Question
ABCD is a quadrilateral such that diagonal AC bisects the angles $\angle\text{A}$ and $\angle\text{C}.$ Prove that AB = AD and CB = CD.
✓
Answer
In $\triangle\text{ABC}$ and $\triangle\text{ADC},$
$\angle\text{BAC}=\angle\text{DAC}$ $\big($AC bisects $\angle\text{A}\big)$
$\text{AC = AC}$ (common)
$\angle\text{BCA}=\angle\text{DCA}$ $\big($AC bisects $\angle\text{C}\big)$
$\therefore\triangle\text{ABC}\cong\triangle\text{ADC}$ (by ASA congruence criterion)
$\Rightarrow\text{AB = AD}$ and $\text{CB = CD}$ (C.P.C.T.)
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