Question

In the given figure, $AB=AC=CD$ and $\angle ADC={38}^{\circ }$.

Calculate :

(i) $\angle ABC$

(ii) $\angle BEC$

Answer 1

Given: $AB=AC=CD$ and $\angle ADC=$ ${38}^{\circ }$.

(i) $AC=CD$

$\angle CAD=\angle CDA={38}^{\circ }$ [Angles opposite to equal sides]

$\angle ACD={180}^{\circ }-{38}^{\circ }-{38}^{\circ }={104}^{\circ }$ [Sum of all angles in a triangle is ${180}^{\circ }$]

$\angle ACB={180}^{\circ }-{104}^{\circ }={76}^{\circ }$ [Linear pair]

$AB=AC$

$\angle ABC=\angle ACB={76}^{\circ }$ [Angles opposite to equal sides]

(ii) $\angle BAC={180}^{\circ }-{76}^{\circ }-{76}^{\circ }={28}^{\circ }$

$\angle BAC=\angle BEC={28}^{\circ }$ (angles in the same segment are equal, subtended by BC as a common chord)