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9.6 Secants, Tangents and Angle Measures Geometry Objectives • Use angles formed by tangents and chords to solve problems in geometry. • Use angles formed by lines that intersect a circle to solve problems. Using Tangents and Chords • Measure of an angle inscribed in a circle is half the measure of its intercepted arc. A C D B m ADB = ½m AB Theorem 9.11 • If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. B C 1 2 A m1= ½m AB m2= ½m ABC Finding Angle and Arc Measures m • Line m is tangent to the circle. Find the measure of the red angle or arc. • Solution: m1= ½ AB m1= ½ (150°) m1= 75° B 1 A 150° Finding Angle and Arc Measures S • Line m is tangent to the circle. Find the measure of the red angle or arc. 130° P • Solution: m RSP = 2(130°) m RSP = 260° R Finding an Angle Measure BC is tangent to the circle. Find m CBD (9x + 20)° • Solution: m CBD = ½ m DAB 5x = ½(9x + 20) 10x = 9x +20 x = 20 D mCBD = 5(20°) = 100° C A 5x° B D m1 = ½ ( m CD + m AB) 1 A 2 B C m2 = ½ ( m BC+ m AD) Finding the Measure of an Angle Formed by Two Chords 106° P • Find the value of x • Solution: x° = ½ (m QR +m PS) x° = ½ (106° + 174°) x = 140 S Q x° R 174° E Using Theorem 9.13 200° • Find the value of x D F m GHF = ½ (m EGD - m GF ) 72° = ½ (200° - x°) 144 = 200 - x° - 56 = -x 56 = x x° G H 72° Using Theorem 9.13 M Because MN and MLN make a whole circle, m MLN =360°-92°=268° L 92° • Find the value of x m GHF = ½ (m MLN - m MN) = ½ (268 - 92) = ½ (176) = 88 N x° P Practice Practice m1 = ½ ( 40 + 52) =46 m2 = ½ ( 134) = 67 m3 = ½ ( 100 – 70) = 15 Practice 100 = ½ ( 130 + x) 200 = 130+ x X = 70 50 = ½ ( (360 – x) -x) 100 = 360- 2x 260 = 2x X = 130 20 = ½ ( 70 – x) 40 = 70-x X = 30 CD = CQD = 120 E = ½ ( AD -BC) 25 = ½ (x -30) 50 = x – 30 X = 80 AB = 360-30 – 120 – 80 = AB = 130 QDC = (180- 120) / 2 = 30 360 = 140+ 2y + y +2y 360= 140 +5y 220 = 5y Y = 44 Y = 44 2 * 44 = 88 Y = 44 2 * 44 = 88 BCD = ½( AE – BD) BCD = ½( 140-44) BCD = 48 A = FB = 50 BCA = ½ * FB = 25 ABC = 180- 50 -25 = 105 GBC =180-105 =75 360 = 4x – 50 +x + x + 25+ x – 15 + 50 360=7x +10 350 = 7x X = 50 FHE = ½( 35 + 50) FHE = 42.5 X = 50 CFD = ½*50 = 25