chapter 3-5 proving lines parallel. lesson 3-5 ideas/vocabulary recognize angle conditions that...
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Chapter 3-5Chapter 3-5
Proving Lines ParallelProving Lines Parallel
Lesson 3-5 Ideas/Vocabulary
• Recognize angle conditions that occur with parallel lines.• Prove that two lines are parallel based on given angle
relationships.
Standard 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. (Key)Standard 16.0 Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line. (Key)
Transitive property of Parallels• If two lines are parallel to the same line, then
they are parallel to each other.• If p // q and q // r, then p // r.
p
q
r
Reminders from Section 1We will use these same theorems to
prove the lines are parallel given certain angle information.
Corresponding Angle TheoremIf two parallel lines are cut by a transversal, then corresponding
angles are congruent.
// lines corresponding s are
Corresponding Angle Theorem
Alternate Interior Angle TheoremIf two parallel lines are cut by a
transversal, then alternate interior angles are congruent.
// lines Alt. Int. s are
Alternate Interior Angle Theorem
Alternate Exterior Angle TheoremIf two parallel lines are cut by a
transversal, then alternate exterior angles are congruent.
// lines Alt. Ext. s are
Alternate Exterior Angle Theorem
Consecutive Interior Angle TheoremIf two parallel lines are cut by a
transversal, then consecutive interior angles are supplementary.
// lines Consec. Int. s are Supp.
Consecutive Interior Angle Theorem
1
2
m1 + m2 = 180
Two Theorem• If two lines are perpendicular to the same
line, then they are parallel to each other.• If m p and n p, then m // n.
p
m
n
Lesson 3-5 Postulates
Animation: Construct a Parallel Line Through a Point not on Line
Lesson 3-5 Theorems
Lesson 3-5 Example 1
Determine which lines, if any, are parallel.
Identify Parallel Lines
77oConsec. Int. s are supp.
a//bAlt. Int. s are not
a is not // c
Consec. Int. s are not supp.
b is not // c
A B
C D
0% 0%0%0%
Lesson 3-5 CYP 1
A. A
B. B
C. C
D. D
I only
II only
III only
I, II, and III
Determine which lines, if any are parallel.I. e || fII. e || gIII. f || g
ALGEBRA Find x and m ZYN so that || .
Lesson 3-5 Example 2
Solve Problems with Parallel Lines
Explore From the figure, you know that m WXP = 11x – 25 and m ZYN = 7x + 35. You also know that WXP and ZYN are alternate exterior angles.
Lesson 3-5 Example 2
If Alt. Ext. angles are , then the lines will be // m WXP = m ZYN Alternate exterior thm.11x – 25 = 7x + 35 Substitution4x – 25 = 35 Subtract 7x from each side.
4x = 60 Add 25 to each side.x = 15 Divide each side by 4.
ALGEBRA Find x and m ZYN so that || .
Lesson 3-5 Example 2
Solve Problems with Parallel Lines
Now use the value of x to find m ZYN.
Answer: x = 15, m ZYN = 140
m ZYN = 7x + 35 Original equation= 7(15) + 35 x = 15= 140 Simplify.
A B
C D
0% 0%0%0%
Lesson 3-5 CYP 2
A. A
B. B
C. C
D. D
ALGEBRA Find x so that || .x = 60
x = 9
x = 12
x = 12
Lesson 3-5 Example 3
Prove Lines Parallel
Prove: r || s
Given: ℓ || m
Lesson 3-5 Example 3
Prove Lines Parallel
2. 2. Consecutive Interior Angle Theorem
5. 5. Substitution 6. 6. Definition of supplementary
angles7. 7. If consecutive interior angles
theorem
1. 1. Given
Proof:Statements Reasons
4. 4. Definition of congruent angles
3. 3. Definition of supplementary
angles
Given x || y and , can you use theCorresponding Angles Postulate to prove a || b?
A. A
B. B
C. C
A B
C
0% 0%0%
Lesson 3-5 CYP 3
yes
no
not enough informationto determine
Lesson 3-5 Example 4
Determine whether p || q.
Slope and Parallel Lines
slope of p:
slope of q:
Answer: Since the slopes are equal, p || q.
A B
C
0% 0%0%
A. A
B. B
C. C
Lesson 3-5 CYP 4
Yes, r is parallel to s.
No, r is not parallel to s.
It cannot be determined.
Determine whether r || s.
HomeworkChapter 3-5• Pg 175
1 – 5, 7 – 19, 23 (proof), 24(proof), 37, 50 – 52
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