To determine the measure of angle $C$ in this problem, let's analyze it step by step.

Given Information:

1. The quadrilateral $ABCD$ is inscribed in a circle (cyclic quadrilateral).

2. In a cyclic quadrilateral, the opposite angles are supplementary, meaning: $\text{Angle A} + \text{Angle C} = 180^\circ \quad \text{and} \quad \text{Angle B} + \text{Angle D} = 180^\circ.$

3. The angles are expressed as follows:

   • $\angle A = 130 - x$,
   • $\angle B = 4x + 35$,
   • $\angle C = 95 - 2x$,
   • $\angle D = 7x - 20$.

We are tasked to find the measure of angle $C$.

Step 1: Use the cyclic quadrilateral property

From the property of a cyclic quadrilateral: $\angle A + \angle C = 180^\circ.$ Substitute the given expressions for $\angle A$ and $\angle C$: $(130 - x) + (95 - 2x) = 180.$

Step 2: Solve for $x$

Simplify the equation: $130 - x + 95 - 2x = 180.$ Combine like terms: $225 - 3x = 180.$ Subtract 225 from both sides: $-3x = -45.$ Divide by $-3$: $x = 15.$

Step 3: Find the measure of $\angle C$

The expression for $\angle C$ is: $\angle C = 95 - 2x.$ Substitute $x = 15$: $\angle C = 95 - 2(15).$ Simplify: $\angle C = 95 - 30 = 65^\circ.$

Final Answer:

The measure of angle $C$ is 65°.

Correct option: C. 65°

Do you want a detailed explanation for any step? Let me know!

Related Questions:

1. What are the measures of the other angles $A$, $B$, and $D$?
2. Why are opposite angles in a cyclic quadrilateral supplementary?
3. How would the solution change if the quadrilateral were not cyclic?
4. Can we verify the solution by solving for all angles using $x = 15$?
5. What is the significance of cyclic quadrilaterals in geometry?

Tip: Always remember that in a cyclic quadrilateral, opposite angles sum up to $180^\circ$. This property is essential for solving such problems!