Chapter 3: Parallel and Perpendicular Lines 3.1 ... · chp 3.notebook October 26, 2014 Example 1: Identify parallel and perpendicular lines Determine whether the lines are parallel,

chp 3.notebook October 26, 2014

Chapter 3: Parallel and Perpendicular Lines 3.1 Relationships Between Linesgoal: Identify relationships between linesVocabulary

Parallel Lines: Two lines are parallel lines if they lie in the same plane and do not intersect

Perpendicular Lines: Two lines are perpendicular lines if they intersect to form a right angle.

Skew lines: Two lines are skew lines if they do not lie in the same plane.

Parallel planes: Two planes are parallel planes if they do not intersect

Line perpendicular to a plane: is a line that intersects a plane in a point and that is perpendicular to every line in the plane that intersects it.


Example 1: Identify parallel and perpendicular lines

Determine whether the lines are parallel, perpendicular or neither.

Solution

a) n and m

b) p and q

c) n and p

p

m

n

q


Followup: In example 1, how do you know that the statement is true?

Lines n and m are parallel

Lines n and p are perpendicular

Lines n and q are not parallel

Lines n and q are not perpendicular


Example 2: Identify Skew LinesDetermine whether the lines are skew.

Solution

a. f and g

b. f and h

f

gh


Checkpoint: USe the diagram shown.1.Name a pair of parallel lines.

2. Name a pair of perpendicular lines.

3. Name a pair of skew lines.

x y

w


Example 3: Identify Relationship in Spacea. Name a plance that appears parallel to plane B.

b. Name a line that is perpendicular to plane B.

Solution B

C

m

n

l


Followup: In example 3.....

Is line l perpendicular to plane C? Explain

Is line n perpendicular to plane C? Explain


Checkpoint: Think of each segment in the diagram as part of a line.

4. Name a line that is skew to VW.

5. Name a plane that appears parallel to plane VXW.

6. Name a line that is perpendicular to plane VXW.

R

Q

T

S

X

W

V


3.2 Theorems About Perpendicular Linesgoal: Bisect an angle.

Theorem 3.1Words: All right angles are __________________.

Symbols: If m A = 90 o and m B = 90o,

then A _____ B

AB


Theorem 3.2Words: If two lines are perpendicular, then they intersect to from _____________ right angles

Symbols: If n m, then m 1 = ____________m 2 = _______, m 3 = _______, and m 4 = _______

1

23

4

n

m


Followup: How do you know that each statement is true?

1. If two lines intersect to form a right angle, then they are perpendicular lines.

2. If two lines are perpendicular, then they intersect to form four right angles.


Example 1: Perpendicular Lines and Reasoning

Solution

In the diagram, r s and r t. Decide whether enough information is given to conclude that the statement is true. Explain your reasoning.

a. 1 ≅ 5

b. 4 ≅ 5

c. 2 ≅ 3

12345

rs

t


Checkpoint: In the diagram, g e and g f. Decide whether enough information is given to conclude that the statement is true. Explain your reasoning.

1. 6 ≅ 10

2. 7 ≅ 10

3. 6 ≅ 8

4. 7 ≅ 11

6

7

10

11

8

9 e

f

g h


Theorem 3.3Words: If two lines intersect to form adjacent congruent angles, then the lineare _______________________.

Symbols: If 1 ≅ 2, then AC _____ BD.

1

2

D

B

C

A

n


Theorem 3.4Words: If two sides of adjacent acute angles are perpendicular, then the angles are _______________________

Symbols: If EF EH, then m 3 + m 4 = _________

34

E

F

H

G


Example 2: Use Theorems about Perpendicular Lines

Solution

In the helicopter at right, are AXBand CXB right angles? Explain.

B

C

D

A

X


Example 3: Use Algebra with Perpendicular Lines

Solution

Check your work!

In the diagram at the right, EF EH andm GEH = 30o. Find the value of y

6yo

30o

E

F

H

G


Checkpoint:Find the value of the variable. Explain.

5. EFG ≅ HFG

6. AB AD

5xo

G

HF

E

9yo36o

A

BC

D


3.3 Angles Formed by Transversalsgoal: Identify angles formed by transversals.Vocabulary

Transversal: A transversal is a line that intersects two or more coplanar lines at different points

Corresponding angles: Two angles are corresponding angles if they occupy corresponding positions.

Alternate interior angles: Two angles are alternate interior angles if they lie between the two lines on the opposite side of the transversal.

Alternate exterior angles: Two angles are alternate exterior angles if they lie outside the two lines on the opposite side of the transversal.

Same side interior angles: Two angles are sameside interior angles if they lie between two lines on the same side of a transversal.


Example 1: Describe Angles Formed by Transversals

Solution

a.) 1 and 2 are _________________________ angles.

b) 3 and 4 are __________________________ angles.

c) 5 and 6 are ___________________________ angles.

