Download 3-1 PPT Properties of Parallel Lines

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
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
The Role of a Transversal
Transversal:
• line that intersects two coplanar
lines at two distinct points.
Pairs of eight angles have special names:
Properties of Parallel Lines
Red arrows indicate parallel lines
Properties of Parallel Lines
Name all the pairs of corresponding angles.
1  5
3  7 2  6
4  8
Are they congruent? Why?
Name all the pairs of Alternate Interior Angles.
3  6
4  5
Are they congruent? Why?
Name all the pairs of Same Side Interior Angles.
3  5
4  6
What is significant about same side interior angles?
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
Given
a||b
1  4
4  3
1  3
If lines are ||, then corresponding angles are congruent.
Vertical angles are congruent.
Transitive Property of Congruence.
 1  3.
Proof of Theorem 3-2: Same Side Interior Angles
If 2 lines are parallel and cut by a transversal, then sameside 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
m1 = m3
Angle Addition Postulate
Corresponding Angles are 
m2 + m1 = 180 Substitute
1 and 2 are suppl. Def of supplementary angles
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
Finding Measures of Angles
Find the measure of the angles
by identifying which postulate
or theorem justifies each
answer.
m1
m2
m3
m4
m5
m1 = 50 Corresponding Angles
m2 = 130 Same Side Interior
m3 = 130 Corresponding Angles
m7
m4 = 130 Vertical Angles are 
m5 = 50 Vertical Angles are 
m6 = 50 Alt Interior Angles are 
m7 = 130 Same side interior angles
m8
m8 = 50 Vertical angles are 
m6
Using Algebra to Find Measures of Angles
Find the values of x and y.
What is the value of x?
x = 70
Corr. Angles
What is the value of y?
70 + 50 + y = 180
y = 60
Angle Addition Postulate
Subtraction Prop. Of Eq
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
x = 45
Same side interior angles = 180
Sub and Div Property of Equality
What is the relation between y and (y - 50)?
y + (y - 50) = 180
2y - 50 = 180
y = 115
Same side interior angles = 180
Simplify
Add and Div Property of Equality