Question
ABCD is a quadrilateral such that diagonal AC bisects the angles A and C. Prove that AB = AD and CB = CD.

Answer

Given in a quadrilateral ABCD, diagonal AC bisects the angles A and C.

To prove $\text{AB}=\text{CD}\ \text{ and }\text{CB}=\text{CD}$

Proof in $\triangle\ \text{ADC}\ \text{and}\ \triangle\ \text{ABC},$

$\angle\ \text{DAC}=\angle\ \text{BAC}$ $\big[\because$ AC is the bisector of $\angle\ \text{A}\ \text{and}\ \angle\ \text{C}\big]$

$\angle\ \text{DCA}=\angle\ \text{BCA}$ $\big[\because$ AC is the bisector of $\angle\ \text{A}\ \text{and}\ \angle\text{C}\big]$

 $\text{and}\ \text{AC}=\text{AC}$ [common side]

$\therefore\ \triangle\text{ADC}\cong\triangle\text{ABC}$ [byASA congruence rule]

$\text{AD}=\text{AB}$ [by CPCT]

$\text{and}\ \text{CD}=\text{CB}$ [by CPCT]

Hence proved.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free