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