Question

a quadrilateral is inscribed in circle o. the angle measures of the quadrilateral, in degrees, are given by the expressions shown in the figure. what is the measure of angle c? a. 15° b. 105° c. 75° d. 85°

Explanation:

Step1: Recall property of cyclic quadrilateral

In a cyclic quadrilateral, the sum of opposite angles is 180 degrees. So, $\angle A+\angle C = 180^{\circ}$ and $\angle B+\angle D=180^{\circ}$. Here, $\angle A=(130 - x)^{\circ}$ and $\angle C=(95 - 2x)^{\circ}$. Then $(130 - x)+(95 - 2x)=180$.

Step2: Simplify the equation

Combine like - terms: $130+95 - x-2x = 180$. So, $225-3x = 180$.

Step3: Solve for x

Subtract 225 from both sides: $-3x=180 - 225=-45$. Then divide both sides by - 3: $x = 15$.

Step4: Find the measure of angle C

Substitute $x = 15$ into the expression for $\angle C$. $\angle C=95-2x$. So, $\angle C=95-2\times15=95 - 30 = 65^{\circ}$.

Answer:

D. $65^{\circ}$