Geometry - Loudoun County Public Schools



Name: _______________________________________________ Date: ____________________

Chapter 3

Parallel

and

Perpendicular Lines

Sections Covered:

3.1 Identify Pairs of Lines and Angles

3.2 Use Parallel Lines and Transversals

3.3 Prove Lines are Parallel

3.4 Find and Use Slopes of Lines

3.5 Write and Graph Equations of Lines

The student will use the relationships between angles formed by two lines cut by a transversal to

a) determine whether two lines are parallel;

b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and

c) solve real-world problems involving angles formed when parallel lines are cut by a transversal.

The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include

a) investigating and using formulas for finding distance, midpoint, and slope;

b) applying slope to verify and determine whether lines are parallel or perpendicular;

Unit 3 Syllabus: Ch 3 Parallel and Perpendicular Lines

|Block |Date |Topic |Homework |

|16 |B 10/16 |3.1 Identifying Lines and Angles |Worksheet: 3.1 and 3.2 Identifying Lines |

| |A 10/17 |3.2 Angle Relationships and Parallel Lines |and Angle Relationships |

|17 |B 10/20 |3.3 Proving Lines Parallel |Worksheet: 3.3 Proving Lines Parallel |

| |A 10/21 | | |

|18 |B 10/22 |Quiz 3.1-3.3 |Khan Academy |

| |A 10/23 | | |

|19 |B 10/24 |3.4 Slope of Lines |Worksheet: 3.4 Slope of Lines |

| |A 10/27 |3.5 Writing Equations of Lines |3.5 Writing Equations of Lines |

| | | | |

|20 |B 10/28 |Review Ch 1 and 2 for Quarter Test |Review Worksheet—Separate from packet |

| |A 10/29 | | |

|21 |B 10/30 |Quarter 1 Benchmark Test |Review Worksheet #1 |

| |A 10/31 | | |

|1 |B 11/5 |Review Day Ch 3 |Review Worksheet #2 |

| |A 11/6 | | |

|2 |B 11/7 |Ch 3 Test |Khan Academy |

| |A 11/8 | | |

***Syllabus subject to change due to weather, pep rallies, illness, etc

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and Monday, Wednesday, and Thursday afternoons.

Notes 3.1 & 3.2: Identifying Lines and Angles and Angle Relationships

_______________________________________________________

| |PARALLEL |PERPENDICULAR |SKEW |

|DEFINITION | | | |

| |LINES- |LINES- |LINES- |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

|EXAMPLE | | | |

|DEFINITION | | | |

| |PLANES- |PLANES- | |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

|EXAMPLE | | | |

Think of each segment in the diagram as part of a line. Which line(s) or plane(s) appear to fit the description?

1. Line(s) parallel to AB

2. Line(s) perpendicular to BF

3. Line(s) skew to CD and containing point E

4. Plane(s) perpendicular to plane ABE

5. Plane(s) parallel to plane ABC

This table defines the five types of angles by their location.

It also states the specific relationships they have when the transversal cuts through parallel lines.

|Type: |Location: |Picture: |

| | | |

|Theorem: If 2 parallel lines are cut by a transversal, then ___________________________ are | |

|_______________. | |

| |

|Type: |Location: |Picture: |

| | | |

|Theorem: If 2 parallel lines are cut by a transversal, then ___________________________ are | |

|_______________. | |

| |

|Type: |Location: |Picture: |

| | | |

|Theorem: If 2 parallel lines are cut by a transversal, then ___________________________ are | |

|_______________. | |

| |

|Type: |Location: |Picture: |

| | | |

|Theorem: If 2 parallel lines are cut by a transversal, then ___________________________ are | |

|_______________. | |

| |

|Type: |Location: |Picture: |

| | | |

|Theorem: If 2 parallel lines are cut by a transversal, then ___________________________ are | |

|_______________. | |

Classify each angle pair as corresponding, alternate interior, alternate exterior, consecutive interior, or consecutive exterior.

a) ∠1 and ∠9 ____________________

b) ∠8 and ∠13 ____________________

c) ∠6 and ∠16 ____________________

d) ∠4 and ∠10 ____________________

e) ∠8 and ∠16 ____________________

f) ∠10 and ∠13 ____________________

Discovery: Lines l and m are parallel. Note: Parallel lines are distinguished by a matching set of arrows on the lines that are parallel. Find the measure of the missing angles by using transparent paper. Then, let’s go back and fill in the theorems.

Key Question: If x = 115°, is it possible for y to equal 115°?

For t he following diagrams, state the type of angles that are given, state their relationship, and then find x.

1. 2. 3.

Find the missing variables.

4. 5. 6.

On Your Own: For t he following diagrams, state the type of angles that are given, state their relationship, and then find x.

1. 2. 3.

4. 5. 6.

