1. Section 3-1 Properties of Parallel Lines SPI 32E: solve problems involving complementary, supplementary, congruent, vertical or adjacent angles given measures expressed algebraically
Objectives:
Identify angles formed by two lines and a transversal
Prove and use properties of parallel lines
Recall Vocabulary
Complementary – sum of angles equal 90º
Supplementary – sum of angles equal 180º
Vertical Angles – are congruent
Adjacent Angles – angle that share a side and vertex

2. The Role of a Transversal
line that intersects two coplanar lines at two distinct points.
Pairs of eight angles have special names:

3. Properties of Parallel Lines
Red arrows indicate parallel lines

4. Properties of Parallel Lines crosses PARALLEL lines!
Theorems Apply
ONLY
when a transversal
crosses PARALLEL lines!
Name all the pairs of corresponding angles.
Corresponding Angles
Postulate
1  5 3  7 2  6 4  8
Are they congruent? Why?
Name all the pairs of Alternate Interior Angles.
Alternate Interior
Angles Theorem
3  6 4  5
Are they congruent? Why?
Name all the pairs of Same Side Interior Angles.
Same side interior
angles are
supplementary
3  5 4  6
What is significant about same side interior angles?

5. Proof of Theorem 3-1: Alternate Interior Angles
If a transversal intersects two parallel lines, then alternate interior angles are congruent.
Given: a||b
Prove: 1  3
a||b
Given
 1  3.
1  4
If lines are ||, then corresponding angles are congruent.
4  3
Vertical angles are congruent.
1  3
Transitive Property of Congruence.

6. Proof of Theorem 3-2: Same Side Interior Angles
If 2 lines are parallel and cut by a transversal, then same-side interior angles are supplementary.
Given: a||b
Prove: 1 and 2 are supplementary
Make a plan:
Prove 1 + 2 = 180
Show 2 + 3 = 180
Show m 1 = m 3
Substitute
a||b
Given
2 +3 = 180
Angle Addition Postulate
m1 = m3
Corresponding Angles are 
m2 + m1 = 180
Substitute
1 and 2 are suppl.
Def of supplementary angles

7. Real-world Connection
This is an airport diagram of Pompano Beach Air Park in Florida.
How can the following help construct an airport runway?
Alternate angles
Corresponding angles
Same side interior angles

8. Finding Measures of Angles
Find the measure of the angles by identifying which postulate or theorem justifies each answer.
m1
m1 = 50 Corresponding Angles
m2
m2 = 130 Same Side Interior
m3
m3 = 130 Corresponding Angles
m4
m4 = 130 Vertical Angles are 
m5
m5 = 50 Vertical Angles are 
m6
m6 = 50 Alt Interior Angles are 
m7
m7 = 130 Same side interior angles
m8
m8 = 50 Vertical angles are 

9. Using Algebra to Find Measures of Angles
Find the values of x and y.
What is the value of x?
x = Corr. Angles
What is the value of y?
y = Angle Addition Postulate
y = Subtraction Prop. Of Eq

10. Using Algebra to Find Measures of Angles
Find the values of x and y. Then find the measures of the angles.
What relationship for x do you notice by looking at the diagram?
2x + 90 = 180
Same side interior angles = 180
x = Sub and Div Property of Equality
What is the relation between y and (y - 50)?
y + (y - 50) = 180
Same side interior angles = 180
2y - 50 = 180
Simplify
y = 115
Add and Div Property of Equality