11.2.1 Inscribed Angles (continued)
An unknown in the expression for the measure of an inscribed angle is determined in this example. The expression for the angle measure is given. It is given that AB is a diameter. Because AB is a diameter, ∠ ACB is inscribed in a semicircle and m ∠ ACB is 90 ° . Equate m ∠ ACB with the given expression and solve for w . The solution yields w = 6.
The measure of an inscribed angle is determined in this example. The expression for the angle measure is given. The expression for the measure of an inscribed angle that subtends the same arc is also given. According to the Inscribed Angle Theorem, m ∠ JKM = m ∠ JLM . Substitute the two expressions for the measures of the angle into the equation and solve for x . The solution yields x = 7. Substitute this value back into the expression for m ∠ JKM and solve. The solution yields m ∠ JKM = 31 ° .
Example 4 Finding Measures in Inscribed Quadrilaterals
If a quadrilateral is inscribed in a circle, then both pairs of opposite angles are supplementary. Look at the two intercepted arcs of opposite angles to understand why this theorem is true. How are the two arcs related?
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