Identify the relationship between the angles.

a. 1 and 2 b. 3 and 4 c. 5 and 6

21

3

4

56


Followup: On each diagram below, label the transversal t. Then label one pair of angles that fits the description.

1 and 6 arecorresponding angles.

2 and 3 arealternate exterior angles.

5 and 7 aresame side interior angles.

4 and 8 arealternate interior angles


Example 2: Identify Angles Formed by Transversals

Solution

List all pairs of angles that fit the description.a. correspondingb. alternate exteriorc. alternate interiord. sameside interior

a. corresponding: 9 and ______, 10 and _____, ______and _______, ______ and _______

b. alternate exterior: 9 and ______, ______ and _______

c. alternate interior: 10 and ______, ______ and ________

d. sameside interior: 10 and ______, ______ and ________

9 1011 12

13 1415 16


Followup: In Example 2, does a transversal intersect two parallel lines?

In the space at the right draw two lines intersected by a transversal t.

How many angles are formed?

Complete the table with the number of pairs of angles formed.

Number of pairs formed

Corresponding angles

Alternate exterior angles

Alternate interior angles

sameside interior angles


Checkpoint:Describe the relationship between the angles in the diagram below.

1. 2 and 7

2. 3 and 5

3. 1 and 5

4. 4 and 5

5. 4 and 8

6. 4 and 6

1 2

3 4

56

78


3.4 Parallel Lines and Transversalsgoal: Find the congruent angles formed when a transversal cuts parallel linesPostulate 8: Corresponding Angles Postulate

Words: If two parallel lines are cut by a transversal, then the corresponding angles are

_________________________________

Symbols: If j k, then the following are true. 1 ≅ _____ 2 ≅ _____

3 ≅ _____ 4 ≅ _____

1 23 4

5 67 8

t

j

k


Example 1: Find Measures of Corresponding Angles

Solution

60o

6

a)135o

5

b)

90o

2

a.) m = _______ b.) m = _________ c.) m = __________

Find the measure of the numbered angle.

c)


Checkpoint: Find the measure of the numbered angles.

120o

1 145o45o

23


Theorem 3.5: Alternate Interior Angles Theorem

Words:

Symbols:

If two parallel lines are cut by a transversal, then alternate interior angles are ___________

465

3

If j k, then the following are true.

3 ≅ _____ 4 ≅ _____


Example 2: Find Measures of Alternate Interior Angles

Solution

RS

PQ35o

Find the measure of PQR.

a. m PQR = _______ b. m PQR = ______ c. m PQR = _______

1200

700

RS

PQ

RS

P Q


Checkpoint: Find the measure of the numbered angle.

4 65o5100o

6


Theorem 3.6: Alternate Exterior Angles Theorem

Words:

Symbols:

If two parallel lines are cut by a transversal, then alternate exterior angles are ___________

2

87

1


1 ≅ _____ 2 ≅ _____


Example 3: Find the measures of Alternate Exterior Angles

Solution

12

Find the measure of 1 and 2

75o

m 2 = _______m 1 + m 2 = _______

m 1 = _______

Answer: m 1 = ________ and m 2 = _______

m 1 + _____ = _______


Checkpoint: Find the measure of the numbered angle.

742o

8 9130o


Theorem 3.7: SameSide Interior Angles Theorem

Words:

Symbols:

If two parallel lines are cut by a transversal, then sameside interior angles are ___________

4

65

3


m 3 + m 5 = _____ m 4 + m 6 = _____


Example 4: Find Measures of SameSide Inerior Angles

Solution

80o

5

Find the measure of the numbered angle.

130o

6

a. m 5 + 80o = ______

m 5 = _______

b. m 6 + 130o = ______

m 6 = _______


Example 5: Use Algebra with Angle Relationships

Solution

(x + 15)oFind the value of x.

125o

Substitute your value for y in the original equation to determine whether it is a solution

(x + 15)o = _________

x = _________


10. 11. 12.

120o(x + 35)o 78o

(x 2)o (2x + 10)o


Sudoku requires students to use deductive reasoning and is a good way to practice using logic and clues to solve a puzzle


3.5 Showing Lines are Parallel goal: Show that two lines are parallel.Vocabulary

Converse: The converse of an ifthen statement is the statement formed by switching the hypothesis and the conclusion.


Example 1: Write the Converse of an IfThen Statement

Solution

Statement: If two segments are congruent, then the two segments have the same length.

a. Write the converse of the true statement above.b. Decide whether the converse is true.

a. Converse: __________________________________________________________________________________________________________

b. The converse is a __________statement.


Checkpoint: Write the converse of the true statement. Then decide whether the converse if true.

