Chapter Five, Part Two

NEW SKILLS: WORKING WITH ANGLES FORMED BY INTERSECTING LINES

A line that intersects two other lines at two distinct points is a transversal. When two

non-parallel lines are intersected by a transversal, they form angles of varying sizes.

Consider the diagram below: t is a transversal that intersects 1 and 2.

2 1

3 4

6 5

7 8

1

2

t

Eight angles are formed.

1 and 5, and 4 and 8, 2 and 6, and 3 and 7 are pairs of

corresponding angles.

1 and 3, and 2 and 4, 5 and 7, and 6 and 8 are pairs of vertically

opposite angles.

3 and 5, and 4 and 6 are pairs of alternate interior angles.

2 and 8, and 1 and 7 are pairs of alternate exterior angles.

3 and 6, and 4 and 5 are pairs of interior angles on the same side of the

transversal.

1 and 8, and 2 and 7 are pairs of exterior angles on the same side of the

transversal.

For more details, see page 198 of MathWorks 10.

transversal: a line that

intersects two or more

lines

corresponding angles:

two angles that occupy

the same relative

position at two different

intersections

vertically opposite

angles: angles created

by intersecting lines that

share only a vertex

alternate interior

angles: angles in

opposite positions

between two lines

intersected by a

transversal and also on

alternate sides of the

same transversal

alternate exterior

angles: angles in

opposite positions outside

two lines intersected by a

transversal

5.3Non-Parallel Lines and Transversals231Chapter 5 Angles and Parallel Lines

Example 1

In the following diagram, identify each of the following, and specify which lines and

transversals you are using.

21

3

4

6 5

7

89

1 2

3

4

a) an interior angle on the same side of the transversal as 6

b) an angle corresponding to 2

c) an angle corresponding to 4

d) an alternate interior angle to 4

SOLUTION

a) Using 1 and 2, with transversal 3, 2 and 6 are interior angles on the same

side of the transversal.

b) Using 1 and 2, with transversal 3, 1 corresponds to 2.

c) Using 1 and 2, with transversal 4, 7 corresponds to 4.

d) Using 3 and 4, with transversal 2, 4 and 9 are alternate interior angles.

A transversal is

not necessarily one

line segment in a

specific drawing.

In this figure, there

are several lines

that intersect other

lines. These can

be considered

transversals.

MathWorks 10 Workbook232

BUILD YOUR SKILLS

1. In the diagram below, identify the relationship between each pair of angles.

2 1

34 6

5

7

8

a) 7 and 8

b) 2 and 7

c) 1 and 6

d) 5 and 7

2. Given the diagram below, identify the following angles.

2 13

4

657

a) an alternate exterior angle to 2

b) an interior angle on the same side of the transversal as 7

c) an alternate interior angle to 4

d) an angle corresponding to 5

233Chapter 5 Angles and Parallel Lines

hfcyberhighHighlight

3. Identify each of the following angles. Name the two lines and the transversal you

are using.

21 3

4

6

5

78

910

3

4

12

a) two angles corresponding to 1

b) an interior angle on the same side of the transversal as 10

c) an alterate interior angle to 5

d) two interior angles on the same side of the transversal as 8

Example 2

In the diagram below, measure and record the sizes of the angles. Identify pairs of equal

angles and state why they are equal.

2 1

43

6 5

7 8

SOLUTION

Use a protractor to measure the angles.

1 and 3 measure 125

2 and 4 measure 55

5 and 7 measure 120

6 and 8 measure 60

These are each pairs of vertically opposite angles.


BUILD YOUR SKILLS

4. Look at the diagram below. Identify two transversals that intersect both AB and AD.

D

AB

C

E

5. In the diagram below, can t be a transversal that intersects 1 and 2? State why

or why not.

12

t



6. In the diagram below, t is a transversal that intersects 1 and 2. Name another pair

of lines and their transversal.

3t

1

2

PRACTISE YOUR NEW SKILLS

1. In the diagram below, where t is the transversal, identify two pairs of each of the

following angles.

2134

6578 t

a) alternate interior angles

b) corresponding angles

c) interior angles on the same side of the transversal



2. A flashlight shines down onto a floor as shown in the diagram below. If the outer

rays are considered to be two lines and the floor is a transversal, name a pair of

corresponding angles.

12

3456

3. In the diagram below, identify which line is a transversal that intersects 1 and 2

that makes the following relationships between the pairs of angles.

a) 1 and 2 a pair of corresponding angles

b) 3 and 4 a pair of alternate interior angles

2

3

4

1

34

1

2




4. In the diagram below, calculate the sizes of each of the interior angles. What is

their sum?

