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Find the measure of c.

A. 120
B. 160
C. 180
D. 140

To find the measure of angle \( c \) in the given circle, we can use the property that the measure of an angle formed by two chords intersecting inside a circle is equal to half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

In the diagram:
- Angle \( e \) is given as \( 20^\circ \).
- Since \( O \) is the center of the circle and the chords intersect at the center, \( c \) is a central angle.

The angles \( a \), \( b \), \( c \), \( d \), and \( e \) form the full circle around point \( O \), thus their sum is \( 360^\circ \). 

The angles at the center are twice the angles at the circumference, so:
- \( \text{Angle } e = 2 \times 20^\circ = 40^\circ \)
- \( \text{Angle } d = c \) (since they are on opposite sides of the central angle \( c \) and they subtend the same arc of the circle)
- \( a + b + c = 180^\circ \) because they form a straight line with the diameter.

Given that \( e = 20^\circ \), the corresponding arc is \( 2 \times 20^\circ = 40^\circ \). Thus, \( c \) is twice the arc opposite it, which is \( 360^\circ - 40^\circ \). 

Now calculate the value of \( c \):
\[
c = \frac{360^\circ - 40^\circ}{2} = \frac{320^\circ}{2} = 160^\circ
\]

Thus, the measure of angle \( c \) is \( 160^\circ \). The correct answer is **B. 160**.

Would you like more details or have any questions?

Here are some additional questions you might find interesting:
1. How do you calculate the measure of an inscribed angle in a circle?
2. What is the relationship between a central angle and its corresponding arc?
3. How can you find the measure of an angle formed by two chords intersecting inside a circle?
4. What are the properties of angles subtended by the same arc in a circle?
5. How can the diameter of a circle influence the angles in the circle?

**Tip:** When working with circle theorems, remember that angles subtended by the same arc in a circle are equal, and the central angle is twice any angle on the circumference subtended by the same arc.

Discover how to find the measure of angle c in a circle using circle theorems. Step-by-step solution with detailed explanations and calculations. Suitable for high school students studying geometry.

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