Notes 3.3: Proving Lines Parallel

_______________________________________________________

Follow along and fill in the missing blanks for each theorem. Then, based on the theorem, use the given theorem to determine if the lines are parallel or not parallel. Provide reasoning.

|Corresponding Angles Converse Theorem: If 2 lines are cut by a transversal so the corresponding angles are __________________, then the lines are |

|__________________. |

|Example: |Non Example: |

| | |

| | |

| | |

| | |

| | |

|Alternate Interior Angles Converse Theorem: If 2 lines are cut by a transversal so the alternate interior angles are __________________, then the lines are|

|__________________. |

|Example: |Non Example: |

| | |

| | |

| | |

| | |

| | |

|Alternate Exterior Angles Converse Theorem: If 2 lines are cut by a transversal so the alternate exterior angles are __________________, then the lines are|

|__________________. |

|Example: |Non Example: |

| | |

| | |

| | |

| | |

| | |

|Consecutive Interior Angles Converse Theorem: If 2 lines are cut by a transversal so the consecutive interior angles are __________________, then the lines|

|are __________________. |

|Example: |Non Example: |

| | |

| | |

| | |

| | |

|Consecutive Exterior Angles Converse Theorem: If 2 lines are cut by a transversal so the consecutive exterior angles are __________________, then the lines|

|are __________________. |

| | |

|Transitive Property of Parallel Lines: |

| |

| |

|Example: | |

| | |

| | |

| | |

Is it possible to prove the lines are parallel or not parallel? If so, state the postulate or theorem you would use. If not, state cannot be determined.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. Find the value of x that makes l // m.

11. a. Find the value of x that makes a // b. b. Find the value of y that makes a // c.

c. Is b // c? Why or why not?

State the postulate or theorem that supports each conclusion.

1. Given: a || b ____________________

Conclusion: [pic]2 ( [pic]7

2. Given: m[pic]4 + m[pic]7 = 180 ____________________

Conclusion: a || b

3. Given: [pic]4 ( [pic]5 ____________________

Conclusion: a || b

Find the values of x and y. Explain your reasoning by stating the proper theorem or postulate.

4. x = y =

_____________________ _____________________

5. x = y =

_____________________ _____________________

6. x = y = _____________________ _____________________

Find the value of x so that n || m. State the theorem or postulate that justifies your solution.

7. 8. 9.

x = x = x =

_____________________ _____________________ _____________________

Can you prove that lines p and q are parallel? If so, state the theorem or postulate that you would use.

10. 11. 12.

_____________________ _____________________ _____________________

Notes 3.4 and 3.5: Writing Equations of Lines

_______________________________________________________

Equations of Lines and Slope

Slope intercept form: Slope Formula:

Graphing and Types of Slopes: Graph the following lines.

|y = 2x + 4 |y = [pic]x – 2 |y = – 3 |x = 5 |

|m = ______ b = ______ |m = _____ b = ______ |Acronym: |Acronym: |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

|type of slope: |type of slope: |type of slope: |type of slope: |

For each equation, rewrite in slope-intercept form and state the m & b values.

| | |[pic] |

|3y – 8x = 2 |9x = 4y – 11 | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | |m=_________b=_________ |

|m=_________b=_________ |m=_________b=_________ | |

Special Types of Lines:

|TYPE OF LINE |PARALLEL LINES |PERPENDICULAR LINES |

|DEFINITION | | |

| | | |

| | | |

| | | |

|SLOPES OF THESE TYPE OF LINES | | |

| | | |

| | | |

| | | |

State the negative reciprocal of the given slope.

1. m = [pic] 2. m = –6 3. m = [pic] 4. m = 9

Find the slope of the given lines.

j passes through

(0, 3) and (3, 1)

m passes through

( –2, 7) and (–6 , 1)

k passes through

(-4, -3) & (0, 3)

Make some conclusions. Make a quick sketch to see what

parallel and perpendicular lines

look like.

Write the equation of a line in slope intercept form:

Steps: 1. Ask yourself “What two letters do I need to write the equation of a line?”

2. Identify which letters you need to still find.

3. If you need m, plug the points into the slope formula.

4. If you need b, plug m and an ordered pair (x, y) into the slope intercept formula and solve for b.

5. Write the equation of a line with the new m and b.

TYPE I: Write the equation of the line that passes through the given y-intercept and given slope.

1. m = 3, b = -3 2. m =[pic], b = 15

TYPE II: Write the equation of the line that passes through the given point and given slope.

3. Passes through (2, 3) and slope is 5.

4. Passes through (6, -5) and slope is [pic]

5. Passes through (5, -2) and slope is 0. Remember: You can always check the b

by graphing. Plot the point and move by

counting the slope till you cross the y-axis.

Type III: Write the equation of a line given two points.

6. Passes through (4, -3) and (3, -6) 7.

TYPE IV: Write the equation of a line given two points and must be parallel or perpendicular to another line.

8. Passes through (3, 2) 9. Passes through (4, 0)

Parallel to [pic] Perpendicular to 2x + y = 1

Practice: Are these equations parallel, perpendicular, or neither?

1. l: [pic] h: [pic] 2. q: [pic] w: [pic]

3. Which lines are //? Which are ⊥? A graph may help.

x = 4

y = –4

y = 4x

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SOL G.2

SOL G.3

B

E

D

A

H

G

C

F

J

N

L

M

P

K

B

E

D

A

H

G

C

F

B

E

D

A

H

G

C

F

J

N

L

M

P

K

115°

x°

y°

z°

l

m

75°

(5x – 10)°

(10y – 25)°

60°

(3x + 15)°

(–5y – 10)°

120°

3x°

(6y – 12)°

100°

(x – 10)°

(2y + 24)°

2

6

j

k

3

6

l

k

25°

j

k

105°

130°

92°

l

k

88°

j

k

28°

l

k

106°

137°

96°

j

k

84°

H

K

J

I

u

t

p

92°

l

k

88°

l

k

105°

75°

A

B

C

D

122°

l

k

58°

m

55°

55°

E

F

G

H

I

a

b

6

5

8

7

1

2

4

3

130

x

y

65

y

x

80

y

x

n

m

n

5x-18

5x

n

m

8x-5

7x+13

5x+23

3x+48

m

q

p

q

p

p

q

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