21

3

465

85

112

1

2

t

5. Calculate the sizes of the six angles indicated in the figure.

2

1

3

4

6 5

120

70

1

2

t




Parallel Lines and Transversals 5.4 NEW SKILLS: WORKING WITH ANGLES FORMED BY PARALLEL LINESINTERSECTED BY A TRANSVERSAL

If two parallel lines are intersected by a transversal:

the alternate interior angles are equal;

the corresponding angles are equal; and

the interior angles on the same side of the transversal are supplementary.

If you know that, given two lines cut by a transversal:

alternate interior angles are equal; or

corresponding angles are equal; or

interior angles on the same side of the transversal are supplementary;

then you can conclude that the lines are parallel.

For more details, see page 209 of MathWorks 10.

Example 1

Consider the diagram below, in which 1 is parallel to 2. What are the measures of the

three indicated angles? Explain how you reached your answers.

t

12

4 = 122

213

SOLUTION

1 measures 122 because it corresponds to 4 .

2 measures 58 because it forms a straight angle with 4.

3 measures 58 because it is vertically opposite 2.


ALTERNATIVE SOLUTION

There may be more than one reason for stating why two angles are equal.

3 is 58 because it forms a straight angle with 4.

2 is 58 because it is vertically opposite 3.

1 is 122 because it is an interior angle on the same side of a transversal as 3.

BUILD YOUR SKILLS

1. In the diagram below, 1 is parallel to 2. State the measures of the indicated angles

and explain your reasoning.

21

4 371

118

1

2

2. What are the measures of the interior angles in the trapezoid shown below? (Hint:

Be careful of the order in which you calculate the angles.)

2

1

3

4

BA

CD

68

The order in

which you find the

angle measures

is important in

explaining your

reasoning.



3. Quadrilateral ABCD is a parallelogram in which B measures 74. Determine the

measures of the other angles and state your reasons.

74B

A

CD

Example 2

Given the diagram below, identify all the pairs of parallel lines and explain your

selection.

1

2

3

456

110

100

100

70

80

95

SOLUTION

6 is parallel to 4. If you consider 1 to be a transversal, 100 and 80 are interior

angles on the same side of the transversal.

1 is parallel to 2. If 6 is a transversal, the two 100 angles are corresponding angles.


BUILD YOUR SKILLS

4. Find a pair of parallel lines in the following diagram. On the diagram, mark all the

angles necessary to determine this.

1 23

4

6

5103

68

62

77

5. What size must 1 be if 1 is parallel to 2?

1

123

1



6. The two vertical pipes in the diagram need to be moved to be parallel to each other.

By what angle must the plumber move the second pipe?

Pipe 1

106

78

Pipe 2

Example 3

Given parallelogram ABCD, determine the values of B, C, and D in that order,

stating your reason for each measure.

BA

CD

130

SOLUTION

In a parallelogram, opposite sides are parallel.

AD is parallel to BC, and they are intersected by transversal AB. B is an

interior angle on the same side of the transversal as A. Therefore, it is

complementary to A.

180 130 = 50

B is 50.


AB is parallel to DC, and they are intersected by transversal BC. You know that B

is 50. C is an interior angle on the same side of the transversal as B, so they are

complementary. C measures 130.

AB is parallel to DC, and they are intersected by transversal AD. You know that A

measures 130 and is complementary to D. Therefore, D measures 50.

BUILD YOUR SKILLS

7. If 1 and 2 are parallel and are intersected by transversals t1 and t2, what are the

measures of the indicated angles? Solve for the measures in the given order, stating

your reasoning.

2

2 1

3

4

83

54

1

t

t2

SOLVING ANGLE MEASURESAngle Measure Reason

1 =

2 =

3 =

4 =

You can use the

notation AD || BC to

indicate that AD and

BC are parallel.



8. In the diagram below, if the side of the house and the side of the shed are parallel, what are

the measures of 1 and 2?

58

1

2

9. A plumber must install pipe 2 parallel to pipe 1. He knows that 1 is 53. What is the

measure of 2?

Pipe 1 1

Pipe 2

2



PRACTISE YOUR NEW SKILLS

1. Given the diagram below, where 1 is parallel to 2, find the measures of the

indicated angles and state your reasons.

1

2

2

1112

3

460

5

2. In the diagram below, the top of the bridge is parallel to the deck, and the brace in

the middle is vertical, perpendicular to the deck, determine the size of 1 and 2.

57

2

1




3. Identify the pairs of parallel lines in the following diagram. (Hint: The lines can be

extended.)

1 2

3

4

6

5

46

46

132

13248

4. Examine the following diagram. By how many degrees do the studs need to be

moved in order to be parallel to each other? What direction do they need to move

in? (The studs are indicated by the capital letters.)

BA C

D

E

F

89 91

134

135




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Chapter Five, Part Two