1.6 Describing Pairs of Angles - Mesa Public Schools · 1.6 Day 1 Warm-up Solve. 1. 4x − 0 ......
Transcript of 1.6 Describing Pairs of Angles - Mesa Public Schools · 1.6 Day 1 Warm-up Solve. 1. 4x − 0 ......
Geometry1.6 Describing Pairs of Angles
August 26, 20161.6 Describing Pairs of Angles
1.6 Day 1 Warm-up
Solve.
1. 4x − 0 = 12 2. 7 = −11c − 4
3. 11 = −19x − 8 4. 7 = 5n + 5 − 4n
5. 3x + 2 + 8 = 2x − 5 6. x + 5 + 6x + 17 = x − 2
Essential Question
What angle relationships occur when two
lines intersect?
August 26, 20161.4 Perimeter and Area in the Coordinate Plane
What You Will Learn
• Identify complementary and supplementary angles.
• Identify linear pairs, vertical angles, and adjacent angles.
• Solve problems with angle relationship properties.
August 26, 20161.6 Describing Pairs of Angles
Adjacent Angles
August 26, 20161.6 Describing Pairs of Angles
A
B
CO
Adjacent angles have the
same vertex, O, and one side
in common, OB. They share
no interior points.
There are THREE angles:
AOB or BOA
BOC or COB
AOC or COA
You cannot use the label
O, since it would be
unclear which angle that is.
RST and VST are NOT adjacent angles.
August 26, 20161.6 Describing Pairs of Angles
R
S T
V
Why not?
They overlap.
Linear Pair
August 26, 20161.6 Describing Pairs of Angles
1 2
Two adjacent angles are a linear pair if their noncommon
sides are opposite rays.
Common Side
Noncommon sides
1 & 2 are
a linear pair.
Linear Pair Property
August 26, 20161.6 Describing Pairs of Angles
The sum of the angles of a linear pair is 180°.
70° ?110°
Complementary Angles
August 26, 20161.6 Describing Pairs of Angles
Two angles are complementary if their sum is 90°.
65°
25°
These angles are
complementary
and adjacent.
Complementary Angles
August 26, 20161.6 Describing Pairs of Angles
Two angles are complementary if their sum is 90°.
These angles are
complementary
and
nonadjacent.
30°
60°
Supplementary Angles
August 26, 20161.6 Describing Pairs of Angles
Angles are supplementary if their sum is 180°.
70° 110°
These angles are
supplementary and
adjacent and a linear pair.
Supplementary Angles
August 26, 20161.6 Describing Pairs of Angles
Angles are supplementary if their sum is 180°.
The angles are
supplementary
and nonadjacent.
80° 100°
Example 1
August 26, 20161.6 Describing Pairs of Angles
In the figure, name a pair of complementary angles, a
pair of supplementary angles, and a pair of adjacent
angles.
Example 2
August 26, 20161.6 Describing Pairs of Angles
Vertical Angles
August 26, 20161.6 Describing Pairs of Angles
1 23
4
Two angles are vertical
angles if their sides form
two pairs of opposite rays.
1 & 2 are
vertical angles.
3 & 4 are
vertical angles.
Vertical Angles Property
August 26, 20161.6 Describing Pairs of Angles
Vertical Angles are congruent.
60°?60°
Example 3
August 26, 20161.6 Describing Pairs of Angles
1
2 3
45
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
Example 4
August 26, 20161.6 Describing Pairs of Angles
50° 12
3
Find the measure of the three angles.
These angles are vertical angles.
Vertical angles are congruent.
50°
These angles form a
linear pair. The sum is
180°.
130°
These are vertical
angles, and
congruent.
130°
Example 5
August 26, 20161.6 Describing Pairs of Angles
A B
DC
E
(4x + 30)°
(6x – 10)°
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°.
