3.2 Properties of Parallel Lines

Ms. Kelly

Fall 2010

Standards/Objectives:

• Objectives:

• State and apply a postulate or theorems about parallel lines

Postulate 10 Corresponding Angles Postulate

• If two parallel lines are cut by a transversal, then corresponding angles are congruent.

1

2

1 ≅ 2

Theorem 3.2 Alternate Interior Angles

• If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

3

4

3 ≅ 4

Theorem 3.3 Same-Side Interior Angles

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

5

6

5 + 6 = 180°

Alternate Exterior Angles

• If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.

7

8

7 ≅ 8

Theorem 3.4 Perpendicular Transversal

• If a transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other.

j k

j

h

k

Example 1: Proving the Alternate Interior Angles Theorem

• Given: p ║ q

• Prove: 1 ≅ 2

1

2

3

Proof

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.

• A. m 6 B. m 7

• C. m 8 D. m 9

6

75

8

9

Solutions:

a. m 6 = m 5 = 65°

b. m 7 = 180° - m 5 =115°

c. m 8 = m 5 = 65°

d. m 9 = m 7 = 115°

Ex. 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?

120°

j k

1

Solution

1. m 1 + 120° = 180°

2. m 1 = 60°

1. Consecutive Interior angles Theorem

2. Subtraction POE

Ex. 4: Using properties of parallel lines

• Use the properties of parallel lines to find the value of x.

125°

4(x + 15)°

3.2 Day 2In the four squares below, 4 of the 5 theorems/postulates will be used heavily for proofs

Postulate 10 Theorem 3-2

Theorem 3-3 Theorem 3-4

12 21

12

2

1

Let’s review Example 1: Theorem 3-2 Proving the Alternate Interior Angles Theorem

• Given: p ║ q

• Prove: 1 ≅ 2

1

2

3

Proof

Statements:

1. p ║ q

2. 1 ≅ 3

3. 3 ≅ 2

4. 1 ≅ 2

Reasons:

1. Given

2. Corresponding Angles Postulate (Postulate 10)

3. Vertical Angles Theorem (Theorem 2-3)

4. Transitive Property of Congruence

You try (we try):

Given: K || n; transversal t cuts k and n.

Prove: <1 is supplementary to <4

1

4 2

Solution

Let’s use what we know about our theorems

Statements Reasons

1. k || n; transversal t cuts k and n 1. Given

2. 1 ≅ 2 2. Theorem 3-2 (alt. int. angles)

3. 4 + 2 = 180 3. Angle Addition Postulate

4. 4 + 1 = 180 4. Substitution Prop

5. 4 is supplementary to 1 4. Def. of supplementary angles

Open your book to page 80

• Complete 2 through 9

• Your word bank:– Post 10– Thm 3-2– Thm 3-3– Thm 3-4– Vertical Angles thm

Complete on your own

• #20 and #21 on page 82

• Ask yourself the following questions:– What am I proving (what kind of angles are

they)?– How do I get there using the other theorems

and postulates?

Now onto algebraic examples!!!!!Review of page 80 10-13

10.Angles 4, 5, 8 = 130; angles 2, 3, 6, 7 = 50

11.Angles 4, 5, 8 = x; angles 2, 3, 6, 7 = 180-x

12. 60

13. 100

In the next few examples, the markings are the most important thing when it comes to finding the angle values!

Algebraic Example 1

Algebraic Example 2

Algebraic Example 3

Algebraic Problems – you try

Closure – will be collected and gradedOn a small piece of paper, answer the

following:

1.What theorem discusses same-side interior angles that are supplementary?

2.Postulate 10 discusses…….

3.What theorem discusses alternate interior angles?

4.Solve:

Groupwork

• Please complete the worksheet in your group and hand in for a grade. Then you may start your homework.

Homework

• Page 81 8-12, 15, 16