9 6 Secants Tangents and Angle Measures Geometry

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. 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. m 1= ½m AB m 2= ½m ABC

Finding Angle and Arc Measures • Line m is tangent to the circle. Find the measure of the red angle or arc. • Solution: m 1= ½ AB m 1= ½ (150°) m 1= 75° 150°

Finding Angle and Arc Measures • Line m is tangent to the circle. Find the measure of the red angle or arc. • Solution: m RSP = 2(130°) m RSP = 260° 130°

Finding an Angle Measure is tangent to the circle. Find m CBD (9 x + 20)° • Solution: m CBD = ½ m DAB 5 x = ½(9 x + 20) 10 x = 9 x +20 x = 20 D m CBD = 5(20°) = 100° 5 x°

m 1 = ½ ( m CD + m AB) m 2 = ½ ( m BC+ m AD)

Finding the Measure of an Angle Formed by Two Chords 106° • Find the value of x • Solution: x° = ½ (m QR +m PS) x° = ½ (106° + 174°) x = 140 x° 174°

Using Theorem 9. 13 200° • Find the value of x m GHF = ½ (m EGD - m GF ) 72° = ½ (200° - x°) 144 = 200 - x° - 56 = -x 56 = x x° 72°

Using Theorem 9. 13 Because MN and MLN make a whole circle, m MLN =360°-92°=268° • Find the value of x m GHF = ½ (m MLN - m MN) = ½ (268 - 92) = ½ (176) = 88 92° x°

Practice

Practice m 1 = ½ ( 40 + 52) =46 m 2 = ½ ( 134) = 67 m 3 = ½ ( 100 – 70) = 15

Practice 100 = ½ ( 130 + x) 200 = 130+ x X = 70 50 = ½ ( (360 – x) -x) 100 = 360 - 2 x 260 = 2 x 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+ 2 y + y +2 y 360= 140 +5 y 220 = 5 y 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 = 4 x – 50 +x + 25+ x – 15 + 50 360=7 x +10 350 = 7 x X = 50 FHE = ½( 35 + 50) FHE = 42. 5 X = 50 CFD = ½*50 = 25