11.5 Inscribed Angles

Goal Use properties of inscribed angles.

Key Vocabulary Inscribed Angle Intercepted Arc Inscribed Circumscribed

Theorems 11.7 Inscribed Angle Theorem 11.8 Inscribed Triangle 11.9 Inscribed Quadrilateral

Inscribed Angle Inscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle). Examples: 3 1 2 4 No! Yes! No! Yes!

Intercepted Arc Intercepted Arc: An angle intercepts an arc if and only if each of the following conditions holds: 1. The endpoints of the arc lie on the angle. 2. All points of the arc, except the endpoints, are in the interior of the angle. 3. Each side of the angle contains an endpoint of the arc.

Theorem - Inscribed Angles Theorem 11.7 (Inscribed Angle Theorem): The measure of an inscribed angle equals ½ the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle). C A B mACB = ½m or 2 mACB = m

Inscribed Angle Theorem The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Y 110 Inscribed Angle 55 X Z Intercepted Arc

Example 1a A. Find mX. Answer: mX = 43

Example 1b B. = 2(252) or 104

Your Turn: A. Find mC. A. 47 B. 54 C. 94 D. 188

Your Turn: B. A. 47 B. 64 C. 94 D. 96

Example 2 In and Find the measures of the numbered angles.

Example 2 First determine Arc Addition Theorem Simplify. 20 40 108 Simplify. Subtract 168 from each side. Divide each side by 2.

Example 2 So, m 20 40 108

Example 2 20 40 108 Answer:

Your Turn: A. 30 B. 60 C. 15 D. 120

Your Turn: A. 110 B. 55 C. 125 D. 27.5

Your Turn: A. 30 B. 80 C. 40 D. 10

Your Turn: A. 110 B. 55 C. 125 D. 27.5

Your Turn: A. 110 B. 55 C. 125 D. 27.5

Comparing Measures of Inscribed Angles Find mACB, mADB, and mAEB. The measure of each angle is half the measure of the intercepted arc. All three angles intercept the same arc, arc AB, whose measure is 60˚. So the measure of each angle is 30°.

Congruent Inscribed Angles If two inscribed angles of a circle intercept the same arc or congruent arcs, then the inscribed angles are congruent.

Example 3 Find mR. R  S R and S both intercept . mR  mS Definition of congruent angles 12x – 13 = 9x + 2 Substitution x = 5 Simplify. Answer: So, mR = 12(5) – 13 or 47.

Your Turn: Find mI. A. 4 B. 25 C. 41 D. 49

Example 4: Find the value of x and y in the figures ° x 50˚ A B C E F y ° 40˚ x 50˚ A B C D E

Polygons Inscribed Polygon: A polygon inside the circle whose vertices lie on the circle. Circumscribed Polygon : A polygon whose sides are tangent to a circle.

Angles of Inscribed Polygons Theorem 11.8: An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. i.e. If AC is a diameter of ⊙O, then the mABC = 90° o

Example 5 Find mB. ΔABC is a right triangle because C inscribes a semicircle. mA + mB + mC = 180 Angle Sum Theorem (x + 4) + (8x – 4) + 90 = 180 Substitution 9x + 90 = 180 Simplify. 9x = 90 Subtract 90 from each side. x = 10 Divide each side by 9. Answer: So, mB = 8(10) – 4 or 76.

Your Turn: Find mD. A. 8 B. 16 C. 22 D. 28

Lesson 4 Ex4 Ex. 6:

Your Turn: A. 45 B. 90 C. 180 D. 80

Your Turn: A. 17 B. 76 C. 60 D. 42

Your Turn: A. 17 B. 76 C. 60 D. 42

Your Turn: A. 73 B. 30 C. 60 D. 48

Angles of Inscribed Polygons Theorem 11.9: If a quadrilateral is inscribed in a ⊙, then its opposite s are supplementary. i.e. Quadrilateral ABCD is inscribed in ⊙O, thus A and C are supplementary and B and D are supplementary. D A C B O

Theorem 11.9 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

Example 7: Quadrilateral QRST is inscribed in If and find and Draw a sketch of this situation. Answer:

Your Turn: Quadrilateral BCDE is inscribed in If and find and Answer:

Example 8 An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mS and mT.

Example 8 Since TSUV is inscribed in a circle, opposite angles are supplementary. S + V = 180 S + V = 180 S + 90 = 180 (14x) + (8x + 4) = 180 S = 90 22x + 4 = 180 22x = 176 x = 8 Answer: So, mS = 90 and mT = 8(8) + 4 or 68.

Your Turn: An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mN. A. 48 B. 36 C. 32 D. 28

Review Problems

Find the measure of the inscribed angle or the intercepted arc. Example 1 Find Measures of Inscribed Angles and Arcs Find the measure of the inscribed angle or the intercepted arc. a. b. SOLUTION mNMP = mNP 2 1 The measure of an inscribed angle is half the measure of its intercepted arc. a. Substitute 100° for . mNP (100°) 2 1 = = 50° Simplify. 44

Example 1 b. b. 1 mZYX = mZWX 2 1 105° = mZWX 2 210° = mZWX Find Measures of Inscribed Angles and Arcs b. b. mZYX = mZWX 2 1 The measure of an inscribed angle is half the measure of its intercepted arc. 105° = mZWX 2 1 Substitute 105° for mZYX. 210° = mZWX Multiply each side by 2. 45

Your Turn: Find the measure of the inscribed angle or the intercepted arc. 1. ANSWER mBAC = 45° 2. ANSWER mDEF = 80° 3. ANSWER mKNP = 240°

Find the values of x and y. Example 2 Find Angle Measures Find the values of x and y. SOLUTION Because ∆ABC is inscribed in a circle and AB is a diameter, it follows from Theorem 11.8 that ∆ABC is a right triangle with hypotenuse AB. Therefore, x = 90. Because A and B are acute angles of a right triangle, y = 90 – 50 = 40. 47

Your Turn: Find the values of x and y in C. 4. ANSWER x = 90; y = 55 5. ANSWER x = 45; y = 90 6. ANSWER x = 30; y = 90

Find the values of y and z. Example 3 Find Angle Measures Find the values of y and z. SOLUTION Because RSTU is inscribed in a circle, by Theorem 11.9 opposite angles must be supplementary. S and U are opposite angles. R and T are opposite angles. mS + mU = 180° mR + mT = 180° 120° + y° = 180° z° + 80° = 180° y = 60 z = 100 49

Your Turn: Find the values of x and y in C. 7. ANSWER x = 85; y = 80 8. ANSWER x = 90; y = 90 9. ANSWER x = 130; y = 100

66˚

54˚

43˚

90˚

54˚

50˚

54˚

36˚

50˚

72˚

180˚

In the figure, find the value of x.

Find the value of x. X = 21.5

Assignment Pg. 617 – 619; #1 – 29 odd, 33 – 49 odd