3.3 Parallel Lines & Transversals

Standards/Objectives:

Standard 3: Students will learn and apply geometric concepts.

Objectives:

• Prove and use results about parallel lines and transversals.

• Use properties of parallel lines to solve real-life problems, such as estimating the Earth’s circumference

Homework

• Pgs. 146-148 #1-30

• REMINDER: There is a quiz after 3.3. If you want a glance at what it kind of looks like, check out pg. 149. You will be doing this for homework next class meeting.

Postulate 15 Corresponding Angles Postulate

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

1

2

1 ≅ 2

Theorem 3.4 Alternate Interior Angles

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

3

4

3 ≅ 4

Theorem 3.5 Consecutive Interior Angles

• If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

5

6

5 + 6 = 180°

Theorem 3.6 Alternate Exterior Angles

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

7

8

7 ≅ 8

Theorem 3.7 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. Tell which postulate or theorem you use.

• A. m 6 B. m 7

• C. m 8 D. m 9

6

75

8

9

Solutions:

a. m 6 = m 5 = 65°• Vertical Angles Theorem

b. m 7 = 180° - m 5 =115°• Linear Pair postulate

c. m 8 = m 5 = 65°• Corresponding Angles Postulate

d. m 9 = m 7 = 115° • Alternate Exterior Angles Theorem

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)°

Proof

Statements:

1. m4 = 125°

2. m4 +(x+15)°=180°

3. 125°+(x+15)°= 180°

4. x = 40°

Reasons:

1. Corresponding Angles Postulate

2. Linear Pair Postulate

3. Substitution POE

4. Subtraction POE

NOTE:

• You must show all your work. Check your syllabus . . . it tells you everything I expect. We are moving into the next quarter shortly, and I expect that your work will be even more professional, neat, organized, and will show even at a casual glance that you did your homework. IF IT EVEN LOOKS COPIED . . . NO CREDIT!!!!