Question

# $$ABCD$$ is a quadrilateral such that diagonal $$AC$$ bisects the angles $$\angle A$$ and $$\angle C$$ prove that $$AB = AD$$ and $$CB = CD$$.

Solution

## By considering $$\triangle ABC$$ and $$\triangle ADC$$We know that $$AC$$ bisects at $$\angle A$$So we get$$\angle BAC = \angle DAC$$We know that $$AC$$ is common i.e. $$AC = AC$$From the figure, we know that $$AC$$ bisects at $$\angle C$$$$\angle BCA = \angle DCA$$By $$ASA$$ congruence criterion we get$$\triangle ABC \cong \triangle ADC$$Therefore, it is proved that $$AB = AD$$ and $$CB = CD (c. p. c. t)$$.MathematicsRS AgarwalStandard IX

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