1. If two angles have the same measure, then the two angles are congruent.

2. If 3 and 4 are complementary, then m 3 + 4 = 90o.

3. If 1 and 2 are right angles, then 1 ≅ 2.


Postulate 9: Corresponding Angles Converse

Words:

Symbols:

If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are _____________________.

s

r5

1

If 1 ≅ 5, then r _____ s


Example 2: Apply Corresponding Angles Converse

Solution

GE

DB

110o

Is enough information given to conclude that BD EG? Explain. 600

1000B

E

C D

110o F G B

E

C D

F G800

a. The 110o angles are corresponding and congruent. By the

____________________________________________, BD EG.

b. ______________________________ information is given to conclude BD EG.

c. You can conclude that _________________ o. So by the

___________________________________, BD EG


Checkpoint: Is enough information given to conclude that

500R

X

ST

Y Z

1300X

R

XS

TY

Z

R

ZT

85o

RT XZ?

85o


Theorem 3.8: Alternate Interior Angles Converse

Words:

Symbols:

If two lines are cut by a transversal sothat alternate interior angles are congruent, then the lines are ___________

45

if 4 ≅ 5, then r ____ s


Theorem 3.9: Alternate Exterior Angles Converse

Words:

Symbols:

If two lines are cut by a transversal so that alternate exterior angles are congruent,then the lines are ___________________.

8

1

If 1 ≅ 8 then r ____ s


Example 3: Identify parallel lines

Solution

m

n

Does the diagram give enough information to conclude that m n?

n

m

a. Yes. You can use _________________ to conclude m n.

b. No. Not enough information is given to conlcude ___________.


Checkpoint: Complete the following exercise.

7. Does the diagram give enough information to conclude that c d? Explain.

c

d


Theorem 3.10: SameSide Interior Angles ConverseWords:

Symbols:

If two lines are cut by a transversal so that sameside interior angles aresupplementary, then the lines are _____________.

5

3

If m 3 + m 5 = _____ , then r s

FOLLOW UP: This lesson gives you four ways to show that two lines are

__________________________________


Example 4: Use Same Side Interior Angles Converse

Solution

2xo

Find the value of x so that j k.

80o

Substitute your value for y in the original equation to determine whether it is a solution

2xo + 80o = _________

2x = _________

x = _________

Lines j and k will be parallel if themarked angles are ____________

j

k


8. 9. 10.

70o

2xo2xo

4xo

(x + 22)ov

w

v

w

v

w


3.6 Using Perpendicular and Parallel Linesgoal: Construct parallel and perpendicular lines. Use properties of parallel and perpendicular lines

Vocabulary

≅

Construction:


Postulate 10: Parallel Postulate

Words:

Symbols:

If there is a line and a point not on the line, then there is exactly one line through the point

______________ to the given line.

If P is not on l, then there exists a line m through P such that __________________.

m

l

P


Postulate 11: Perpendicular Postulate

Words:

Symbols:

If there is a line and a point not on the line, then there is exactly one line through the point________________ to the given line.

If P is not on l, then there exists a line m through P such that __________________.

m

l

P


Theorem 3.11

Words:

Symbols:

If two lines are ______________ to the same line, then they are parallel to each other.

If q r and r s, then __________.

q

sr


Theorem 3.12

Words:

Symbols:

In a plane, if two lines are _____________________to the same line, then they are parallel to each other.

If m p and n p, then __________.

m

p

Pn


Example 2: Use Properties of Parallel Lines

Solution

In the diagram at the right, each rung on the ladder is parallel to the rung immediately below it, and the bottom rung is parallel to the ground. Explain why the top rung is parallel to the ground.

You are given that i ___ and m ___. By Theorem 3.11, _______.Since l n and ____ f, it follows that ______. So the top rung is parallel to the ground.

f

nml


Example 3: Use Properties of Parallel Lines

Solution

A B

C D

By Theorem 3.12, AB and CD will be parallel if AB and CD are both ____________

to AC. For this to be true, angle BAC must measure ________.

Find the value of x that makes AB CD

(2x + 2)o

(2x + 2)o = ________

2x = _________

x = ________


Checkpoint: Complete the following exercises.

1. explaine why a c.

2. Find th value of x that makes d e.


Checkpoint: Complete the following exercises.

1. explaine why a c.

2. Find th value of x that makes d e.

cb

a

(5x+10)o

de


Ways to Show that Two Lines are ParallelCorresponding Angles Converse

Show that a pair of corresponding angles are_________________________.

Alternate Interior Angles Converse

Show that a pair of alternate interior angles are___________________________________.

Alternate Exterior Angles Converse

Show that a pair of alternate exterior anglesare __________________________________

SameSide Interior Angles Converse

Show that a pair of sameside interior angles are ______________________________.

Theorem 3.11

Show that both lines are ______________________ to a third line

Theorem 3.12

In a plane, show that both lines are___________________ to a third line.

Documents

Chapter 3: Parallel and Perpendicular Lines 3.1 ... · chp 3.notebook October 26, 2014 Example 1: Identify parallel and perpendicular lines Determine whether the lines are parallel,