Chapter 3Student Notes

Chapter 3 TestFriday, October 12 th

3.1Parallel Lines and Transversals

Parallel Lines

A B

C D

Skew Lines and Parallel Planes

Two lines are skew if they

l

ml and m are ________

Examples

1. Name all segments that are parallel to AD

2. Name all segments that intersect AD

3. Name all segments that are skew to AD

4. Name all planes that are parallel to plane ABC.

Answers:

5. ___________________

6. ___________________

7. ___________________

8. ___________________

H

GF

E

D C

BA

t

Transversal – ___________________________

Exterior Angles – _____________________

Interior Angles – _____________________

lm

1 2 3 4

5 6 7 8

lm

t

1 2 3 4

5 6 7 8

Consecutive Interior Angles – _____________________

Alternate Exterior Angles – _____________________

Alternate Interior Angles – _____________________

Corresponding Angles – _____________________

1 2 3 4 8 7 6 5

9 10 11 12 16 17 18 19

r

s

Name the transversal that forms each pair of angles.

Then name the special name for each pair.

1. 3 & 112. 11 & 173. 17 & 14. 2 & 35. 4 & 6

____________________

__________________________________________________________________________________________

Transversal Special Angle Pair Name

pq

3-2 Angles and Parallel Lines

1 2 3 4

5 6 7 8

m

nt

If m ║ n , then the following relationships

exists:

1 2 3 4

5 6 7 8

m

nt

If m ║ n , then:

Corresponding ’s

Alternate Interior ’s

Alternate Exterior ’s

Consecutive Interior ’s supplementary

If m1 = 70o, find the others.

70o

1 23 4

5 67 8

More Examples

t

16 151413

12 11109

8 7

65

34

21

s

DC

BA1. The value of x, if m3 = 4x + 6 and m11 = 126.

If line AB is parallel to line CD and s is parallel to t, find:

2. The value of x, if m1 = 100 and m8 = 2x + 10.

3. The value of y, if m11 = 3y – 5 and m16 = 2y + 20.

Important Notes:•When the lines are parallel;

• The acute angles ____________________.

• The obtuse angles ___________________.

• One acute angle is _______________ to one obtuse angle.

1 2 3 4

5 6 7 8

m

nt

1

30o

36o

Find the measure of angle 1.

1

140o

30o

Find the measure of angle 1.

Find the value of x and y.

(5y + 10)o

(10y + 5)o

(5x)o

(5x + 7)0

(8x + 4)0

(2y)0(5x + 12)0

(6y + 8)0

(6x + 4)0

Find x and y.

3-3 Slopes of Lines

Slope of , andǁ ⊥ lines

Determine if each pair of lines are ǁ , , or neither.⊥

1. Line 1, m = -2 Line 2, m = ½

2. Line 3, m = 3 Line 4, m = 3

3. Line 5, m = 4/3 Line 6, m = 3/4

4. Line 7, m = -1 Line 8, m = 1

Find the slope of each line.

1. l2. m3. Any line ǁ to l.

4. Any line to ⊥ m.

lm

Slope of a LineThe slope of the non-vertical line

through the points and is

m =

The slope of a vertical line ____________.

The slope of a horizontal line is _______.

Find the slope of the line through the given points.

(-4, 7) and (3, 7)

Examples

Find the slope of the line through the given points.

(3, -1) and (3, 2)

Examples

Find the slope of the line through the given points.

(1, -4) and (2, 5)

Examples

Find the slope of the line through the given points.

(-2, 5) and (1, -1)

Examples

Given each pair of points, Determine if AB ǁ CD, AB CD, or neither.⊥

1. A(-3, -2) B(9, 1) C(3, 6) D(5, -2)

2. A(5, -4) B(10, 0) C(9, -8) D(5, -13)

lm

m(l) = m(m) = m(s) = m(r) =

rs

1. m = 3, passes through (2, 1)

2. Passes through (-4, -5) the line that passes through MN, M(-1, -3), N(-3, 4)

Graph each line described below.

m(MN) =

m() =

3-5 Proving Lines Parallel

Postulate 3-4

lm

t

if , then

______.

If ___________________________________________________ corresponding angles are congruent, then the

_________________.

Theorem 3-5

lm

t

if , then

______.

If ________________________________________________________ alternate exterior angles are congruent, then the

___________________.

Theorem 3-6

lm

t

if

, then

______.

1

2

If __________________________________________________________ consecutive interior angles are supplementary,

then ____________________.

Theorem 3-7

lm

t

if , then

______.

If ____________________________________________________ alternate interior angles are congruent, then

________________.

Theorem 3-8

lm

t

if , then

______.

Determine which pair of lines is parallel and why.

1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16

p

q

rs 1. 1 8

2. 7 12

3. 11 9

4. m 6 + 10 = 180

Find x so that l || m

110o

(5x +10)o

l

m

Find x so that l || m

(5x + 15)o

(6x -10)o

l

m

Find x so that l || m

(5x–7)o(7x–5)ol

m

Find x so that l || m(7x–1)o

l

m

3.6Perpendiculars and Distance

How would you measure the distance from Fishersville to the Beach?

Fishersville

Beach

Draw the segment that represents the distance from P to AB.

P

BA

P

BA

Draw the segment that represents the distance from P to AB.

P

BA

P

BA

Draw the segment that represents the distance from P to AB.

PB

A

P

B

A