geometry section 1-5 1112
TRANSCRIPT
Section 1-5Angle Relationships
Sunday, September 28, 14
Essential Questions
• How do you identify and use special pairs of angles?
• How do you identify perpendicular lines?
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Vocabulary1. Adjacent Angles:
2. Linear Pair:
3. Vertical Angles:
4. Complementary Angles:
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Vocabulary1. Adjacent Angles: Two angles that share a vertex and
side but no interior points
2. Linear Pair:
3. Vertical Angles:
4. Complementary Angles:
Sunday, September 28, 14
Vocabulary1. Adjacent Angles: Two angles that share a vertex and
side but no interior points
2. Linear Pair: A pair of adjacent angles where the non-shared sides are opposite rays
3. Vertical Angles:
4. Complementary Angles:
Sunday, September 28, 14
Vocabulary1. Adjacent Angles: Two angles that share a vertex and
side but no interior points
2. Linear Pair: A pair of adjacent angles where the non-shared sides are opposite rays
3. Vertical Angles: Nonadjacent angles that are formed when two lines intersect; only share a vertex
4. Complementary Angles:
Sunday, September 28, 14
Vocabulary1. Adjacent Angles: Two angles that share a vertex and
side but no interior points
2. Linear Pair: A pair of adjacent angles where the non-shared sides are opposite rays
3. Vertical Angles: Nonadjacent angles that are formed when two lines intersect; only share a vertex
4. Complementary Angles: Two angles with measures such that the sum of the angles is 90 degrees
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Vocabulary5. Supplementary Angles:
6. Perpendicular:
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Vocabulary5. Supplementary Angles: Two angles with measures
such that the sum of the angles is 180 degrees
6. Perpendicular:
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Vocabulary5. Supplementary Angles: Two angles with measures
such that the sum of the angles is 180 degrees
6. Perpendicular: When two lines, segments, or rays intersect at a right angle
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Adjacent Angles
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Adjacent Angles
A
B
C
D
Sunday, September 28, 14
Adjacent Angles
A
B
C
D
m∠ADB + m∠BDC = m∠ADC
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Linear Pair
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Linear Pair
1 2
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Linear Pair
1 2
m∠1+ m∠2 =180°
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Vertical Angles
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Vertical Angles
12
34
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Vertical Angles
∠1≅ ∠3
12
34
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Vertical Angles
∠1≅ ∠3
12
34
∠2 ≅ ∠4
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Example 1Name a pair that satisfies each condition.
a. A pair of vertical angles
b. A pair of adjacent angles
c. A linear pair
Sunday, September 28, 14
Example 1Name a pair that satisfies each condition.
a. A pair of vertical angles
∠AGF and ∠DGC
b. A pair of adjacent angles
c. A linear pair
Sunday, September 28, 14
Example 1Name a pair that satisfies each condition.
a. A pair of vertical angles
∠AGF and ∠DGC
b. A pair of adjacent angles
∠AGF and ∠FGE
c. A linear pair
Sunday, September 28, 14
Example 1Name a pair that satisfies each condition.
a. A pair of vertical angles
∠AGF and ∠DGC
b. A pair of adjacent angles
∠AGF and ∠FGE
c. A linear pair
∠AGB and ∠BGD
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Example 1
d. What is the relationship between the following angles?
∠FGB and ∠BGD
Sunday, September 28, 14
Example 1
d. What is the relationship between the following angles?
∠FGB and ∠BGD
These are adjacent angles
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Example 2Find answers to satisfy each condition.
a. Name two complementaryangles
b. Name two supplementaryangles
Sunday, September 28, 14
Example 2Find answers to satisfy each condition.
a. Name two complementaryangles
∠TWU and ∠UWV
b. Name two supplementaryangles
Sunday, September 28, 14
Example 2Find answers to satisfy each condition.
a. Name two complementaryangles
∠TWU and ∠UWV
b. Name two supplementaryangles
∠SWT and ∠TWV
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Example 2Find answers to satisfy each condition.
c. If , find m∠RWS = 72° m∠UWV
d. If , find m∠RWS = 72° m∠RWV
Sunday, September 28, 14
Example 2Find answers to satisfy each condition.
m∠UWV = 72°
c. If , find m∠RWS = 72° m∠UWV
d. If , find m∠RWS = 72° m∠RWV
Sunday, September 28, 14
Example 2Find answers to satisfy each condition.
m∠UWV = 72°
c. If , find m∠RWS = 72° m∠UWV
(vertical angles)
d. If , find m∠RWS = 72° m∠RWV
Sunday, September 28, 14
Example 2Find answers to satisfy each condition.
m∠UWV = 72°
c. If , find m∠RWS = 72° m∠UWV
(vertical angles)
d. If , find m∠RWS = 72° m∠RWV
180° − 72°
Sunday, September 28, 14
Example 2Find answers to satisfy each condition.
m∠UWV = 72°
c. If , find m∠RWS = 72° m∠UWV
(vertical angles)
m∠RWV =108°
d. If , find m∠RWS = 72° m∠RWV
180° − 72°
Sunday, September 28, 14
Example 2Find answers to satisfy each condition.
m∠UWV = 72°
c. If , find m∠RWS = 72° m∠UWV
(vertical angles)
m∠RWV =108°
d. If , find m∠RWS = 72° m∠RWV
(supplementary angles) 180° − 72°
Sunday, September 28, 14
Example 2Find answers to satisfy each condition.
f. If , find m∠UWV = 47° m∠UWT
e. Name two perpendicular segments
Sunday, September 28, 14
Example 2Find answers to satisfy each condition.
