Question

In the given figure $\angle ABP=\angle ACQ,\angle BAC={68}^{\circ }$. Find the measure of $\angle ABC$.

• A. ${124}^{\circ }$
• B. ${56}^{\circ }$
• C. ${60}^{\circ }$
• D. ${52}^{\circ }$

Answer 1

The correct option is B ${56}^{\circ }$


$\text{Given:}\angle ABP=\angle ACQ$

$\angle ABP+\angle ABC={180}^{\circ }\text{(Linear pair) ... (i)}$
$\angle ACQ+\angle ACB={180}^{\circ }\text{(Linear pair) ... (ii)}$

From (i) and (ii),
$\angle ABP+\angle ABC=\angle ACQ+\angle ACB$
$\text{But,}\angle ABP=\angle ACQ$
$\text{So,}\angle ABC=\angle ACB$

$\text{In}\Delta ABC$,
$\angle ABC+\angle ACB+\angle BAC={180}^{\circ }$
(sum of interior angles of a triangle is 180${}^{\circ }$)
$\Rightarrow 2\angle ABC={180}^{\circ }-{68}^{\circ }(\because \angle ABC=\angle ACB)$
$\Rightarrow 2\angle ABC={112}^{\circ }$
$\Rightarrow \angle ABC={56}^{\circ }$