Example 5
August 26, 20161.6 Describing Pairs of Angles
A B
DC
E
(4x + 30)°
(6x – 10)°
Linear pair AEB and
BEC means:
(4x + 30) + (6x – 10) = 180
10x + 20 = 180
10x = 160
x = 16
Then AEB = 4(16) + 30 = 94
and BEC = 6(16) – 10 = 86
94°
86°
94°
86°
Your Turn
August 26, 20161.6 Describing Pairs of Angles
Work through these two problems.
145°1
23
1. Find the measure
of 1, 2, 3.
(2x – 4)°(5x + 30)°
A B
C
2. Find the measure
of ABC.
Your Turn Solutions
August 26, 20161.6 Describing Pairs of Angles
145°1
23
(2x – 4)°(5x + 30)°
A B
C
145°
35°
35°
180°
5x + 30 + 2x – 4 = 180
7x + 26 = 180
7x = 154
x = 22
mABC = 5(22) + 30
= 140°
Example 6
August 26, 20161.6 Describing Pairs of Angles
A
B
C
D
6x°
(3x + 45)°
Solve for x, then find the angle measures.
Solution:
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
E
6(15) = 90°
3(15) + 45 = 90°
Example 7
August 26, 20161.6 Describing Pairs of Angles
(5y – 50)°
(4y – 10)°
1
Solve for y, then find m1.
Vertical angles are
congruent, so:
5y – 50 = 4y – 10
y = 40
5(40) – 50 = 150°
150° 1 forms a linear pair with
either of the 150° angles, so
1 is 30°.
30°
Example 8
August 26, 20161.6 Describing Pairs of Angles
(4x + 5)°
(3x + 8)°
Find the measure of each angle.
This is a right angle, the
angles are complementary.
Their sum is 90°.
4x + 5 + 3x + 8 = 90
7x + 13 = 90
7x = 77
x = 11
4(11) + 5 = 49°
49°
3(11) + 8 = 41°
41°
Example 9
August 26, 20161.6 Describing Pairs of Angles
(3x + 8)° (5x – 20)°
Find the value of each variable and the measure of each
labeled angle.
3x + 8 =5x – 20
-2x = -28
x = 14 3(14) + 8 = 50°
50°50°
130°
August 26, 20161.6 Describing Pairs of Angles
1. Solve for x.
August 26, 20161.6 Describing Pairs of Angles
(6x + 10)(4x + 40)
6 10 4 40
2 30
15
x x
x
x
2. Solve for x.
August 26, 20161.6 Describing Pairs of Angles
(5x + 5)(12x – 12)
(12 12) (5 5) 180
17 7 180
17 187
11
x x
x
x
x
3. Solve for x.
August 26, 20161.6 Describing Pairs of Angles
(7x + 2)
( 8) (7 2) 90
8 10 90
8 80
10
x x
x
x
x
4. Solve for x & y.
August 26, 20161.6 Describing Pairs of Angles
(7x + 4)
(13x + 16)
(9y + 3)
(5y 5)
(7 4) (13 16) 180
20 20 180
20 160
8
(9 3) (5 5) 180
14 2 180
14 182
13
x
y y
x
x
x
y
y
y
x
5. Solve for x.
August 26, 20161.6 Describing Pairs of Angles
A is supplementary to B.
mA = (2x + 10)
mB = (3x 5)
2x + 10 + 3x 5 = 180
5x + 5 = 180
5x = 175
x = 35
Quick Review
Two angles are complementary if their sum is 90.
August 26, 20161.6 Describing Pairs of Angles
What do you know about supplementary angles?
Two angles are supplementary if their sum is 180.
What do you know about complementary angles?
Quick Review
August 26, 20161.6 Describing Pairs of Angles
1 2
4
3
Which angles are Vertical Angles and what do you
know about them?
1 2 & 3 4
Quick Review
August 26, 20161.6 Describing Pairs of Angles
1 2
4
3
Which angles are linear with ∠4 and what do
we know about them?
m1 + m4 = 180
m4 + m2 = 180
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?
August 26, 20161.4 Perimeter and Area in the Coordinate Plane
Assignment
August 26, 20161.6 Describing Pairs of Angles