SW and TW
f. If , find m∠UWV = 47° m∠UWT
e. Name two perpendicular segments
Sunday, September 28, 14
Example 2Find answers to satisfy each condition.
SW and TW
f. If , find m∠UWV = 47° m∠UWT
90° − 47°
e. Name two perpendicular segments
Sunday, September 28, 14
Example 2Find answers to satisfy each condition.
SW and TW
m∠UWT = 43°
f. If , find m∠UWV = 47° m∠UWT
90° − 47°
e. Name two perpendicular segments
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Example 2Find answers to satisfy each condition.
SW and TW
m∠UWT = 43°
f. If , find m∠UWV = 47° m∠UWT
(complementary angles) 90° − 47°
e. Name two perpendicular segments
Sunday, September 28, 14
Example 3Find x and y so that AE ⊥ CF .
Sunday, September 28, 14
Example 3Find x and y so that AE ⊥ CF .
2x + 14 + 2x = 90
Sunday, September 28, 14
Example 3Find x and y so that AE ⊥ CF .
2x + 14 + 2x = 904x + 14 = 90
Sunday, September 28, 14
Example 3Find x and y so that AE ⊥ CF .
2x + 14 + 2x = 904x + 14 = 90−14 −14
Sunday, September 28, 14
Example 3Find x and y so that AE ⊥ CF .
2x + 14 + 2x = 904x + 14 = 90−14 −14
4x = 76
Sunday, September 28, 14
Example 3Find x and y so that AE ⊥ CF .
2x + 14 + 2x = 904x + 14 = 90−14 −14
4x = 764 4
Sunday, September 28, 14
Example 3Find x and y so that AE ⊥ CF .
2x + 14 + 2x = 904x + 14 = 90−14 −14
4x = 764 4x = 19
Sunday, September 28, 14
Example 3Find x and y so that AE ⊥ CF .
2x + 14 + 2x = 904x + 14 = 90−14 −14
4x = 764 4x = 19
8y + 10 = 90
Sunday, September 28, 14
Example 3Find x and y so that AE ⊥ CF .
2x + 14 + 2x = 904x + 14 = 90−14 −14
4x = 764 4x = 19
8y + 10 = 90−10 −10
Sunday, September 28, 14
Example 3Find x and y so that AE ⊥ CF .
2x + 14 + 2x = 904x + 14 = 90−14 −14
4x = 764 4x = 19
8y + 10 = 90−10 −10
8y = 80
Sunday, September 28, 14
Example 3Find x and y so that AE ⊥ CF .
2x + 14 + 2x = 904x + 14 = 90−14 −14
4x = 764 4x = 19
8y + 10 = 90−10 −10
8y = 808 8
Sunday, September 28, 14
Example 3Find x and y so that AE ⊥ CF .
2x + 14 + 2x = 904x + 14 = 90−14 −14
4x = 764 4x = 19
8y + 10 = 90−10 −10
8y = 808 8y = 10
Sunday, September 28, 14
Example 4Find the measures of two supplementary angles so that the difference between the measures of the two angles
is 44.
x° (180 − x)°
Sunday, September 28, 14
Example 4Find the measures of two supplementary angles so that the difference between the measures of the two angles
is 44.
x° (180 − x)°
(180 − x) − x = 44
Sunday, September 28, 14
Example 4Find the measures of two supplementary angles so that the difference between the measures of the two angles
is 44.
x° (180 − x)°
(180 − x) − x = 44180 − 2x = 44
Sunday, September 28, 14
Example 4Find the measures of two supplementary angles so that the difference between the measures of the two angles
is 44.
x° (180 − x)°
(180 − x) − x = 44180 − 2x = 44−180 −180
Sunday, September 28, 14
Example 4Find the measures of two supplementary angles so that the difference between the measures of the two angles
is 44.
x° (180 − x)°
(180 − x) − x = 44180 − 2x = 44−180 −180−2x = −136
Sunday, September 28, 14
Example 4Find the measures of two supplementary angles so that the difference between the measures of the two angles
is 44.
x° (180 − x)°
(180 − x) − x = 44180 − 2x = 44−180 −180−2x = −136−2 −2
Sunday, September 28, 14
Example 4Find the measures of two supplementary angles so that the difference between the measures of the two angles
is 44.
x° (180 − x)°
(180 − x) − x = 44180 − 2x = 44−180 −180−2x = −136−2 −2
x = 68°
Sunday, September 28, 14
Example 4Find the measures of two supplementary angles so that the difference between the measures of the two angles
is 44.
x° (180 − x)°
(180 − x) − x = 44180 − 2x = 44−180 −180−2x = −136−2 −2
x = 68°
180 − 68
Sunday, September 28, 14
Example 4Find the measures of two supplementary angles so that the difference between the measures of the two angles
is 44.
x° (180 − x)°
(180 − x) − x = 44180 − 2x = 44−180 −180−2x = −136−2 −2
x = 68°
180 − 68 =112°
Sunday, September 28, 14
Example 4Find the measures of two supplementary angles so that the difference between the measures of the two angles
is 44.
x° (180 − x)°
(180 − x) − x = 44180 − 2x = 44−180 −180−2x = −136−2 −2
x = 68°
180 − 68 =112°
The two angles are 68° and 112°
Sunday, September 28, 14
Problem Set
Sunday, September 28, 14
Problem Set
p. 50 #1-41 odd, 49
“A ship in port is safe, but that’s not what ships are built for.” - Grace Murray Hopper
Sunday, September 28, 14