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

SQH

HW: Pg. 87 #1-15 odd, 19, 25, 27, 29MK: 3.3 Makeup Homework from website

1. Review: || Line Postulate & Theorems

SQH

a

b

t

If the lines are || then corr are ∠ 's ≅.

If the lines are || then alt int are ∠ 's ≅.

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)

SQH

3.3 Homework Index

1 - 16

18 - 19

SQH

25 - Proof

27 - 29

HW: Pg. 87 #1-15 odd, 19, 25, 27, 29MK: 3.3 Makeup Homework from website

3.3 Homework, p. 87 (1 - 16)

SQH

Use the information given to name the || segments. If there are no || segments, write none.

1. ∠2 ≅ ∠9

3. m∠1=m∠8 =90

5. m∠2 =m∠5

7.

2. ∠6 ≅ ∠7

4. ∠5 ≅ ∠7

6. ∠3 ≅ ∠11

8. m∠10 =m∠11m∠1=m∠4 =90

9. m∠8 +m∠5 +m∠6 =180

10. FC ⊥ AE& FC ⊥ BD

11. m∠5 +m∠6 =m∠9 +m∠10

12. ∠7 &∠EFB are suppl.

13. ∠2 &∠3 are compl. &

14. m∠2 +m∠3=m∠4

15. m∠7 =m∠3=m∠10

16. m∠4 =m∠8 =m∠1

m∠1=90

3.3 Homework, p. 87 (18 - 19)

SQH

Find the values of x & y that make the red lines parallel & the blue lines parallel.

(x - 40) (x + 40)

y

18.

105x

3x

19.

2y

3.3 Homework, p. 87 (25)

SQH

C

D

B

AE

1 2 3

Given:

Prove:

BE ⊥ DA;CD⊥ DA

∠1≅ ∠2

Proof:

Statements ReasonsBE ⊥ DA;CD⊥ DA 1. Given

2.

3.

4.

5.

6.

25.

1.

2.

3.

4.

5.

6.

3.3 Homework, p. 87 (27 - 29)

SQH

70

40

X

Y

R

T

S

27. 28.

110T

S

X

R

Y

120

29. Find the values of x & y that make the lines shown in red parallel.

30

2x (x - y)

5y

CP Geometry Homework Quiz

Section: Period: Date: Name: Answers:

1

2

3

4

65

Box 1 Box 2 Box 3

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.

Your answer to question 4

SQH

CP Geometry Homework Quiz 3.3A

1. Letter2. Letter3. Letter4. Letter 5. x= 6. y =

Questions 1-4. Write the letter that indicates which segments must be || if…

1. ∠2 ≅∠52. ∠3≅∠113. ∠9 ≅∠54. ∠7 supp∠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 ||

6x96 4y x

SQH

CP Geometry Homework Quiz 3.3B

1. Letter2. Letter3. Letter4. Letter 5. x= 6. y =

Questions 1-4. Write the letter that indicates which segments must be || if…

1. ∠9 ≅∠52. ∠3≅∠113. ∠2 ≅∠54. ∠2 comp∠FBC&m∠EAB=90

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 ||

4x96 4y x

SQH

CP Geometry Homework Quiz 3.3C

1. Letter2. Letter3. Letter4. Letter 5. x= 6. y =

Questions 1-4. Write the letter that indicates which segments must be || if…

1. ∠3≅∠112. ∠2 ≅∠53. ∠9 ≅∠54. ∠2 comp∠FBC&m∠EAB=90

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 ||

3x96 4y x

SQH