Question

Question 68

Fill in the blanks to make the statement true.

In the given figure, AB = AD and $\angle BAC=\angle DAC$. Then,
i) $\Delta$ ___ $\cong \Delta ABC$
ii) BC = ___
iii) $\angle BCA=$ ___
iv) Line segment AC bisects ___ and ___

Answer 1

In $\Delta ABC$ and $\Delta ADC$,
AB = AD [given]
AC = AC [common side]
$\angle BAC=\angle DAC$ [given]
By SAS congruence criterion,
i) $\Delta ADC\cong \Delta ABC$
ii) BC = DC [By CPCT]
iii) Also, $\angle BCA=\angle DCA$ [by CPCT]
iv) Line segment AC bisects $\angle BAD$ and $\angle BCD$
Since, $\angle BAC=\angle DAC$
and $\angle BCA=\angle DCA$[ Since two triangles are congruent by SAS congruence criterion ; therefore their correspondence angles are equal.]