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Circles Part 2 Geometry Central Angle Definition: An angle whose vertex lies on the center of the circle. Central Angle (of a circle) Central Angle (of a circle) Lesson 8-5: Angle Formulas NOT A Central Angle (of a circle) 2 Central Angle Theorem The measure of a center angle is equal to the measure of the intercepted arc. Y Intercepted Arc Center Angle Example: Given AD is the diameter, find the value of x and y and z in the figure. O 110 B 25 A C x y O 55 z D Z x 25 y 180 (25 55 ) 180 80 100 z 55 Lesson 8-5: Angle Formulas 3 Example: Find the measure of each arc. 4x + 3x + (3x +10) + 2x + (2x-14) = 360° 14x – 4 = 360° B 14x = 364° x = 26° 2x-14 C 4x E 4x = 4(26) = 104° 2x 3x 3x = 3(26) = 78° 3x+10 A D 3x +10 = 3(26) +10= 88° 2x = 2(26) = 52° 2x – 14 = 2(26) – 14 = 38° Lesson 8-5: Angle Formulas 4 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. C B ADC is the intercepted arc of ABC O A Lesson 8-5: Angle Formulas D 5 Intercepted Arc • Intercepted arcs for 1 K M – KM and KL 1 L 2 • Intercepted arcs for 2 O N – NO and NPO P 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). ABC is an inscribed angle. No! B O Examples: 1 C A D 3 2 Yes! No! Lesson 8-5: Angle Formulas 4 Yes! 7 Using Inscribed Angles • An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arcinscribed angle that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle. intercepted arc Theorem: Measure of an Inscribed Angle A • If an angle is inscribed in a circle, then its measure is one half the measure of its intercepted arc. mADB = ½m AB C D B Ex. : Finding Measures of Arcs and Inscribed Angles • Find the measure of the blue arc or angle. S R m QTS = 2mQRS = T 2(90°) = 180° Q Ex. : Finding Measures of Arcs and Inscribed Angles • Find the measure of the blue arc. m ZWX = 2mZYX = 2(115°) = 230° W Z 115 Y X Ex. : Finding Measures of Arcs and Inscribed Angles • Find the measure of the blue arc or angle. m NMP = ½ m NP N 100° M P ½ (100°) = 50° Corollary • 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. • B is a right angle if and only if AC is a diameter of the circle. A O B C Using Corollary B • Find the value of x. • AB is a diameter. So, C is a right angle and mC = 90° • 2x° = 90° • x = 45 Q A 2x° C Corollary: A • If two inscribed angles of a circle intercept the same arc, then the angles are congruent. • C D D B C Ex. : Comparing Measures of Inscribed Angles A • Find mACB, mADB, and mAEB. 60 E The measure of each angle is half the measure of AB m AB = 60°, so the measure of each angle is 30° B D C Ex. : Finding the Measure of an Angle G • It is given that mE = 75°. What is mF? • E and F both intercept GH , so E F. So, mF = mE = 75° E 75° F H Ex. : Using the Measure of an Inscribed Angle • Theater Design. When you go to the movies, you want to be close to the movie screen, but you don’t want to have to move your eyes too much to see the edges of the picture. Ex. : Using the Measure of an Inscribed Angle • If E and G are the ends of the screen and you are at F, mEFG is called your viewing angle. Ex. : Using the Measure of an Inscribed Angle • You decide that the middle of the sixth row has the best viewing angle. If someone else is sitting there, where else can you sit to have the same viewing angle? Ex. : Using the Measure of an Inscribed Angle • Solution: Draw the circle that is determined by the endpoints of the screen and the sixth row center seat. Any other location on the circle will have the same viewing angle. Theorem • If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. B A 1 C m1 = ½ m( BC - AC ) Theorem • If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. P 2 Q R m2 = ½ m(PQR - PR ) Theorem X W • If a tangent and a secant, two tangents 3 Z or two secants intercept in the EXTERIOR of a circle, Y then the measure of the angle formed is one half the difference of the measures of the m3 = ½ m( XY - WZ ) intercepted arcs. Exterior Angle Theorem The measure of the angle formed is equal to ½ the difference of the intercepted arcs. x y 1 x y m1 2 x y 2 x y m2 2 x y 3 x y m3 2 E Ex. : Using Theorem 200° • Find the value of x Solution: D F x° mGHF = ½ m(EDG G - GF ) Apply Theorem 72° = ½ (200° - x°) 144 = 200 - x° - 56 = -x 56 = x H 72° Substitute values. Multiply each side by 2. Subtract 200 from both sides. Divide by -1 to eliminate negatives. Ex. : Using Theorem M Because MN and MLN make a whole circle, m MLN =360°-92°=268° L • Find the value of x Solution: mGHF = ½ m(MLN - m MN ) = ½ (268 - 92) = ½ (176) = 88 92° x° N Apply Theorem 10.14 Substitute values. Subtract Multiply P D Theorem • If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 1 A 2 B m1 = ½ m CD + m AB m2 = ½ m BC + m AD C Ex. : Finding the Measure of an Angle Formed by Two Chords P 106° • Find the value of x S Q x° R • Solution: x° = ½ (mPS +m RQ ) x° = ½ (106° + 174°) x = 140 174° Apply Theorem Substitute values Simplify