Question

calculator
quadrilateral abcd is inscribed in this circle.
what is the measure of angle c?
enter your answer in the box.

Explanation:

Step1: Use the property of cyclic quadrilaterals

In a cyclic quadrilateral, the sum of opposite angles is \(180^{\circ}\). So, \(\angle A+\angle C = 180^{\circ}\) and \(\angle B+\angle D=180^{\circ}\). We use \(\angle B+\angle D = 180^{\circ}\) to find \(x\).

$$3x+(x + 20)=180$$

Step2: Solve the equation for \(x\)

Combine like - terms: \(3x+x+20 = 180\), which simplifies to \(4x+20=180\).
Subtract 20 from both sides: \(4x=180 - 20\), so \(4x=160\).
Divide both sides by 4: \(x=\frac{160}{4}=40\).

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

We know that \(\angle A=(2x + 38)^{\circ}\) and \(\angle A+\angle C = 180^{\circ}\). First, find \(\angle A\): \(\angle A=2\times40 + 38=80+38 = 118^{\circ}\).
Then, \(\angle C=180-\angle A\). So \(\angle C=180-(2x + 38)\). Substituting \(x = 40\), \(\angle C=180-(2\times40 + 38)=180-(80 + 38)=180 - 118=62^{\circ}\).

Answer:

\(62^{\circ}\)