Question

In the figure given above, $ABCD$ is a cyclic quadrilateral in which $\angle BAD={75}^{\circ },\angle ABD={58}^{\circ }$ and $\angle ADC={77}^{\circ }.$ Find the measure of $\angle CBD.$

Answer 1

Given $ABCD$ is a cyclic quadrilateral and opposite angles of a cyclic quadrilateral are supplementary.

Let us calculate the measure of $\angle ABC,$

$\begin{array}{cc}\angle ADC+\angle ABC& ={180}^{\circ }\\ \angle ABC& ={180}^{\circ }-{77}^{\circ }\\ & ={103}^{\circ }\end{array}$

By using $\angle ABC,$

$\begin{array}{cc}\angle ABC& =\angle CBD+\angle ABD\\ \angle CBD& =\angle ABC-\angle ABD\\ & ={103}^{\circ }-{58}^{\circ }\\ & ={45}^{\circ }\end{array}$

Hence, $\angle CBD={45}^{\circ }.$