Question

calculator
quadrilateral ( a b c d ) is inscribed in a circle.
what is the measure of angle ( a ) ?
enter your answer in the box.
( m angle a=square^{circ} )

Explanation:

Step1: Use the property of inscribed quadrilaterals

In a cyclic quadrilateral, the sum of opposite angles is \(180^{\circ}\). Here, \(\angle A\) and \(\angle C\) are related by the inscribed - angle theorem. The measure of an inscribed angle is half the measure of its intercepted arc. But also, for a cyclic quadrilateral \(ABCD\) inscribed in a circle, \(\angle A+\angle C = 180^{\circ}\) (the sum of opposite angles of a cyclic quadrilateral is \(180^{\circ}\)). Given \(\angle A=(4x + 5)^{\circ}\) and \(\angle C=(x + 15)^{\circ}\).
So, \((4x + 5)+(x + 15)=180\).

Step2: Solve the equation for \(x\)

Combine like terms: \(4x+x+5 + 15=180\), which simplifies to \(5x+20 = 180\).
Subtract \(20\) from both sides: \(5x=180 - 20=160\).
Divide both sides by \(5\): \(x=\frac{160}{5}=32\).

Step3: Find the measure of \(\angle A\)

Substitute \(x = 32\) into the expression for \(\angle A\). \(\angle A=(4x + 5)^{\circ}\).
\(\angle A=4\times32+5\).
First, calculate \(4\times32 = 128\), then \(128+5=133\).

Answer:

\(133\)