3.23.2 Use Parallel Lines and TransversalsBell Thinger
ANSWER alternate interior
1. 5, 7
2. 3, 6
3. 1, 8
ANSWER alternate exterior
ANSWER corresponding
Identify the type of angles.
3.2
3.2Example 1
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.
3.2
3.2Example 2ALGEBRA
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
3.2Guided Practice
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.
Vertical Angles Congruence Theorem.
Corresponding Angles Postulate.m 5 =105°
Alternate Exterior Angles Theoremm 8 =105°
m 4 = 105°
ANSWER
3.2
Use the diagram.
If m 3 = 68° and m 8 = (2x + 4)°, what is the value of x? Show your steps.
2.
Guided Practice
m 3 = m 7
68 + 2x + 4 = 180
2x + 72 = 180
2x = 108
x = 54
m 7 + m 8 = 180
ANSWER
3.2Example 3
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
3.2
GIVEN : p q
PROVE : 1 2
STATEMENTS REASONS
p q1. 1. Given
2. Corresponding Angles Postulate
3. Vertical Angles Congruence Theorem
Transitive Property of Congruence
4.
Example 3
2. 1 3
3. 3 2
4. 1 2
3.2Example 4 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?
Science
SOLUTIONBecause 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°.
3.2Guided Practice
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
3.2
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?
Guided Practice
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
3.2Exit Slip
What theorem justifies each statement?
2. 4 and 6 are supplementary.
3. If m 2 = 115 , find m 7.o
115ANSWERo
Alt. Interior Thm.sANSWER
Consec. Interior Thm.s
ANSWER
1. 3 6
3.24. Find the values of x and y.
17.5, 35ANSWER
Exit Slip
3.2
82ANSWERo
5. The figure shows a plant trellis .
If m 1 = 82o , find m 2.
Exit Slip
3.26. If m 1 = 105o , find all other angle measures.
3.2
Homework
Pg 161-164#22, 23, 24, 31, 41