Question

In the given figure, $AB=AC$ and $\angle GBC=\angle HCB={90}^{\circ }$.
Which of the following statements is true?

• A. BG = CB and AG = AH
• B. AG = AB and BG = CH
• C. BG = CH and AG = AH
• D. $\angle BAC={120}^{\circ }$

Answer 1

The correct option is C BG = CH and AG = AH

(i) $\Delta BAC$ is isosceles ….. (given)

(ii) $\angle ABC=\angle ACB=x^{\circ }$ (say) (base angles of an isosceles triangle)

(iii) $\angle GBC=\angle HCB={90}^{\circ }$ .. (given)

(iv) $\therefore \angle GBA=\angle HCA=90-x^{\circ }$

In $\Delta GBA$ and $\Delta HCA$

(v) $\angle GBA=\angle HCA$ (from (iv))

(vi) $AB=AC$ …(given)

(vii) $\angle GAB=\angle HAC$ .….(vertically opposite angles are equal)

(viii) $\Delta GBA\cong \Delta HCA$ …… (ASA postulate)

$\Rightarrow BG=CHandAG=AH$ ...(CPCT)