1. 9.6 Secants, Tangents and Angle Measures
Geometry

2. 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.

3. Using Tangents and Chords
Measure of an angle inscribed in a circle is half the measure of its intercepted arc.
m ADB = ½m AB

4. 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

5. 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°

6. 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°

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

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

10. 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°

11. Using Theorem 9.13 Find the value of x 72° = ½ (200° - x°)
m GHF = ½ (m EGD - m GF ) x°
72°

12. Using Theorem 9.13 Find the value of x = ½ (268 - 92) = ½ (176) = 88
Because MN and MLN make a whole circle, m MLN =360°-92°=268° x°
92°
Find the value of x
= ½ ( )
= ½ (176)
= 88
m GHF = ½ (m MLN - m MN)

13. Practice

14. Practice m1 = ½ ( 40 + 52) =46 m2 = ½ ( 134) = 67

15. Practice 100 = ½ ( 130 + x) 200 = 130+ x X = 70 50 = ½ ( (360 – x) -x)

16. CD = CQD = 120
 E = ½ ( AD -BC)
25 = ½ (x -30)
50 = x – 30
X = 80
AB = – 120 – 80 =
AB = 130
QDC = ( ) / 2 = 30

17. 360 = y + y +2y
360= y
220 = 5y
Y = 44
Y = 44
2 * 44 = 88
Y = 44
2 * 44 = 88
BCD = ½( AE – BD)
BCD = ½( )
BCD = 48

18. A = FB = 50
BCA = ½ * FB = 25
ABC = = 105
GBC = =75
360 = 4x – 50 +x + x x –
360=7x +10
350 = 7x
X = 50
X = 50
CFD = ½*50 = 25
FHE = ½( )
FHE = 42.5