a parallelogram and $AP$ and $CQ$ are perpendicular from vertices $A$ and $C$ on diagonal $BD$ respectively.
To Prove :
  1. $\triangle APB  \cong \triangle CQD$
  2. $AP = CQ.$
Proof :
  1. In $\triangle APB$ and $\triangle CQD$
    $AB = CD . . . [$Opp. sides of $|| \ gm \ \text{ABCD}]$
    $\angle ABP =  \angle CDQ . . .[$Alternate interior angles for $AB \ ||\ CD]$
    $\therefore \angle APB =  \angle CQD . . .[$Each $90^\circ]$
    $\triangle APB  \cong \triangle CQD . . . [$By $\text{AAS}$ rule$]$
  2. As $\triangle APB  \cong \triangle CQD . . .[$As proved above$]$
    $\therefore AP = CQ . . .[c.p.c.t.]$

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