3-1 Properties of Parallel Lines Objectives: 1) Identify angles formed by two lines and a transversal.

2) Prove and use properties of parallel

Objectives: 1) Identify angles formed by two lines and a transversal. 2) Prove and use properties of parallel.

5. The sum of m1 and twice its complement is 146. Find m1.

6. The measures of two supplementary angles are in the ratio 2:3.

Find the measures.

m1 = 34°

72° and 108°

Objectives: 1) Identify angles formed by two lines and a transversal. 2) Prove and use properties of parallel.

Lesson Notes: pages 49-51 (textbook pages 127-130)

A transversal is ___________________

________________________________

a line that intersects

two coplanar lines at two distinct points.

l

m

t

Alternate interior angles are ___________________

______________________________________

Same-side interior angles are _____________

______________________________________

Corresponding angles are _________________

______________________________________

______________________________________

nonadjacent interior s

that lie on opposite sides of the transversal.

interior angles

that lie on the same side of the transversal.

s that lie on the same

side of the transversal and in corresponding

positions relative to the coplanar lines.

1

2

3

4

5 6

7 8

5, 6, 7, and 8 are called exterior angles.

A two-column proof is a display that shows the steps

to prove a theorem. The first column shows the steps

and the second column shows the reason for each step.

Objectives: 1) Identify angles formed by two lines and a transversal. 2) Prove and use properties of parallel.

The same-size angles that appear to be formed by the vapor trails suggest the postulate and theorems below.

Objectives: 1) Identify angles formed by two lines and a transversal. 2) Prove and use properties of parallel.

4 5

m4 + m6 = 180

Applying Properties of Parallel Lines

In the diagram of Lafayette Regional Airport, the black segments are runways and the gray areas are taxiways and terminal buildings. Compare 2 and the angle vertical to 1. Classify the angles as alternate interior angles, same-side interior angles, or corresponding angles.

alternate interior angles

alternate interior angles

Objectives: 1) Identify angles formed by two lines and a transversal. 2) Prove and use properties of parallel.

Finding Measures of Angles

corresponding angles

42°

Corresponding Angles Postulate

180° Angle Addition Postulate

42° 42° m2 180°

42° 138°

65° Alternate Interior Angles

40° Alternate Interior Angles

180° Angle Addition

65° 40° 180°

75° ( or APE)

Objectives: 1) Identify angles formed by two lines and a transversal. 2) Prove and use properties of parallel.

2 and 3 are same-side interior angles.

m3= 42°; by the Alternate Exterior s Theorem.

m4= 138°; by the Corresponding s Postulate

m5= 138°; by the Vertical s Theorem

=42°

138° m6= 42°; by the Corresponding s Postulate

m7= 138°; by the Alternate Interior s Theorem

m8= 138°; by the Vertical s Theorem.

y + y – 50 = 180

2y = 230

y = 115

2x = 90

x = 45

115°,90°, 90°, and 65°

(3 and 42°-angle)

(4 and 2)

(5 and 4)

(6 and 42°-angle)

(7and 4)

(8 and 7)

2x + 90 = 180

42°

138°=

=42°

138°

138°=

Objectives: 1) Identify angles formed by two lines and a transversal. 2) Prove and use properties of parallel.

Practice 3-1 Properties of Parallel Lines

Find m1 and then m2. Justify each answer.

m1= 100° ; Alternate Interior s Theorem

m2= 100° ; Vertical s Theorem m1= 75° ; Alternate Interior s Theorem

m2= 135° ; Alternate Exterior s Theorem

m2= 75° ; Corresponding s Postulate

m1= 135° ; Corresponding s Postulate

Objectives: 1) Identify angles formed by two lines and a transversal. 2) Prove and use properties of parallel.

Algebra Find the value of x. Then find the measure of each angle.

(using Same-side Interior s Theorem)

x + x - 26 = 180°

2x = 206°

x = 103°

= 103°

= 77°

(using Same-side Interior s Theorem)

18072

+ xx

1802

15

x

15

2

15

2

x = 24°

(using Corresponding s Postulate)

x + 55 = 3x - 5

60 = 2x

x = 30°

= 12°

= 168° = 85°

= 85°

Practice 3-1 Properties of Parallel Lines

Objectives: 1) Identify angles formed by two lines and a transversal. 2) Prove and use properties of parallel.

Practice 3-1 Properties of Parallel Lines

(Proof for the Alternate Interior s Theorem) 1 ≅ 2

1 ≅ 2

1 ≅ 3 Corresponding Angles Postulate

Vertical Angles Theorem

Substitution Property

Objectives: 1) Identify angles formed by two lines and a transversal. 2) Prove and use properties of parallel.

Page 132 textbook

STATEMENTS REASONS

1. a ⃦ b , 1 ≅ 4 1. Given

2. 1 & 2 are supplementary s.

2. Same-side Interior s Theorem

3. 3 & 4 are supplementary s.

3. Same-side Interior s Theorem

4. 2 ≅ 3 4. Congruent Supplements Theorem

Supplements of ≅ angles (or of the same angle) are ≅.

If a transversal intersects two parallel lines, then same-side interior angles are supplementary.

STATEMENTS REASONS

1. a ⃦ b 1. Given

2. Corresponding s

Postulate 3. 3 & 2 are supplementary s. 3. Linear Pair Postulate

4. 3 & 2 are supplementary s.

4. Substitution Property

Linear pair is a pair of two adjacent s that form a straight line.

Linear Pair Postulate: If two s form a linear pair, then they are supplementary.

2. 1 ≅ 3

Properties of parallel lines can be summarized as follows:

If two parallel lines are cut by a transversal, then…

We use these statements (postulate or theorem) to prove angles are either congruent or supplementary.

•same-side interior (or same-side exterior) angles are supplementary.

•alternate interior (or alternate exterior) angles are congruent.

•corresponding angles are congruent.

Quiz follows…