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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