College Math

3.5 Theorems of Circle Theorem 1: If a central angle and an inscribed angle share the same endpoints, the central angle will measure twice the inscribed angle.

∠ POQ = 2* ∠ PRQ Theorem 2: All angles that have their vertex on the circumference and share the same endpoints have equal measures. That is, ∠ PRQ = ∠ PSQ = ∠ PTQ Appl cation: The application of these two theorems can be seen in the under given example. Find the measures of angles x and y in the given diagram. In the above circle, ∠PRQ = 54 0 , using the second theorem, ∠PSQ will be equal to ∠PRQ as both angles are subtended by the same arc PQ on the circumference of the circle. Moreover, the ∠POQ will be double of ∠PRQ as per the first theorem, that the angle subtended by same arc will be double of the angle subtended on the circumference of the circle. Thus, = 54° and = 2 ∗ 54 = 108°

Angles subtended by the diameter, that is, angles in a semicircle, are right angles.

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Theorem 3: