Download - Chapter Five, Part Two
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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
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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
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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
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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.
MathWorks 10 Workbook234
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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
235Chapter 5 Angles and Parallel Lines
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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
MathWorks 10 Workbook236
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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
237Chapter 5 Angles and Parallel Lines
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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
MathWorks 10 Workbook238
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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.
239Chapter 5 Angles and Parallel Lines
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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.
MathWorks 10 Workbook240
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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.
241Chapter 5 Angles and Parallel Lines
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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
MathWorks 10 Workbook242
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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.
243Chapter 5 Angles and Parallel Lines
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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.
MathWorks 10 Workbook244
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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
245Chapter 5 Angles and Parallel Lines
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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
MathWorks 10 Workbook246
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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
247Chapter 5 Angles and Parallel Lines
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