Warm-Up Exercises
Identify the type of angles.
1.
5,
7
corresponding
ANSWER
2.
3,
6
alternate interior
ANSWER
3.
1,
ANSWER
8
alternate exterior
Warm-Up1Exercises
EXAMPLE
Identify congruent angles
The measure of three of the numbered angles is 120°.
Identify the angles. Explain your reasoning.
SOLUTION
By the Corresponding Angles Postulate, m 5 = 120°.
Using the Vertical Angles Congruence Theorem,
m 4 = 120°. Because 4 and 8 are corresponding
angles, by the Corresponding Angles Postulate, you
know that m 8 = 120°.
Warm-Up2Exercises
EXAMPLE
Use properties of parallel lines
ALGEBRA
Find the value of x.
SOLUTION
By the Vertical Angles Congruence Theorem,
m 4 = 115°. Lines a and b are parallel, so you can
use the theorems about parallel lines.
m
4 + (x+5)° = 180°
115° + (x+5)° = 180°
x + 120 = 180
x = 60
Consecutive Interior Angles Theorem
Substitute 115° for m
4.
Combine like terms.
Subtract 120 from each side.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
Use the diagram.
1. If m 1 = 105°, find m 4, m 5, and m 8. Tell
which postulate or theorem you use in each
case.
ANSWER
m
4 = 105°
Vertical Angles Congruence Theorem.
m
5 =105°
Corresponding Angles Postulate.
m
8 =105°
Alternate Exterior Angles Theorem
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
Use the diagram.
2. If m 3 = 68° and m 8 = (2x + 4)°, what is the
value of x? Show your steps.
ANSWER
m
7+m
m
8 = 180
3= m
68 + 2x + 4 = 180
2x + 72 = 180
2x = 108
x = 54
7
Warm-Up3Exercises
EXAMPLE
Prove the Alternate Interior Angles Theorem
Prove that if two parallel lines are cut by a transversal,
then the pairs of alternate interior angles are congruent.
SOLUTION
Draw a diagram. Label a pair of alternate interior
angles as 1 and 2. You are looking for an angle
that is related to both 1 and 2. Notice that one
angle is a vertical angle with 2 and a corresponding
angle with 1. Label it 3.
GIVEN :
p q
PROVE :  1  2
Warm-Up3Exercises
EXAMPLE
Prove the Alternate Interior Angles Theorem
STATEMENTS
1.
p
q
REASONS
1. Given
Corresponding
Angles Postulate
2.
1  3
2.
3.
3  2
3. Vertical Angles Congruence
4.
1  2
Theorem
4.
Transitive Property of
Congruence
Warm-Up4Exercises
EXAMPLE
Solve a real-world problem
Science
When sunlight enters a drop of rain, different colors
of light leave the drop at different angles. This
process is what makes a rainbow. For violet light,
m 2 = 40°. What is m 1? How do you know?
Warm-Up4Exercises
EXAMPLE
Solve a real-world problem
SOLUTION
Because the sun’s rays are parallel, 1 and 2 are
alternate interior angles. By the Alternate Interior
Angles Theorem, 1
2. By the definition of
congruent angles, m 1 = m 2 = 40°.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
3. In the proof in Example 3, if you use the third
statement before the second statement, could
you still prove the theorem? Explain.
SAMPLE ANSWER
Yes; 3 and 2 congruence is not dependent on
the congruence of 1 and 3.
Warm-Up
Exercises
GUIDED
PRACTICE
4.
for Examples 3 and 4
WHAT IF?
Suppose the diagram in Example 4 shows yellow
light leaving a drop of rain. Yellow light leaves the
drop at an angle of 41°. What is m
1 in this
case? How do you know?
ANSWER
41°;
1 and 2 are alternate interior angles. By the
Alternate Interior Angles Theorem, 1
2. By the
definition of congruent angles, m 1 = m 2 = 41°.
Daily
Homework
Quiz
Warm-Up
Exercises
What theorem justifies each statement?
1.
3
6
ANSWER
2.
4 and
ANSWER
3. If m
Alt. Interior
Thm.
6 are supplementary.
Consec. Interior
o
2 = 115 , find m
ANSWER
s
115
o
7.
s
Thm.
Daily
Homework
Quiz
Warm-Up
Exercises
4. Find the values of x and y.
ANSWER
17.5, 35
Daily
Homework
Quiz
Warm-Up
Exercises
5. The figure shows a plant trellis .
If m
o
1 = 82 , find m
ANSWER
82
o
2.