University:
High School
Course:
High School Mathematics
Academic year:

2023
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120

Pages:

3
Author:

spleodairrh

10.4 Inscribed Angles & Polygons In your own words, write the meaning of each vocabulary term: inscribed angle: An angle on the circle. intercepted arc: The arc made by the inscribed angle. subtend: Another word for "connected to" or "opposite to" circumscribed circle: A circle on the outside. Inscribed Polygon: A polygon with endpoints on the circle. Core Concepts Inscribed Angle and Intercepted Arc An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. An arc that lies between two lines, rays, or segments is called an intercepted arc. If the endpoints of a chord or arc lie on the sides of an inscribed angle, then the chord or arc is said to subtend the angle. 10.4 Theorems Theorem 10.10: Measure of an Inscribed Angle Theorem The measure of an inscribed angle is one-half the measure of its intercepted arc. m∠ADB = 1/2 mAB Theorem 10.11: Inscribed Angles of a Circle Theorem If two inscribed angles of a circle intercept the same arc, then the angles are congruent. m∠ADB = m∠ACB Theorem 10.12: Inscribed Right Triangle Theorem If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. m∠ADB = 90° if and only if AC is a diameter of the circle. Theorem 10.13: Inscribed Quadrilateral Theorem A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. D, E, F, and G lie on OC if and only if m∠D + m∠F = 180° and m∠E + m∠G = 10.4 Extra Practice In Exercises 1-5, use the diagram to find the indicated measure. m∠C = 62° Since △ABC is a right triangle, 180° = 90° + 28° + x° 180° = 118° + x° 62° = x° m∠C = 28° Since △ABC is a right triangle, 180° = 90° + 73° + x° 180° = 163° + x° 17° = x° BC = 8.2 Since △ABC is a right triangle, 7.3² + 3.9² = BC² 53.29 + 15.21 = BC² 68.5 = BC² 8.2765 ≈ BC m∠AC = 56° Since the rule is inscribed angle = 1/2 arc, 2 * 28° = Arc 56° = Arc m∠AB = 124° Since 1/2 AB = x° (from previous problem), 2 * 28° = AB 56° = AB Name two pairs of congruent angles. ∠BAC ≅ ∠BDC ∠ACD ≅ ∠ABD Find the value of each variable. Using the opposite angle rule, m + 65 = 180° m = 115° and n + 100 = 180° n = 80°