Transcript CPG 3.3 - Proving Lines Parallel

Section 3.3: Proving Lines are Parallel
1. Review of the Parallel Lines Postulate & Theorems.
2. Converses of Parallel Lines Postulate & Theorems
3. Proof of the Converse of the Alt Int Angles Theorem
4. Two more ways to prove lines are parallel
5. Example 1
6. Example 2
7. Parallel & Perpendicular Through a Point Theorems
HW: Pg. 87 #1-15 odd, 19, 25, 27, 29
MK: 3.3 Makeup Homework from website
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1. Review: || Line Postulate & Theorems
t
a
b
When you know the lines are parallel…
Corr 's Post:
If the lines are || then corr 's are  .
Alt int 'sThm: If the lines are || then alt int 'sare  .
Ss int 's Thm: If the lines are || then ss int 's are supplementary.
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2. Converses of Parallel Lines Postulate & Theorems
t
a
b
When you don’t know the lines are parallel…
Converse of Corr 'sPost: If corr 's are  then the lines are ||.
Converse of Alt int 'sThm: If alt int 's are  then the lines are ||.
Converse of Ss int 's Thm: If ss int 's are supplementary
then the lines are ||.
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3. Proof of the Converse of the Alt Int Angles Theorem
If alternate interior angles are congruent, then the lines are ||.
t
Given: 1  2
3
Prove: a || b
a
2
1
b
Proof:
Statements:
Reasons:
1. 1  2
2. 2  3
3. 1  3
1. Given
2. Vertical Angles Theorem
3. Transitive Property of Congruence
4. a || b
4. Converse of Corr Angles Post.
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4. Two more ways to prove lines are parallel
a
b
 to Same Line Theorem (3.7):
c
If 2 lines in a plane are  to the same line, then those lines are ||.
k || w & p || w :
k
p
w
|| to Same Line Theorem (3.10):
If 2 lines are || to the same line, then those lines are ||.
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5. Example 1
a
Find the value of x that would make a || b.
3
46
b
4x+10
4
1. Angles of interest: 3& 4
2. They are ss int ' s.
3. If ss int are supplementary then the lines are ||.
so m3  m4  180
46 + (4x+10) = 180
4x + 56 = 180
4x = 124
x = 31
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6. Example 2
If 1  2 which lines are || ?
a
b
1
c
2
d
1. Put a dot on both sides of each angle.
2. Highlight all lines with a dot.
3. The transversal has 2 dots; the lines each have one.
Since corr 's (1& 2) are  c || d.
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7. Parallel & Perpendicular Through a Point Theorems
parallel
perpendicular
|| Thru a Point Theorem (3.8):
Through a point not on a line, there exists exactly one line
|| to the given line.
 Thru a Point Theorem (3.9):
Through a point not on a line, there exists exactly one line 
to the given line.
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3.3 Summary
The 5 ways to prove that lines are parallel:
1.
2.
3.
4.
5.
Show a pair of corresponding angles are congruent (11)
Show a pair of alternate interior angles are congruent (3.5)
Show a pair of same-side interior angles are supplementary (3.6)
Show that both lines are perpendicular to a 3rd line (3.7)
Show that both lines are parallel to a 3rd line (3.10)
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3.3 Homework Index
HW: Pg. 87 #1-15 odd, 19, 25, 27, 29
MK: 3.3 Makeup Homework from website
1 - 16
18 - 19
25 - Proof
27 - 29
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3.3 Homework, p. 87 (1 - 16)
Use the information given to name the ||
segments. If there are no || segments, write none.
1. 2  9
2. 6  7
3. m1  m8  90
4. 5  7
5. m2  m5
6. 3  11
7. m1  m4  90
8. m10  m11
9. m8  m5  m6  180
13. 2 & 3 are compl. & m1  90
10. FC  AE & FC  BD
14. m2  m3  m4
11. m5  m6  m9  m10
15. m7  m3  m10
12. 7 & EFB are suppl.
16. m4  m8  m1
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3.3 Homework, p. 87 (18 - 19)
Find the values of x & y that make the red lines parallel & the blue lines
parallel.
18.
19.
(x - 40) (x + 40)
y
3x
105
2y
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3.3 Homework, p. 87 (25)
C
25.
Given: BE  DA;CD  DA
B
Prove: 1  2
1
D
Proof:
Statements
1. BE  DA;CD  DA
Reasons
1. Given
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
2 3
A
E
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3.3 Homework, p. 87 (27 - 29)
27.
28.
Y
X
R
40
X
S
120 T
70
T
110
R
Y
S
29. Find the values of x & y that make the lines shown in red parallel.
30
5y
2x(x - y)
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CP Geometry Homework Quiz
Section:
Period:
Date:
Name:
Answers:
1
2
Box 1
Box 2
Box 3
Your answer
to question 4
3
4
5
6
Instructions:
All red fields are required. Name must be FIRST & LAST. One point deduction if
anything missing.
Boxes are for showing work. (If calculations are required, write the formula first.)
Put all answers in the spaces to the right. If an answer does not fit, put it in a Box
& draw an arrow to it (as showing in the example above.)
Do not copy the problem or drawing from the board onto your HWQ form.
IF YOU WERE ABSENT: Fill in all red fields; write “ABSENT on <date you were
absent>” in Box 1. If you do not have one, ask for a Makeup Form. H Q
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CP Geometry Homework Quiz 3.3A
Questions 1-4. Write the letter that
indicates which segments must be || if…
1. 2  5
2. 3  11
3. 9  5
4. 7 sup p EFB
A. AB || FC
B. AE || BD
C. FB || EC
D. None of these
Questions 5 & 6. Find the values of x &
y that make the red & blue lines ||
6x
96 4y
x
1.
2.
3.
4.
5.
6.
Letter
Letter
Letter
Letter
x=
y=
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CP Geometry Homework Quiz 3.3B
Questions 1-4. Write the letter that
indicates which segments must be || if…
1. 9  5
2. 3  11
3. 2  5
A. AB || FC
B. AE || BD
C. FB || EC
4. 2 comp FBC & mEAB  90 D. None of these
Questions 5 & 6. Find the values of x &
y that make the red & blue lines ||
4x
96 4y
x
1.
2.
3.
4.
5.
6.
Letter
Letter
Letter
Letter
x=
y=
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CP Geometry Homework Quiz 3.3C
Questions 1-4. Write the letter that
indicates which segments must be || if…
1. 3  11
2. 2  5
3. 9  5
A. AB || FC
B. AE || BD
C. FB || EC
4. 2 comp FBC & mEAB  90 D. None of these
Questions 5 & 6. Find the values of x &
y that make the red & blue lines ||
3x
96 4y
x
1.
2.
3.
4.
5.
6.
Letter
Letter
Letter
Letter
x=
y=
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