Download Non-Parallel Lines and Transversals

Chapter 5 Angles and Parallel Lines
Non-Parallel Lines and Transversals
231
5.3
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
transversal: a line that
non-parallel lines are intersected by a transversal, they form angles of varying sizes.
intersects two or more
Consider the diagram below: t is a transversal that intersects ℓ1 and ℓ2.
t
lines
corresponding angles:
two angles that occupy
the same relative
2
3
ℓ1
5
6
ℓ2
7
1
4
8
position at two different
intersections
vertically opposite
angles: angles created
by intersecting lines that
share only a vertex
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.
alternate interior
angles: angles in
opposite positions
between two lines
intersected by a
transversal and also on
alternate sides of the
• ∠3 and ∠5, and ∠4 and ∠6 are pairs of alternate interior angles.
same transversal
• ∠2 and ∠8, and ∠1 and ∠7 are pairs of alternate exterior angles.
alternate exterior
angles: angles in
• ∠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.
opposite positions outside
two lines intersected by a
transversal
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MathWorks 10 Workbook
Example 1
A transversal is
not necessarily one
line segment in a
I n the following diagram, identify each of the following, and specify which lines and
transversals you are using.
ℓ1
ℓ2
specific drawing.
In this figure, there
are several lines
that intersect other
1
6
2
8
lines. These can
be considered
5
9
ℓ3
7
transversals.
4
3
ℓ4
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
SOL U T ION
a) Using ℓ1 and ℓ2, with transversal ℓ3, ∠2 and ∠6 are interior angles on the same
side of the transversal.
b) Using ℓ1 and ℓ2, with transversal ℓ3, ∠1 corresponds to ∠2.
c) Using ℓ1 and ℓ2, with transversal ℓ4, ∠7 corresponds to ∠4.
d) Using ℓ3 and ℓ4, with transversal ℓ2, ∠4 and ∠9 are alternate interior angles.
Chapter 5 Angles and Parallel Lines
BUILD YOUR SKILLS
1. In the diagram below, identify the relationship between each pair of angles.
2 1
5 8
7 3
4 6
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 1
3
7 4
5 6
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
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MathWorks 10 Workbook
3. Identify each of the following angles. Name the two lines and the transversal you
are using.
ℓ2
ℓ1
1
2 3
5 4
6
8 10
9
7
ℓ3
ℓ4
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
3
4
5
6
7
8
SOL U T ION
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.
Chapter 5 Angles and Parallel Lines
BUILD YOUR SKILLS
4. Look at the diagram below. Identify two transversals that intersect both AB and AD.
A
B
C
D
E
5. In the diagram below, can t be a transversal that intersects ℓ1 and ℓ2? State why
or why not.
t
ℓ1
ℓ2
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MathWorks 10 Workbook
6. In the diagram below, t is a transversal that intersects ℓ1 and ℓ2. Name another pair
of lines and their transversal.
t
ℓ3
ℓ1
ℓ2
PRACTISE YOUR NEW SKILLS
1. In the diagram below, where t is the transversal, identify two pairs of each of the
following angles.
1
4
2
3
5 6
8 7
t
a) alternate interior angles
b) corresponding angles
c) interior angles on the same side of the transversal
Chapter 5 Angles and Parallel Lines
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.
2
1
6
5
4
3
3. In the diagram below, identify which line is a transversal that intersects ℓ1 and ℓ2
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
4
ℓ1
ℓ2
1
3
ℓ4
ℓ3
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MathWorks 10 Workbook
4. In the diagram below, calculate the sizes of each of the interior angles. What is
their sum?
t
2
3
1
85°
ℓ1
5
6
4
112°
ℓ2
5. Calculate the sizes of the six angles indicated in the figure.
t
1
2
120°
3
ℓ1
6
ℓ2
4
70°
5
Chapter 5 Angles and Parallel Lines
Parallel Lines and Transversals
NEW SKILLS: WORKING WITH ANGLES FORMED BY PARALLEL LINES
INTERSECTED 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 ℓ1 is parallel to ℓ2. What are the measures of the
three indicated angles? Explain how you reached your answers.
ℓ1
ℓ2
3
1
4 = 122°
t
2
SOL U T ION
∠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.
5.4
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MathWorks 10 Workbook
A LT ERN AT I V E SOL U T ION
There may be more than one reason for stating why two angles are equal.
The order in
which you find the
∠3 is 58° because it forms a straight angle with ∠4.
angle measures
∠2 is 58° because it is vertically opposite ∠3.
is important in
explaining your
∠1 is 122° because it is an interior angle on the same side of a transversal as ∠3.
reasoning.
BUILD YOUR SKILLS
1. In the diagram below, ℓ1 is parallel to ℓ2. State the measures of the indicated angles
and explain your reasoning.
71°
4
1
2
118°
3
ℓ1
ℓ2
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.)
D
2
3
68° 1
A
4
B
C
Chapter 5 Angles and Parallel Lines
3. Quadrilateral ABCD is a parallelogram in which ∠B measures 74°. Determine the
measures of the other angles and state your reasons.
B
A
74°
C
D
Example 2
Given the diagram below, identify all the pairs of parallel lines and explain your
selection.
100°
80°
110°
ℓ2
100°
70°
ℓ6
ℓ1
95°
ℓ5
ℓ3
ℓ4
SOL U T ION
ℓ6 is parallel to ℓ4. If you consider ℓ1 to be a transversal, 100° and 80° are interior
angles on the same side of the transversal.
ℓ1 is parallel to ℓ2. If ℓ6 is a transversal, the two 100° angles are corresponding angles.
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MathWorks 10 Workbook
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.
ℓ1
ℓ2
ℓ3
62°
103°
77°
ℓ4
ℓ5
ℓ6
68°
5. What size must ∠1 be if ℓ1 is parallel to ℓ2?
ℓ1
ℓ
123°
1
Chapter 5 Angles and Parallel Lines
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?
78°
106°
Pipe 2
Pipe 1
Example 3
Given parallelogram ABCD, determine the values of ∠B, ∠C, and ∠D in that order,
stating your reason for each measure.
A
B
130°
D
C
SOL U T ION
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°.
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MathWorks 10 Workbook
You can use the
notation AD || BC to
indicate that AD and
BC are parallel.
B is parallel to DC, and they are intersected by transversal BC. You know that ∠B
A
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 ℓ1 and ℓ2 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.
ℓ2
ℓ1
1
2
54°
t
t2
4
83°
3
SOLVING ANGLE MEASURES
Angle Measure
∠1 =
∠2 =
∠3 =
∠4 =
Reason
Chapter 5 Angles and Parallel Lines
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 2
2
Pipe 1
1
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MathWorks 10 Workbook
PRACTISE YOUR NEW SKILLS
1. Given the diagram below, where ℓ1 is parallel to ℓ2, find the measures of the
indicated angles and state your reasons.
ℓ1
1
4
2 112°
60°
ℓ2
3
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
Chapter 5 Angles and Parallel Lines
3. Identify the pairs of parallel lines in the following diagram. (Hint: The lines can be
extended.)
ℓ2
ℓ1
132°
48°
ℓ4
ℓ3
46°
ℓ5
46°
132°
ℓ6
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.)
A
89°
B
C
91°
134°
D
135°
E
F
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