# 9.10.2. The Relation between Inscribed and Central Angles Subtended by the Same Arc

Given the figure below, find .

• A
• B
• C

### Example

Given the figure below, find .

### Solution

Since is parallel to , it must be true that and are supplementary angles. Suppose the measure of is . The measure of would then be . The measure of an inscribed angle is half the measure of the arc that subtends it. This means that the measure of is half the measure of arc . Use this fact to solve for the measure of in terms of as follows:

Also, the measure of is half the measure of . Now use this fact to solve for the measure of in terms of .

Since and , the sum of the measures of arcs and is . Also, the measure of a central angle is equal to the measure of the arc that subtends it, so the measure of arc arc is . Next, use this fact to find the measure of arc .

The measure of an inscribed angle is half the measure of the arc that subtends it. This means that the measure of is half the measure of arc . Finally, use this fact to solve for the measure of .

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