Download Geometry 1.6 Describing Pairs of Angles

Geometry
1.6 Describing Pairs of Angles
1.6 Describing Pairs of Angles
May 25, 2017
1.6 Essential Question
How can you describe complementary
and supplementary angles and use these
descriptions to find angle measures?
1.4 Perimeter and Area in the Coordinate Plane
May 25, 2017
What You Will Learn
• Identify complementary and supplementary
angles.
• Identify linear pairs and vertical angles.
1.6 Describing Pairs of Angles
May 25, 2017
Recall from Last Lesson
1
1.6 Describing Pairs of Angles
Sometimes, for clarity
and convenience, we
will use a single
number inside the
angle to name it.
This is 1.
May 25, 2017
More than One Angle
1
1.6 Describing Pairs of Angles
2
3
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Adjacent Angles
Adjacent angles have the
same vertex, O, and one side
in common, OB. They share
no interior points.
A
B
O
There are THREE angles:
C
You cannot use the label
O, since it would be
unclear which angle that is.
1.6 Describing Pairs of Angles
AOB or BOA
BOC or COB
AOC or COA
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RST and VST are NOT
adjacent angles.
R
V
S
1.6 Describing Pairs of Angles
T
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Complementary Angles
Two angles are complementary if their sum is 90°.
These angles are
complementary AND
adjacent.
65°
25°
1.6 Describing Pairs of Angles
May 25, 2017
Complementary Angles
Two angles are complementary if their sum is 90°.
These angles are
complementary
AND
NONADJACENT.
30°
60°
1.6 Describing Pairs of Angles
Explain why:
May 25, 2017
Supplementary Angles
Angles are supplementary if their sum is 180°.
These angles are
adjacent AND
supplementary (and a
linear pair).
70°
1.6 Describing Pairs of Angles
110°
May 25, 2017
Supplementary Angles
Angles are supplementary if their sum is 180°.
The angles are
nonadjacent and
supplementary.
Explain why:
80°
1.6 Describing Pairs of Angles
100°
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Example 1
In the figure, name a pair of complementary angles, a
pair of supplementary angles, and a pair of adjacent
angles.
1.6 Describing Pairs of Angles
May 25, 2017
Example 2
1.6 Describing Pairs of Angles
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Linear Pair
Two adjacent angles are a linear pair if their noncommon
sides are opposite rays.
Common Side
1 & 2 are
a linear pair.
1
A
1.6 Describing Pairs of Angles
2
B
Noncommon sides
C
May 25, 2017
Linear Pair Property
The sum of the angles of a linear pair is 180°.
70°
1.6 Describing Pairs of Angles
110°
?
May 25, 2017
Vertical Angles
Two angles are vertical
angles if their sides form
two pairs of opposite rays.
1
3
2
4
1 & 2 are
vertical angles.
3 & 4 are
vertical angles.
1.6 Describing Pairs of Angles
May 25, 2017
Vertical Angles Property
Vertical Angles are congruent.
60°
1.6 Describing Pairs of Angles
?60°
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Example 3
a. Are 1 and 2 a linear pair?
Yes
b. Are 4 and 5 a linear pair?
No
c. Are 3 and 5 vertical angles?
No
d. Are 1 and 3 vertical angles?
Yes
1.6 Describing Pairs of Angles
2
1
5
3
4
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Example 4
Find the measure of the three angles.
130°
2
50°
These are vertical
angles, and
congruent.
1.6 Describing Pairs of Angles
1 50°
These angles form a
linear pair. The sum is
180°.
3
130°
These angles are vertical angles.
Vertical angles are congruent.
May 25, 2017
Example 5
A
B
(4x + 30)°
E
(6x – 10)°
D
C
Solve for x, then
find the measure of
each angle.
AEB and BEC
form a linear pair.
What do we know about the sum of the angles of
a linear pair? The sum is 180°.
1.6 Describing Pairs of Angles
May 25, 2017
Example 5
94°
B
Linear pair AEB and
(4x + 30)°
BEC means:
E
(6x – 10)°
86°
86° (4x + 30) + (6x – 10) = 180
94°
10x + 20 = 180
10x = 160
C
D
x = 16
Then AEB = 4(16) + 30 = 94
and BEC = 6(16) – 10 = 86
A
1.6 Describing Pairs of Angles
May 25, 2017
Your Turn
Work through these two problems.
C
145° 1
3
2
1. Find the measure
of 1, 2, 3.
1.6 Describing Pairs of Angles
(5x + 30)° (2x – 4)°
A
B
2. Find the measure
of ABC.
