Question

find ( mangle a ).
( mangle a=) ( circ )

Explanation:

Step1: Find the measure of arc \(BD\)

The sum of the measures of the arcs of a circle is \(360^{\circ}\). Let the measure of arc \(BD\) be \(x\). Then \(x + 136^{\circ}+88^{\circ}=360^{\circ}\). So \(x=360^{\circ}-(136^{\circ} + 88^{\circ})=136^{\circ}\).

Step2: Use the property of a cyclic quadrilateral

The sum of opposite angles of a cyclic quadrilateral is \(180^{\circ}\). In cyclic quadrilateral \(ABCD\), \(\angle A+\angle C = 180^{\circ}\). First, find \(\angle C\). The measure of an inscribed angle is half the measure of its intercepted arc. The arc intercepted by \(\angle C\) is arc \(BD\) (measure \(136^{\circ}\)), so \(m\angle C=\frac{1}{2}\times136^{\circ} = 68^{\circ}\).
Since \(\angle A+\angle C=180^{\circ}\), then \(m\angle A=180^{\circ}-m\angle C\). Substitute \(m\angle C = 68^{\circ}\), we get \(m\angle A=180^{\circ}- 68^{\circ}=112^{\circ}\)

Answer:

\(112\)