geometry section 10-6 1112
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Section 10-6Secants, Tangents, and Angle Measures
Monday, May 21, 2012
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
Vocabulary & Theorems
1. Secant:
Theorem 10.12 - Two Secants:
Monday, May 21, 2012
A line that intersects a circle in exactly two points
Vocabulary & Theorems
1. Secant:
Theorem 10.12 - Two Secants:
Monday, May 21, 2012
A line that intersects a circle in exactly two points
Vocabulary & Theorems
1. Secant:
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
Vocabulary & Theorems
Theorem 10.13 - Secant and Tangent:
Monday, May 21, 2012
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
Vocabulary & Theorems
Theorem 10.14 - Exterior Intersection:
Monday, May 21, 2012
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
Example 1
Find x. a.
Monday, May 21, 2012
Example 1
Find x. a.
m∠FDE = 180 − m∠EDH
Monday, May 21, 2012
Example 1
Find x. a.
m∠FDE = 180 − m∠EDH
m∠EDH =76 + 882
Monday, May 21, 2012
Example 1
Find x. a.
m∠FDE = 180 − m∠EDH
m∠EDH =76 + 882
=1642
Monday, May 21, 2012
Example 1
Find x. a.
m∠FDE = 180 − m∠EDH
m∠EDH =76 + 882
=1642
= 82°
Monday, May 21, 2012
Example 1
Find x. a.
m∠FDE = 180 − m∠EDH
m∠EDH =76 + 882
=1642
= 82°
m∠FDE = 180 − 82
Monday, May 21, 2012
Example 1
Find x. a.
m∠FDE = 180 − m∠EDH
m∠EDH =76 + 882
=1642
= 82°
m∠FDE = 180 − 82 = 98°
Monday, May 21, 2012
Example 1
Find x. a.
m∠FDE = 180 − m∠EDH
m∠EDH =76 + 882
=1642
= 82°
m∠FDE = 180 − 82 = 98°
x = 98Monday, May 21, 2012
Example 1
Find x. b.
Monday, May 21, 2012
Example 1
Find x. b.
x = 180 − m∠VZW
Monday, May 21, 2012
Example 1
Find x. b.
x = 180 − m∠VZW
m∠VZW =96 + 622
Monday, May 21, 2012
Example 1
Find x. b.
x = 180 − m∠VZW
m∠VZW =96 + 622
=1582
Monday, May 21, 2012
Example 1
Find x. b.
x = 180 − m∠VZW
m∠VZW =96 + 622
=1582
= 79°
Monday, May 21, 2012
Example 1
Find x. b.
x = 180 − m∠VZW
m∠VZW =96 + 622
=1582
= 79°
x = 180 − 79
Monday, May 21, 2012
Example 1
Find x. b.
x = 180 − m∠VZW
m∠VZW =96 + 622
=1582
= 79°
x = 180 − 79 = 101
Monday, May 21, 2012
Example 1
Find x. c.
Monday, May 21, 2012
Example 1
Find x. c.
60 = x + 252
Monday, May 21, 2012
Example 1
Find x. c.
60 = x + 252
120 = x + 25
Monday, May 21, 2012
Example 1
Find x. c.
60 = x + 252
120 = x + 25
x = 95
Monday, May 21, 2012
Example 2
Find each measure.
a. m∠QPS when mPTS = 250°
Monday, May 21, 2012
Example 2
Find each measure.
a. m∠QPS when mPTS = 250°
m∠QPS = 1
2mPTS
Monday, May 21, 2012
Example 2
Find each measure.
a. m∠QPS when mPTS = 250°
m∠QPS = 1
2mPTS
=12(250)
Monday, May 21, 2012
Example 2
Find each measure.
a. m∠QPS when mPTS = 250°
m∠QPS = 1
2mPTS
=12(250) = 125°
Monday, May 21, 2012
Example 2
Find each measure.
b. mBD
Monday, May 21, 2012
Example 2
Find each measure.
b. mBD
mBD = 360 − 2m∠ADB
Monday, May 21, 2012
Example 2
Find each measure.
b. mBD
mBD = 360 − 2m∠ADB= 360 − 2(108)
Monday, May 21, 2012
Example 2
Find each measure.
b. mBD
mBD = 360 − 2m∠ADB= 360 − 2(108)
= 360 − 216
Monday, May 21, 2012
Example 2
Find each measure.
b. mBD
mBD = 360 − 2m∠ADB= 360 − 2(108)
= 360 − 216
= 144°
Monday, May 21, 2012
Example 3
a. mBC when m∠AED = 62°
Find each measure.
Monday, May 21, 2012
Example 3
a. mBC when m∠AED = 62°
Find each measure.
m∠AED =
mABD − mBC
2
Monday, May 21, 2012
Example 3
a. mBC when m∠AED = 62°
Find each measure.
m∠AED =
mABD − mBC
2
62 = 141− x2
Monday, May 21, 2012
Example 3
a. mBC when m∠AED = 62°
Find each measure.
m∠AED =
mABD − mBC
2
62 = 141− x2
124 = 141− x
Monday, May 21, 2012
Example 3
a. mBC when m∠AED = 62°
Find each measure.
m∠AED =
mABD − mBC
2
62 = 141− x2
124 = 141− x
−17 = −x
Monday, May 21, 2012
Example 3
a. mBC when m∠AED = 62°
Find each measure.
m∠AED =
mABD − mBC
2
62 = 141− x2
124 = 141− x
−17 = −xx = 17
Monday, May 21, 2012
Example 3
a. mBC when m∠AED = 62°
Find each measure.
m∠AED =
mABD − mBC
2
62 = 141− x2
124 = 141− x
−17 = −xx = 17 mBC = 17°
Monday, May 21, 2012
Example 3
b. mXYZFind each measure.
Monday, May 21, 2012
Example 3
b. mXYZFind each measure.
m∠W =
mXYZ − mXZ
2
Monday, May 21, 2012
Example 3
b. mXYZFind each measure.
m∠W =
mXYZ − mXZ
2
40 = mXYZ −140
2
Monday, May 21, 2012
Example 3
b. mXYZFind each measure.
m∠W =
mXYZ − mXZ
2
40 = mXYZ −140
2
80 = mXYZ −140
Monday, May 21, 2012
Example 3
b. mXYZFind each measure.
m∠W =
mXYZ − mXZ
2
40 = mXYZ −140
2
80 = mXYZ −140
mXYZ = 220°Monday, May 21, 2012
Check Your Understanding
p. 731 #1-7
Monday, May 21, 2012
Problem Set
Monday, May 21, 2012
Problem Set
p. 732 #9-29 odd, 41, 47
"I hate quotations. Tell me what you know."– Ralph Waldo Emerson
Monday, May 21, 2012
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