9-4 Inscribed Angles Find each measure. 3. 1. SOLUTION: SOLUTION: If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its Here, is a semi-circle. So, intercepted arc. Therefore, If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its ANSWER: intercepted arc. So, 30 Therefore, 2. ANSWER: 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If , what is ? SOLUTION: If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc. Therefore, ANSWER: 126 SOLUTION: If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc. Therefore, ANSWER: 72 eSolutions Manual - Powered by Cognero Page 1 9-4 Inscribed Angles ALGEBRA Find each measure. 5. SOLUTION: Given: Prove: SOLUTION: If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. So, Therefore, ANSWER: Proof: 54 Statements (Reasons) 1. (Given) 2. (If minor arcs are , their 6. corresponding chords are .) 3. intercepts . intercepts . intercepts . intercepts . (Def. of intercepted arc) 4. (Inscribed of same arc are .) 5. (ASA) SOLUTION: ANSWER: If two inscribed angles of a circle intercept the same Given: arc or congruent arcs, then the angles are congruent. Prove: So, Therefore, ANSWER: 36 Proof: 7. PROOF Write a two-column proof. Statements (Reasons) Given: 1. (Given) Prove: 2. (If minor arcs are , their corresponding chords are .) 3. intercepts . intercepts . intercepts . intercepts . (Def. of eSolutions Manual - Powered by Cognero Page 2 9-4 Inscribed Angles intercepted arc) 9. x 4. (Inscribed of same arc are .) 5. (ASA) STRUCTURE Find each value. 8. SOLUTION: An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. So, Since The sum of the measures of the angles of a triangle is SOLUTION: 180. So, An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. So, The sum of the measures of the angles of a triangle is 180. So, ANSWER: 25 10. and Therefore, ANSWER: 62 SOLUTION: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Solve the two equations to find the values of x and y. Use the values of the variables to find and . ANSWER: 122; 80 eSolutions Manual - Powered by Cognero Page 3 9-4 Inscribed Angles Find each measure. 13. 11. SOLUTION: SOLUTION: The sum of the measures of the central angles of a If an angle is inscribed in a circle, then the measure circle with no interior points in common is 360. So, of the angle equals one half the measure of its intercepted arc. Therefore, ANSWER: If an angle is inscribed in a circle, then the measure 162 of the angle equals one half the measure of its intercepted arc. Therefore, 12. ANSWER: 70 14. SOLUTION: If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc. Therefore, SOLUTION: If an angle is inscribed in a circle, then the measure ANSWER: of the angle equals one half the measure of its 46 intercepted arc. Therefore, ANSWER: 48 eSolutions Manual - Powered by Cognero Page 4 9-4 Inscribed Angles 15. ALGEBRA Find each measure. 17. SOLUTION: Here, SOLUTION: The arc is a semicircle. So, If two inscribed angles of a circle intercept the same If an angle is inscribed in a circle, then the measure arc or congruent arcs, then the angles are congruent. of the angle equals one half the measure of its So, intercepted arc. So, ANSWER: Therefore, 32 ANSWER: 140 18. 16. SOLUTION: If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. SOLUTION: So, Here, The arc is a semicircle. So, Therefore, Then, If an angle is inscribed in a circle, then the measure ANSWER: of the angle equals one half the measure of its 34 intercepted arc. Therefore, ANSWER: 66 eSolutions Manual - Powered by Cognero Page 5 9-4 Inscribed Angles 19. PROOF Write the specified type of proof. 