3-1 PROPERTIES OF PARALLEL LINES

SWBAT:• Identify angles formed by two lines and a

transversal• Prove and use properties of parallel lines

Transversal• A transversal is a line that

intersects two coplanar

lines at two distinct

points

• What is the transversal here?• Line t

• Eight angles are formed by a transversal intersecting two coplanar lines

Corresponding Angles

Angles that lie on the same side of the transversal and in corresponding positions relative to the two intersected lines

(Think Same Quadrants in coordinate plane)• Which angles here are

corresponding?

Alternate Interior Angles

Alternate Interior Angles are non adjacent interior angles that lie on opposite sides of the transversal.• Which angles are alternate

interior? • <1 and <2• <3 and <4

Same Side (Consecutive) Interior Angles

Angles that lie on the same side of the transversal between the two lines intersected by the transversal• Which angles fit this

description?• <1 and <4 • <3 and <2

Alternate Exterior Angles

Alternate Exterior Angles are non adjacent exterior angles that lie on opposite sides of the transversal.

These angles will lie outside the two lines.

• Which angles are alternate exterior?

• <5 and <8• <7 and <6

Same Side (Consecutive) Exterior Angles

Angles that lie on the same side of the transversal outside the two lines intersected by the transversal• Which angles fit this

description?• <5 and <7 • <6 and <8

Finding Angle Pairs

Special Relationships with Parallel Lines!

Parallel Lines: Two coplanar lines that do not intersect.

When the two coplanar lines that are intersected by the transversal are parallel, special relationships exist between the different angle types.

Let’s explore what these relationships are…

• In the diagram above, the m<1 = 110°• Find the measure of the other 7 angles.

Example 1

Example 1B

Remember…

•A Postulate is an accepted statement of fact…

•Theorems are created by using Postulates to prove them.

Two Column Proof

Proof of Theorem 3-1:Given: a || b

 

Prove: <1 ≅ <3

 

Statements Reasons

1.) a || b 1.

  

2 ) <1 ≅ <4 2.

  

3.) <4 ≅ <3 3.

  

4.) <1 ≅ <3 4.

 

Given

Corresponding Angles

Substitution

Vertical Angles

The above Proof shows that by using the Corresponding Angle Postulate, that Alternate Interior Angles are Congruent!

Proof of Theorem 3-2:Given: a || b

 

Prove: m<1 + m<2 = 180

 

Statements Reasons

1.) a || b 1.

  

2 ) m<3 + m<2 = 180 2.

  

3.) <3 ≅ <1 3.

  

4.) m<1 + m<2 = 180 4.

 

Given

Supplementary Angles

Corresponding Angles

Substitution

The above Proof shows that by using the Corresponding Angle Postulate, that Same Side Interior Angles are Supplementary!

Ex 2:1)

Proof of Theorem 3-3: Given: a || b

 

Prove: <1 ≅ <4

 

Statements Reasons

1.) a || b 1.

  

2 ) <1 ≅ <2 2.

  

3.) <2 ≅ <4 3.

  

4.) <1 ≅ <4 4.

 

Given

Corresponding Angles

Substitution

Vertical Angles

The above Proof shows that by using the Corresponding Angle Postulate, that Alternate Exterior Angles are Congruent!

Proof of Theorem 3-4:Given: a || b

 

Proof: m<1 + m<2 = 180

 

Statements Reasons

1.) a || b 1.

  

2 ) m<3 + m<2 = 180 2.

  

3.) <3 ≅ <1 3.

 

 4.) m<1 + m<2 = 180 4.

 

Given

Supplementary Angles

Corresponding Angles

Substitution

The above Proof shows that by using the Corresponding Angle Postulate, that Same Side Exterior Angles are Supplementary!

Example 3

Find the Measure of Each Angle

Using Algebra to Find Angle Measures

Solve for x and y.

Solve For x. Find and Explain how to find measure of each angle.