May 25, 2017
Your Turn Solutions
180° C
145°
35° 3
1
2
145°
1.6 Describing Pairs of Angles
35°
(5x + 30)° (2x – 4)°
A
B
5x + 30 + 2x – 4 = 180
7x + 26 = 180
7x = 154
x = 22
mABC = 5(22) + 30
= 140°May 25, 2017
Essential Question
How can you describe complementary
and supplementary angles and use these
descriptions to find angle measures?
1.4 Perimeter and Area in the Coordinate Plane
May 25, 2017
Assignment
1.6 Describing Pairs of Angles
May 25, 2017
1.6 Describing
Pairs of Angles
Day 2
Essential Question
When two lines intersect, how do you
know if two angles are congruent or
supplementary and how do you use this
information to find angle measures?
1.4 Perimeter and Area in the Coordinate Plane
May 25, 2017
Quick Review
4
1
2
3
Vertical Angles are congruent.
1  2 & 3  4
1.6 Describing Pairs of Angles
May 25, 2017
Quick Review
4
1
2
3
The angles of a Linear Pair are Supplementary
m1 + m4 = 180
1.6 Describing Pairs of Angles
m4 + m2 = 180
May 25, 2017
Quick Review
• Two angles are supplementary if their sum is 180.
• Two angles are complementary if their sum is 90.
1.6 Describing Pairs of Angles
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Example 6
Solve for x, then find the angle measures.
Solution:
B
6(15) = 90°
A
6x° E
(3x + 45)°
3(15) + 45 = 90°
1.6 Describing Pairs of Angles
D
C
AEB and DEA are a
linear pair. The sum of
the angles in a linear pair
is 180°.
6x + (3x + 45) = 180
9x = 135
x = 15
May 25, 2017
Example 7
Solve for y, then find m1.
5(40) – 50 = 150°
(5y – 50)°
30° 1
(4y – 10)°
150°
1.6 Describing Pairs of Angles
Vertical angles are
congruent, so:
5y – 50 = 4y – 10
y = 40
1 forms a linear pair with
either of the 150° angles, so
1 is 30°.
May 25, 2017
Example 8
Find the measure of each angle.
4x + 5 + 3x + 8 = 90
49°
(4x + 5)°
41°
(3x + 8)°
This is a right angle, the
angles are complementary.
Their sum is 90°.
1.6 Describing Pairs of Angles
7x + 13 = 90
7x = 77
x = 11
4(11) + 5 = 49°
3(11) + 8 = 41°
May 25, 2017
Example 9
Find the value of each variable and the measure of each
labeled angle.
5x + 4y = 130
5(14) + 4y = 130
130°
50° (3x + 8)°
(5x – 20)°
70 + 4y = 130
50°
4y = 60
(5x + 4y)°
130°
y = 15
3x + 8 = 5x – 20
-2x = -28
3(14) + 8 = 50°
x = 14
1.6 Describing Pairs of Angles
May 25, 2017
1.6 Describing Pairs of Angles
May 25, 2017
1. Solve for x.
(4x + 40)
(6x + 10)
6 x  10  4 x  40
2 x  30
x  15
1.6 Describing Pairs of Angles
May 25, 2017
2. Solve for x.
(12x – 12)
(5x + 5)
(12 x  12)  (5 x  5)  180
17 x  7  180
17 x  187
x  11
1.6 Describing Pairs of Angles
May 25, 2017
3. Solve for x.
(7x + 2)
1.6 Describing Pairs of Angles
( x  8)  (7 x  2)  90
8 x  10  90
8 x  80
x  10
May 25, 2017
4. Solve for x & y.
(7 x  4)  (13 x  16)  180
20 x  20  180
(7x + 4) (9y + 3)
(13x + 16) (5y  5)
20 x  160
x 8
(9 y  3)  (5 y  5)  180
14 y  2  180
1.6 Describing Pairs of Angles
14 y  182
y  13
May 25, 2017
5. Solve for x.
A is supplementary to B.
mA = (2x + 10)
mB = (3x  5)
2x + 10 + 3x  5 = 180
5x + 5 = 180
5x = 175
x = 35
1.6 Describing Pairs of Angles
May 25, 2017
Essential Question
When two lines intersect, how do you
know if two angles are congruent or
supplementary and how do you use this
information to find angle measures?
1.4 Perimeter and Area in the Coordinate Plane
May 25, 2017
Assignment
1.6 Describing Pairs of Angles
May 25, 2017