21. paragraph proof Given: Prove: SOLUTION: If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. ∠ ∠ ∠ Since A and B both intercept arc CD , m A = SOLUTION: m∠B. Proof: Given means that Therefore, m∠A = 5(4) or 20. Since and ANSWER: , the equation becomes 20 . Multiplying each side of the 20. equation by 2 results in . ANSWER: Proof: Given means that . Since and SOLUTION: , the equation becomes If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. Multiplying each side of the Since ∠C and ∠D both intercept arc AB, m∠C = m∠D. equation by 2 results in . Therefore, m∠C = 5(10) – 3 or 47. ANSWER: 47 eSolutions Manual - Powered by Cognero Page 6 9-4 Inscribed Angles 22. two-column proof ALGEBRA Find each value. Given: Prove: 23. x SOLUTION: Statements (Reasons): SOLUTION: 1. (Given) An inscribed angle of a triangle intercepts a diameter 2. (Inscribed intercepting same arc or semicircle if and only if the angle is a right angle. are .) So, 3. (Vertical are .) The sum of the measures of the angles of a triangle is 4. (AA Similarity) 180. So, ANSWER: Statements (Reasons): 1. (Given) 2. (Inscribed intercepting same arc ANSWER: are .) 30 3. (Vertical are .) 4. (AA Similarity) 24. SOLUTION: An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. So, The sum of the measures of the angles of a triangle is 180. So, Therefore, ANSWER: 60 eSolutions Manual - Powered by Cognero Page 7 9-4 Inscribed Angles 25. x STRUCTURE Find each measure. 27. SOLUTION: SOLUTION: An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. If a quadrilateral is inscribed in a circle, then its So, opposite angles are supplementary. The sum of the measures of the angles of a triangle is 180. So, Therefore, ANSWER: 135 ANSWER: 28. 12.75 26. SOLUTION: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. SOLUTION: An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. Therefore, So, The sum of the measures of the angles of a triangle is ANSWER: 180. So, 80 Therefore, ANSWER: 51.75 eSolutions Manual - Powered by Cognero Page 8 9-4 Inscribed Angles 29. Proof: By arc addition and the definitions of arc SOLUTION: measure and the sum of central angles, If a quadrilateral is inscribed in a circle, then its . Since by Theorem 10.6 opposite angles are supplementary. and , but , so or Therefore, 180. This makes ∠C and ∠A supplementary. ANSWER: Because the sum of the measures of the interior 106 angles of a quadrilateral is 360, . But 30. , so , making them supplementary also. ANSWER: Given: Quadrilateral ABCD is inscribed in . Prove: and are supplementary. and are supplementary. SOLUTION: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Proof: By arc addition and the definitions of arc Therefore, measure and the sum of central angles, ANSWER: . Since by Theorem 10.6 93 and 31. PROOF Write a paragraph proof for Theorem 9.9. , but SOLUTION: , so Given: Quadrilateral ABCD is inscribed in . Prove: and are supplementary. or 180. This makes ∠C and ∠A supplementary. and are supplementary. Because the sum of the measures of the interior angles of a quadrilateral is 360, eSolutions Manual - Powered by Cognero Page 9 9-4 Inscribed Angles . But 33. , so , making them supplementary also. SOLUTION: Since all the sides of the stop sign are congruent chords of the circle, all the corresponding arcs are SIGNS: A stop sign in the shape of a regular congruent. There are eight adjacent arcs that make octagon is inscribed in a circle. Find each up the circle, so their sum is 360. Thus, the measure measure. of each arc joining consecutive vertices is of 360 or 45. Since ∠RLQ is inscribed in the circle, its measure equals one half the measure of its intercepted arc QR. 32. Therefore, the measure of ∠RLQ is 22.5. SOLUTION: Since all the sides of the stop sign are congruent chords of the circle, the corresponding arcs are ANSWER: congruent. There are eight adjacent arcs that make 22.5 up the circle, so their sum is 360. Thus, the measure 34. of each arc joining consecutive vertices is of 360 or 45. SOLUTION: If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its Therefore, measure of arc NPQ is 135. intercepted arc. So, ANSWER: The sum of the measures of the central angles of a 135 circle with no interior points in common is 360. Here, 360 is divided into 8 equal arcs and each arc measures Then, Therefore, ANSWER: 112.5 eSolutions Manual - Powered by Cognero Page 10 9-4 Inscribed Angles 35. SOLUTION: If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc. So, SOLUTION: The sum of the measures of the central angles of a a.