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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