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3.2 Use Parallel Lines and Transversals
Objectives
• Use the properties of
parallel lines to determine
congruent angles
• Use algebra to find
angle measures
Postulate 15 Corresponding s Postulate
• If 2 lines are cut by a transversal, then each pair of corres. s is .
• i.e. If l m, then 12.
l
m
1
2
Theorem 3.1 Alternate Interior s Theorem
• If 2 lines are cut by a transversal, then each pair of alternate interior s is .
• i.e. If l m, then 12.
l
m
1
2
Theorem 3.2 Alternate Exterior s Theorem
• If 2 lines are cut by a transversal, then the pairs of alternate exterior s are .
• i.e. If l m, then 12.
l m
1
2
Theorem 3.3 Consecutive Interior s Theorem
• If 2 lines are cut by a transversal, then each pair of consecutive int. s is supplementary.
i.e. If l m, then 1 & 2 are supplementary or m1 + m2 = 180°.
l
m 1
2
• If a transversal is to one of 2 lines, then it is to the other.
• i.e. If l m, & t l, then t m.
1
2
Theorem 3.11 Transversal Theorem
l
m
t
EXAMPLE 1 Identify congruent angles
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°.
The measure of three of the numbered angles is 120°. Identify the angles. Explain your reasoning.
EXAMPLE 2 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.
Consecutive Interior Angles Theoremm 4 + (x+5)° = 180°
Substitute 115° for m 4.115° + (x+5)° = 180°
Combine like terms.x + 120 = 180
Subtract 120 from each side.x = 60
GUIDED PRACTICE for Examples 1 and 2
Vertical Angles Congruence Theorem.
Corresponding Angles Postulate.m 5 =105°
Alternate Exterior Angles Theoremm 8 =105°
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.
m 4 = 105°
ANSWER
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.
m 3 = m 7
68 + 2x + 4 = 180
2x + 72 = 180
2x = 108
x = 54
m 7 + m 8 = 180ANSWER
EXAMPLE 3 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
EXAMPLE 3 Prove the Alternate Interior Angles Theorem
STATEMENTS REASONS
p q1. 1. Given
2. 1 ∠ 3 2. Corresponding Angles Postulate
3. 3 ∠ 2 3. Vertical Angles Congruence Theorem
4. 1 ∠ 2 Transitive Property of Congruence
4.
EXAMPLE 4 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?
EXAMPLE 4 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°.
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.
Yes; 3 and 2 congruence is not dependent on the congruence of 1 and 3.
SAMPLE ANSWER
GUIDED PRACTICE for Examples 3 and 4
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?
4. WHAT IF?
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°.
ANSWER
Assignment
Pages 181–183 12-22 even; 24 – 29 all; 31-34,36,38,39
Draw diagrams for 38 and 39
