example perpendicular lines and reasoning -...

10/3/10 9:00 PM

3.1 Relationships Between Lines 113

Challenge Fill in the blank with always, sometimes, or never.

39. Two skew lines are __?__ parallel.

40. Two perpendicular lines __?__ intersect.

41. Two skew lines are __?__ coplanar.

42. Multiple Choice Two lines are __?__ lines if they do not lie in the same plane and they do not intersect.

!A perpendicular !B parallel

!C coplanar !D skew

43. Multiple Choice Use the diagram below to determine which of the following statements is false. Think of each segment in the diagram as part of a line.

!F QP^&( and MP^&( are not parallel.

!G MP^&( and NR^&(are skew.

!H JM^&( and KS^&( are perpendicular.

!J Plane KJM and plane QPM are not parallel.

If-Then Statements Identify the hypothesis and the conclusion of the if-then statement. (Lesson 2.5)

44. If the band plays, then each member gets $50.

45. If ma5 ! 120", then a5 is obtuse.

46. If there is a sale, then the store will be crowded.

47. If we can get tickets, then we’ll go to the movies.

Properties of Congruence Use the property to complete thestatement. (Lesson 2.6)

48. Reflexive Property of Congruence: __?__ c aXYZ

49. Symmetric Property of Congruence: If a1 c a2, then __?__c __?__.

50. Transitive Property of Congruence: If AB&* c EF&* and EF&* c ST&*,then __?__ c __?__.

Reciprocals Find the reciprocal. (Skills Review, p. 656)

51. 26 52. #7 53. 10 54. $83

$

Integers Evaluate. (Skills Review, p. 663)

55. 18 % (#3) 56. #4 & 2 57. 17 % (#6) 58. 16 # (#5)

59. #5 % 31 60. 24 # 28 61. (#8)(#10) 62. #25 # 19

Algebra Skills

Mixed Review

Standardized TestPractice

P P

K

S

J

RMN

3.2 Theorems About Perpendicular Lines 115

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

a. a3 c a5

b. a4 c a5

c. a2 c a3

Solution a. Yes, enough information is given. Both angles are right

angles. By Theorem 3.1, they are congruent.

b. Yes, enough information is given. Lines r and t are perpendicular. So, by Theorem 3.2, a4 is a right angle. By Theorem 3.1, all right angles are congruent.

c. Not enough information is given to conclude that a2 c a3.

EXAMPLE 1 Perpendicular Lines and Reasoning

Theorem 3.3

Words If two lines intersect to form adjacent congruent angles, then the lines are perpendicular.

Symbols If a1 c a2, then AC^&( ! BD^&(.

Theorem 3.4

Words If two sides of adjacent acute angles are perpendicular, then the angles are complementary.

Symbols If EF&( ! EH&(, then ma3 ! ma4 " 90#.

THEOREMS 3.3 and 3.4

Perpendicular Lines and Reasoning

LOOK BACKTheorems 3.3 and 3.4refer to adjacent angles.For the definition ofadjacent angles, see p. 68.

Student Help

In the diagram, g ! e and g ! f. Determine whether enoughinformation is given to conclude that the statement is true. Explain.

1. a6 c a10 2. a7 c a10

3. a6 c a8 4. a7 c a11

5. a7 c a9 6. a6 c a11

r

u

t

s

13

2

45

e6

7

11

98

f

g h

10

CD

BA

21

F

E H

G

43



a. a3 c a5

b. a4 c a5

c. a2 c a3






Theorem 3.3



Theorem 3.4






Student Help


1. a6 c a10 2. a7 c a10

3. a6 c a8 4. a7 c a11

5. a7 c a9 6. a6 c a11

r

u

t

s

13

2

45

e6

7

11

98

f

g h

10

CD

BA

21

F

E H

G

43

3.2 and 3.3 Perpendicular Lines and Special Angle Pairs Goal: You will use theorems about perpendicular lines, as well as identify special angle pairs made by transversals Warm up: Name a pair of Skew lines Name a pair of Perpendicular Lines Name a pair of Parallel Lines Notes: Properties of Perpendicular Lines 1) 2) Is there enough information to say the angles are congruent?



a. a3 c a5

b. a4 c a5

c. a2 c a3






Theorem 3.3



Theorem 3.4






Student Help


1. a6 c a10 2. a7 c a10

3. a6 c a8 4. a7 c a11

5. a7 c a9 6. a6 c a11

r

u

t

s

13

2

45

e6

7

11

98

f

g h

10

CD

BA

21

F

E H

G

43

116 Chapter 3 Parallel and Perpendicular Lines

HELICOPTERS Main rotorsof a helicopter may have twoto eight blades. The bladescreate the helicopter’s liftpower.

