Section 3.3: Proving Lines are Parallel Review of the Parallel Lines Postulate & Theorems. Converses of Parallel Lines Postulate & Theorems Proof of the Converse of the Alt Int Angles Theorem Two more ways to prove lines are parallel Example 1 Example 2 Parallel & Perpendicular Through a Point Theorems S Q H HW: Pg. 87 #1-15 odd, 19, 25, 27, 29 MK: 3.3 Makeup Homework from website
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Section 3.3: Proving Lines are Parallel 1. Review of the Parallel Lines Postulate & Theorems. 2. Converses of Parallel Lines Postulate & Theorems 3. Proof.
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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
If the lines are || then ss int are supplementary. ∠ 's
Corr Post:∠ 's
Alt int Thm:∠ 's
Ss int Thm:∠ 's
When you know the lines are parallel…
2. Converses of Parallel Lines Postulate & Theorems
SQH
If corr are then the lines are ||. ∠ 's ≅
When you don’t know the lines are parallel…
Converse of Corr Post:
Converse of Alt int Thm:∠ 's
Converse of Ss int Thm:∠ 's
∠ 's
If alt int are then the lines are ||. ∠ 's ≅
If ss int are supplementary∠ 'sthen the lines are ||.
a
b
t
3. Proof of the Converse of the Alt Int Angles Theorem
a
b
t
1
3
2
Given:
Prove:
∠1≅ ∠2
a || b
Statements: Reasons:
Proof:
1. ∠1≅ ∠2 1. Given
2. ∠2 ≅ ∠3 2. Vertical Angles Theorem
3. ∠1≅ ∠3 3. Transitive Property of Congruence
4. a || b 4. Converse of Corr Angles Post.
If alternate interior angles are congruent, then the lines are ||.
SQH
4. Two more ways to prove lines are parallel
|| to Same Line Theorem (3.10):
If 2 lines are || to the same line, then those lines are ||.
to Same Line Theorem (3.7):
If 2 lines in a plane are to the same line, then those lines are ||.
SQH
a
b
c
⊥
⊥
kpw
k || w & p || w :
5. Example 1
SQH
1. Angles of interest:
m∠3+m∠4 = 18046 + (4x+10) = 180
4x + 56 = 180
4x = 124
a
4x+1046b
Find the value of x that would make a || b.
3 4
∠3&∠4
2. They are ss int ∠ 's.
3. If ss int are supplementary then the lines are ||.
so
x = 31
6. Example 2
SQH
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.
a
1
2
b
c
d
which lines are || ?∠1≅ ∠2If
Since corr ∠ 's (∠1&∠2) are ≅ c || d.
7. Parallel & Perpendicular Through a Point Theorems
|| Thru a Point Theorem (3.8):
Through a point not on a line, there exists exactly one line || to the given line.
parallel
perpendicular
SQH
Thru a Point Theorem (3.9):
Through a point not on a line, there exists exactly one line to the given line.
⊥
⊥
3.3 Summary
The 5 ways to prove that lines are parallel:
1. Show a pair of corresponding angles are congruent (11)2. Show a pair of alternate interior angles are congruent (3.5)3. Show a pair of same-side interior angles are supplementary (3.6)4. Show that both lines are perpendicular to a 3rd line (3.7)5. Show that both lines are parallel to a 3rd line (3.10)