Download 3.3 Parallel Lines & Transversals

3.3 Parallel Lines
& Transversals
Objectives/Assignment
• Prove and use results about parallel
lines and transversals
• Use properties of parallel lines to solve
real-life problems
• Assignment: 1-29 all, Quiz page 149
Goal 1: Properties of Parallel Lines
Postulate 15 - Corresponding
Angles Postulate
• If two parallel lines are cut by a transversal,
then the pairs of corresponding angles are
congruent.
1
2
1 ≅ 2
Theorem 3.4 - Alternate Interior
Angles
• If two parallel lines are cut by a transversal,
then the pairs of alternate interior angles
are congruent.
3
4
3 ≅ 4
Theorem 3.5 - Consecutive Interior
Angles
• If two parallel lines are cut by a transversal,
then the pairs of consecutive interior angles
are supplementary.
5
6
5 + 6 = 180°
Theorem 3.6 - Alternate Exterior
Angles
• If two parallel lines are cut by a transversal,
then the pairs of alternate exterior angles
are congruent.
7
8
7 ≅ 8
Theorem 3.7 - Perpendicular
Transversal
• If a transversal is perpendicular to one of the
two parallel lines, then it is perpendicular to
the other.
j
h
k
jk
Example 1: Proving the Alternate
Interior Angles Theorem
• Given: p ║ q
• Prove: 1 ≅ 2
1
2
3
Proof
Given: p ║ q
Prove: 1 ≅ 2
1
2
3
Statements:
1. p ║ q
2. 1 ≅ 3
3. 3 ≅ 2
4. 1 ≅ 2
Reasons:
1. Given
2. Corresponding
Angles Postulate
3. Vertical Angles
Theorem
4. Transitive Property
of Congruence
Example 2: Using Properties of
Parallel Lines
• Given that m 5 = 65°, find each measure.
Tell which postulate or theorem you use.
• A. m 6
B. m 7
• C. m 8
D. m 9
9
6
5
7
8
Solutions:
a. m 6 = m 5 = 65°
•
9
8
6
Vertical Angles Theorem
7
b. m 7 = 180° - m 5 =115°
•
Linear Pair postulate
c. m 8 = m 5 = 65°
•
Corresponding Angles Postulate
d. m 9 = m 7 = 115°
•
Alternate Exterior Angles Theorem
5
Example 3 - Classifying Leaves
BOTANY—Some plants are classified by
the arrangement of the veins in their
leaves. In the diagram below, j ║ k. What
is m 1?
j
k
120° 1
Solution
1. m 1 + 120° = 180°
1. Consecutive Interior
angles Theorem
2. Subtraction POE
2. m 1 = 60°
k
j
120°
1
Goal 2: Properties of Special Pairs of Angles
Example 4: Using Properties of
Parallel Lines
• Use the properties of parallel lines to find
the value of x.
125°
4
(x + 15)°
Proof
Statements:
1. m4 = 125°
2. m4 +(x+15)°=180°
3. 125°+(x+15)°= 180°
4. x = 40°
Given
125°
4
(x + 15)°
Reasons:
1. Corresponding
Angles Postulate
2. Linear Pair Postulate
3. Substitution POE
4. Subtraction POE