Question

In fig. $\angle ABP=\angle ACQ,\angle BAC={68}^{\circ }$. Measure of $\angle ABC$ is:

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

Answer 1

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

Given: $\angle ABP=\angle ACQ$
By linear pair property,
$\angle ABP+\angle ABC={180}^{\circ }$ $\cdot \cdot \cdot$ (i)
$\angle ACQ+\angle ACB={180}^{\circ }$ $\cdot \cdot \cdot$ (ii)

From (i) and (ii),
$\angle ABP+\angle ABC=\angle ACQ+\angle ACB$
[$\because$ $\angle ABP=\angle ACQ$ ]
So, $\angle ABC=\angle ACB$

In $\Delta ABC$,
$\angle ABC+\angle ACB+\angle BAC={180}^{\circ }$
$\Rightarrow 2\angle ABC={180}^{\circ }-{68}^{\circ }$
$[\because \angle ABC=\angle ACB]$
$\Rightarrow 2\angle ABC={112}^{\circ }$
$\Rightarrow \angle ABC={56}^{\circ }$