Download lesson 1.5 gem

Warm-Up Exercises
1. The sum of two numbers is 90 and one number is
4 times the other. Write an equation and solve to find
the numbers.
ANSWER
2. Find m
ANSWER
x + 4x = 90; 18, 72
ABD. What kind of angle is it?
180° , straight
Warm-Up1Exercises
EXAMPLE
Identify complements and supplements
In the figure, name a pair of
complementary angles, a pair
of supplementary angles, and
a pair of adjacent angles.
SOLUTION
Because 32°+ 58° = 90°,
complementary angles.
BAC and
Because 122° + 58° = 180°,
supplementary angles.
RST are
CAD and
RST are
Because BAC and CAD share a common vertex
and side, they are adjacent.
Warm-Up
Exercises
GUIDED
PRACTICE
1.
for Example 1
In the figure, name a pair of complementary
angles, a pair of supplementary angles, and a
pair of adjacent angles.
ANSWER
FGK and
HGK and
FGK and
GKL,
GKL,
HGK
Warm-Up
Exercises
GUIDED
PRACTICE
2.
for Example 1
Are KGH and LKG adjacent angles ? Are
FGK and FGH adjacent angles? Explain.
ANSWER
No, they do not share a common vertex.
No, they have common interior points.
Warm-Up2Exercises
EXAMPLE
Find measures of a complement and a supplement
a.
Given that
find m 2.
1 is a complement of
2 and m
SOLUTION
a.
You can draw a diagram with complementary
adjacent angles to illustrate the relationship.
m 2 = 90° – m
22°
1 = 90° – 68° =
1 = 68°,
Warm-Up2Exercises
EXAMPLE
Find measures of a complement and a supplement
b.
Given that
find m 3.
3 is a supplement of
4 and m
SOLUTION
b. You can draw a diagram with supplementary
adjacent angles to illustrate the relationship.
m
3 = 180° – m 4 = 180° –56° = 124°
4 = 56°,
Warm-Up3Exercises
EXAMPLE
Find angle measures
Sports
When viewed from the side, the frame of a ballreturn net forms a pair of supplementary angles with
the ground. Find m BCE and m ECD.
Warm-Up3Exercises
EXAMPLE
Find angle measures
SOLUTION
STEP 1
m
Use the fact that the sum of the measures
of supplementary angles is 180°.
BCE + m  ECD = 180° Write equation.
(4x + 8)° + (x + 2)° = 180°Substitute.
5x + 10 = 180
5x = 170
x = 34
Combine like terms.
Subtract 10 from each side.
Divide each side by 5.
Warm-Up3Exercises
EXAMPLE
Find angle measures
SOLUTION
STEP 2
Evaluate: the original expressions when x = 34.
m
BCE = (4x + 8)° = (4 34 + 8)° = 144°
m
ANSWER
ECD = (x + 2)° = ( 34 + 2)° = 36°
The angle measures are 144° and 36°.
Warm-Up
Exercises
GUIDED
PRACTICE
3.
Given that
find m 1.
ANSWER
4.
1 is a complement of
2 and m
82o
Given that 3 is a supplement of
m 3 = 117o, find m 4.
ANSWER
5.
for Examples 2 and 3
4 and
63o
LMN and PQR are complementary
angles. Find the measures of the angles if
m
LMN = (4x – 2)o and m PQR = (9x + 1)o.
ANSWER 26o, 64o
2 = 8o ,
Warm-Up4Exercises
EXAMPLE
Identify angle pairs
Identify all of the linear pairs and all of the
vertical angles in the figure at the right.
SOLUTION
To find vertical angles, look or
angles formed by intersecting lines.
ANSWER
1 and
5 are vertical angles.
To find linear pairs, look for adjacent angles whose
noncommon sides are opposite rays.
ANSWER
1 and 4 are a linear pair.
are also a linear pair.
4 and
5
Warm-Up5Exercises
EXAMPLE
Find angle measures in a linear pair
ALGEBRA
Two angles form a linear pair. The measure of
one angle is 5 times the measure of the other.
Find the measure of each angle.
SOLUTION
Let x° be the measure of one
angle. The measure of the
other angle is 5x°. Then use
the fact that the angles of a
linear pair are supplementary
to write an equation.
Warm-Up5Exercises
EXAMPLE
Find angle measures in a linear pair
xo + 5xo = 180o
6x = 180
x = 30o
ANSWER
Write an equation.
Combine like terms.
Divide each side by 6.
The measures of the angles are 30o
and 5(30)o = 150o.
Warm-Up
Exercises
GUIDED
PRACTICE
6.
For Examples 4 and 5
Do any of the numbered angles in the diagram below
form a linear pair? Which angles are vertical angles?
Explain.
ANSWER
No, no adjacent angles have their noncommon
sides as opposite rays, 1 and 4 , 2 and 5,
3 and 6, these pairs of angles have sides
that from two pairs of opposite rays.
Warm-Up
Exercises
GUIDED
PRACTICE
7.
For Examples 4 and 5
The measure of an angle is twice the measure of
its complement. Find the measure of each angle.
ANSWER
60°,
30°
Daily
Homework
Quiz
Warm-Up
Exercises
1.
1 and
find m
ANSWER
2 are supplementary. If m
2.
o
1 = 97 ,
o
83
o
2.
3 and 4 are complementary angles. If m
Find m 4.
ANSWER
16
o
3= 74,
Daily
Homework
Quiz
Warm-Up
Exercises
3.
Find m
ANSWER
4.
Find m
ANSWER
EFH.
96
o
ABC.
36
o
Daily
Homework
Quiz
Warm-Up
Exercises
5.
Is it possible to draw a figure that contains
exactly one pair of vertical angles? Explain.
ANSWER
No; once you have drawn a pair of vertical angles, you
have drawn two pairs of opposite rays. This
automatically gives another pair of vertical angles.