3'3 Parallel Lines

1
3.3 Parallel Lines Transversals

• Mrs. Spitz
• Fall 2005

2
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
  Earths circumference

3
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.

4
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
5
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
6
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
7
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
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
h
k
j ? k
9
Example 1 Proving the Alternate Interior Angles
Theorem

• Given p q
• Prove ?1 ? ?2

1
2
3
10
Proof

• Statements
• p q
• ?1 ? ?3
• ?3 ? ?2
• ?1 ? ?2

• Reasons
• Given
• Corresponding Angles Postulate
• Vertical Angles Theorem
• Transitive Property of Congruence

11
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

9
6
8
5
7
12
Solutions

• m ?6 m ?5 65
• Vertical Angles Theorem
• m ?7 180 - m ?5 115
• Linear Pair postulate
• m ?8 m ?5 65
• Corresponding Angles Postulate
• m ?9 m ?7 115
• Alternate Exterior Angles Theorem

13
Ex. 3Classifying Leaves

• BOTANYSome plants are classified by the
  arrangement of the veins in their leaves. In the
  diagram below, j k. What is m ?1?

k
j
1
120
14
Solution

• m ?1 120 180
• m ?1 60

• Consecutive Interior angles Theorem
• Subtraction POE

15
Ex. 4 Using properties of parallel lines

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

125
4
(x 15)
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Proof

• Statements
• m?4 125
• m?4 (x15)180
• 125(x15) 180
• x 40

• Reasons
• Corresponding Angles Postulate
• Linear Pair Postulate
• Substitution POE
• Subtraction POE

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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!!!!