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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 $$CB = CD (c. p. c. t)$$.

1539396_1177677_ans_303afc222d4e4244a8d1c5e5d1dce5f5.png

Mathematics
RS Agarwal
Standard IX

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