The document provides definitions and theorems about secants, tangents, and angle measures formed by lines that intersect within or outside of a circle. It includes examples that apply the theorems to find measures of angles and arc lengths. Theorems relate the measures of angles formed to the measures of intercepted arcs when lines intersect inside or outside the circle. Examples work through applications of the theorems to find specific angle measures.
1. Section 10-6
Secants, Tangents, and Angle Measures
Monday, May 21, 2012
2. Essential Questions
How do you find measures of angles
formed by lines intersecting on or inside
a circle?
How do you find measure of angles
formed by lines intersecting outside the
circle?
Monday, May 21, 2012
4. Vocabulary & Theorems
1. Secant: A line that intersects a circle in exactly
two points
Theorem 10.12 - Two Secants:
Monday, May 21, 2012
5. Vocabulary & Theorems
1. Secant: A line that intersects a circle in exactly
two points
Theorem 10.12 - Two Secants: If two secants or
chords intersect in the interior of a circle, then
the measure of an angle formed is half of the
sum of the measure of the arcs intercepted by
the angle and its vertical angle
Monday, May 21, 2012
7. Vocabulary & Theorems
Theorem 10.13 - Secant and Tangent: If a
secant and a tangent intersect at the point of
tangency, then the measure of each angle
formed is half of the measure of its intercepted
arc
Monday, May 21, 2012
9. Vocabulary & Theorems
Theorem 10.14 - Exterior Intersection: If two
secants, a secant and a tangent, or two
tangents intersect in the exterior of a circle,
then the measure of the angle formed is half
the difference of the measures of the
intercepted arcs
Monday, May 21, 2012
26. Example 1
Find x.
c.
x + 25
60 =
2
Monday, May 21, 2012
27. Example 1
Find x.
c.
x + 25
60 =
2
120 = x + 25
Monday, May 21, 2012
28. Example 1
Find x.
c.
x + 25
60 =
2
120 = x + 25
x = 95
Monday, May 21, 2012
29. Example 2
Find each measure.
= 250°
a. m∠QPS when mPTS
Monday, May 21, 2012
30. Example 2
Find each measure.
= 250°
a. m∠QPS when mPTS
1
m∠QPS = mPTS
2
Monday, May 21, 2012
31. Example 2
Find each measure.
= 250°
a. m∠QPS when mPTS
1
m∠QPS = mPTS
2
1
= (250)
2
Monday, May 21, 2012
32. Example 2
Find each measure.
= 250°
a. m∠QPS when mPTS
1
m∠QPS = mPTS
2
1
= (250) = 125°
2
Monday, May 21, 2012
33. Example 2
Find each measure.
b. mBD
Monday, May 21, 2012
34. Example 2
Find each measure.
b. mBD
= 360 − 2m∠ADB
mBD
Monday, May 21, 2012
35. Example 2
Find each measure.
b. mBD
= 360 − 2m∠ADB
mBD
= 360 − 2(108)
Monday, May 21, 2012
36. Example 2
Find each measure.
b. mBD
= 360 − 2m∠ADB
mBD
= 360 − 2(108)
= 360 − 216
Monday, May 21, 2012
37. Example 2
Find each measure.
b. mBD
= 360 − 2m∠ADB
mBD
= 360 − 2(108)
= 360 − 216
= 144°
Monday, May 21, 2012
38. Example 3
Find each measure.
when m∠AED = 62°
a. mBC
Monday, May 21, 2012
39. Example 3
Find each measure.
when m∠AED = 62°
a. mBC
− mBC
mABD
m∠AED =
2
Monday, May 21, 2012
40. Example 3
Find each measure.
when m∠AED = 62°
a. mBC
− mBC
mABD
m∠AED =
2
141 − x
62 =
2
Monday, May 21, 2012
41. Example 3
Find each measure.
when m∠AED = 62°
a. mBC
− mBC
mABD
m∠AED =
2
141 − x
62 = 124 = 141 − x
2
Monday, May 21, 2012
42. Example 3
Find each measure.
when m∠AED = 62°
a. mBC
− mBC
mABD
m∠AED =
2
141 − x
62 = 124 = 141 − x
2
−17 = −x
Monday, May 21, 2012
43. Example 3
Find each measure.
when m∠AED = 62°
a. mBC
− mBC
mABD
m∠AED =
2
141 − x
62 = 124 = 141 − x
2
−17 = −x
x = 17
Monday, May 21, 2012
44. Example 3
Find each measure.
when m∠AED = 62°
a. mBC
− mBC
mABD
m∠AED =
2
141 − x
62 = 124 = 141 − x
2
−17 = −x = 17°
mBC
x = 17
Monday, May 21, 2012
45. Example 3
Find each measure.
b. m XYZ
Monday, May 21, 2012
46. Example 3
Find each measure.
b. m XYZ
− m XZ
m XYZ
m∠W =
2
Monday, May 21, 2012
47. Example 3
Find each measure.
b. m XYZ
− m XZ
m XYZ
m∠W =
2
− 140
m XYZ
40 =
2
Monday, May 21, 2012
48. Example 3
Find each measure.
b. m XYZ
− m XZ
m XYZ
m∠W =
2
− 140
m XYZ
40 =
2
− 140
80 = m XYZ
Monday, May 21, 2012
49. Example 3
Find each measure.
b. m XYZ
− m XZ
m XYZ
m∠W =
2
− 140
m XYZ
40 =
2
− 140
80 = m XYZ
= 220°
m XYZ
Monday, May 21, 2012