The problem asks to identify a central angle and an inscribed angle from the given diagram. It also asks to find the measures of the variables a, b, c, and d based on the diagram. The diagram shows a circle with an angle labeled 108 degrees, and a chord with an angle labeled $2x+21$ degrees inside the circle. The variable c is equal to $2x+21$.

1. Problem Description

The problem asks to identify a central angle and an inscribed angle from the given diagram. It also asks to find the measures of the variables a, b, c, and d based on the diagram. The diagram shows a circle with an angle labeled 108 degrees, and a chord with an angle labeled $2x+21$ degrees inside the circle. The variable c is equal to $2x+21$.

2. Solution Steps

First, we identify a central angle and an inscribed angle. Since the image is incomplete, we are missing the point where the central angle and inscribed angle are located.

We are given that d = 108 degrees. Since the inscribed angle is half the measure of the central angle subtending the same arc, $c = 2x + 21$ should be half of d. Therefore, $2c = d$.

$2(2x+21) = 108$

$4x + 42 = 108$

$4x = 108 - 42$

$4x = 66$

$x = \frac{66}{4}$

$x = 16.5$

So, $c = 2x+21 = 2(16.5) + 21 = 33 + 21 = 54$

The value of a and b cannot be determined from the given image.

3. Final Answer

Name a central angle: d

Name an inscribed angle: c

a = cannot be determined

b = cannot be determined

c = 54

d = 108