Question

Quadrilateral ABCD is inscribed in a circle. find the measure of each of the angles of the quadrilateral. Show the steps.

Answer 1

Given:

To solve the angle, we use the theorem of angles in a cyclic quadrilateral.

The theorem states that the opposite angles in a cyclic quadrilateral are supplementary (i.e. they add up to 180°).

Hence,

`\begin{gathered} \angle A+\angle C=180^0 \\ \angle B+\angle D=180^0 \end{gathered}`

Thus,

`\begin{gathered} \angle A+\angle C=180^0 \\ (x+2)+(x-2)=180^0 \\ x+x+2-2=180 \\ 2x=180 \\ x=(180)/(2) \\ x=90^0 \end{gathered}`

Hence,

`\begin{gathered} \angle A=x+2=90+2 \\ \angle A=92^0 \\ \\ \angle C=x-2=90-2 \\ \angle C=88^0 \\ \\ \angle D=x-10=90-10 \\ \angle D=80^0 \\ \\ \text{Also, } \\ \angle B+\angle D=180^0 \\ \angle B+80=180 \\ \angle B=180-80 \\ \angle B=100^0 \end{gathered}`

Therefore, the measure of each angle of the quadrilateral is;

`\begin{gathered} \angle A=92^0 \\ \angle B=100^0 \\ \angle C=88^0 \\ \angle D=80^0 \end{gathered}`