Aviation In the helicopter at the right, areaAXB and aCXB right angles? Explain.

SolutionIf two lines intersect to form adjacentcongruent angles, as AC&* and BD&* do, then the lines are perpendicular (Theorem 3.3). So, AC&* ! BD&*.

Because AC&* and BD&* are perpendicular, they form four right angles (Theorem 3.2). So, aAXB and aCXB are right angles.

B

C

D

A X

EXAMPLE 2 Use Theorems About Perpendicular Lines

In the diagram at the right,

EF&*( ! EH&*( and maGEH ! 30". Find the value of y.

SolutionaFEG and aGEH are adjacent acute angles and EF&*( ! EH&*(. So, aFEG and aGEH are complementary (Theorem 3.4).

6y" # 30" ! 90" maFEG # maGEH ! 90"

6y ! 60 Subtract 30 from each side.

y ! 10 Divide each side by 6.

ANSWER ! The value of y is 10.

EXAMPLE 3 Use Algebra with Perpendicular Lines

Use Algebra with Perpendicular Lines

Find the value of the variable. Explain your reasoning.

7. aEFGc aHFG 8. AB&( ! AD&*( 9. KJ&( ! KL&(,aJKMc aMKL

F

E

H

G30!

6y !

G

HE F

5x !

B

A D

C36!

9y !

M

L K

J

z !z !

Intersecting Perpendicular Lines Theorems 1) 2)

3.3 Angles Formed by Transversals 121

GoalIdentify angles formed bytransversals.

Key Words• transversal• corresponding angles• alternate interior angles• alternate exterior angles• same-side interior angles

3.33.3 Angles Formed byTransversals

Lines intersected by atransversal can beparallel or not parallel.

Visualize It!

m

n

p

t

A is a line that intersects two or more coplanar lines at different points. For instance, in the diagram below, the blue line t is a transversal.

The angles formed by two lines and a transversal have special names.

Two angles are if they occupy correspondingpositions.

The following pairs of angles are corresponding angles:

a1 and a5 a2 and a6

a3 and a7 a4 and a8

Two angles are if they lie between the twolines on the opposite sides of the transversal.

The following pairs of angles are alternate interior angles:

a3 and a6

a4 and a5

Two angles are if they lie outside the twolines on the opposite sides of the transversal.

The following pairs of angles are alternate exterior angles:

a1 and a8

a2 and a7

Two angles are if they lie between the twolines on the same side of the transversal.

The following pairs of angles are same-side interior angles:

a3 and a5

a4 and a6

same-side interior angles

alternate exterior angles

alternate interior angles

corresponding angles

transversal

1 243

5 687

t

1 243

5 687

t

1 243

5 687

t

1 243

5 687

t

m

n

transversalt

Describe the relationship between the angles.

1. a2 and a7 2. a3 and a5



24

68

13

57

122 Chapter 3 Parallel and Perpendicular Lines

Describe the relationship between the angles.

a. a1 and a2 b. a3 and a4 c. a5 and a6

Solutiona. alternate interior angles

b. alternate exterior angles

c. same-side interior angles

EXAMPLE 1 Describe Angles Formed by Transversals

List all pairs of angles that fit the description.

a. corresponding

b. alternate exterior

c. alternate interior

d. same-side interior

Solutiona. corresponding:

a1 and a5 a2 and a6

a3 and a7 a4 and a8

b. alternate exterior:

a1 and a8 a3 and a6

c. alternate interior:

a2 and a7 a4 and a5

d. same-side interior:

a2 and a5 a4 and a7

EXAMPLE 2 Identify Angles Formed by Transversals

Describe Angles Formed by Transversals

1

2

3

4

5 6

RACE CAR DESIGNTo maximize the speed of arace car, the angles of thefront and rear wings can beadjusted.

Auto Racing

Top view of car

Vocab Word: Transversal – Special Angle Pairs: 1) Corresponding Angles 2) Alternate Interior Angles 3) Consecutive Interior Angles 4) Alternate Exterior Angles

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