- •Study Skills Workshop
- •1.1 An Introduction to the Whole Numbers
- •1.2 Adding Whole Numbers
- •1.3 Subtracting Whole Numbers
- •1.4 Multiplying Whole Numbers
- •1.5 Dividing Whole Numbers
- •1.6 Problem Solving
- •1.7 Prime Factors and Exponents
- •1.8 The Least Common Multiple and the Greatest Common Factor
- •1.9 Order of Operations
- •THINK IT THROUGH Education Pays
- •2.1 An Introduction to the Integers
- •THINK IT THROUGH Credit Card Debt
- •2.2 Adding Integers
- •THINK IT THROUGH Cash Flow
- •2.3 Subtracting Integers
- •2.4 Multiplying Integers
- •2.5 Dividing Integers
- •2.6 Order of Operations and Estimation
- •Cumulative Review
- •3.1 An Introduction to Fractions
- •3.2 Multiplying Fractions
- •3.3 Dividing Fractions
- •3.4 Adding and Subtracting Fractions
- •THINK IT THROUGH Budgets
- •3.5 Multiplying and Dividing Mixed Numbers
- •3.6 Adding and Subtracting Mixed Numbers
- •THINK IT THROUGH
- •3.7 Order of Operations and Complex Fractions
- •Cumulative Review
- •4.1 An Introduction to Decimals
- •4.2 Adding and Subtracting Decimals
- •4.3 Multiplying Decimals
- •THINK IT THROUGH Overtime
- •4.4 Dividing Decimals
- •THINK IT THROUGH GPA
- •4.5 Fractions and Decimals
- •4.6 Square Roots
- •Cumulative Review
- •5.1 Ratios
- •5.2 Proportions
- •5.3 American Units of Measurement
- •5.4 Metric Units of Measurement
- •5.5 Converting between American and Metric Units
- •Cumulative Review
- •6.2 Solving Percent Problems Using Percent Equations and Proportions
- •6.3 Applications of Percent
- •6.4 Estimation with Percent
- •6.5 Interest
- •Cumulative Review
- •7.1 Reading Graphs and Tables
- •THINK IT THROUGH The Value of an Education
- •Cumulative Review
- •8.1 The Language of Algebra
- •8.2 Simplifying Algebraic Expressions
- •8.3 Solving Equations Using Properties of Equality
- •8.4 More about Solving Equations
- •8.5 Using Equations to Solve Application Problems
- •8.6 Multiplication Rules for Exponents
- •Cumulative Review
- •9.1 Basic Geometric Figures; Angles
- •9.2 Parallel and Perpendicular Lines
- •9.3 Triangles
- •9.4 The Pythagorean Theorem
- •9.5 Congruent Triangles and Similar Triangles
- •9.6 Quadrilaterals and Other Polygons
- •9.7 Perimeters and Areas of Polygons
- •THINK IT THROUGH Dorm Rooms
- •9.8 Circles
- •9.9 Volume
- •Cumulative Review
WRITING
99.PHRASES Explain what you think each of these phrases means. How is geometry involved?
a.The president did a complete 180-degree flip on the subject of a tax cut.
b.The rollerblader did a “360” as she jumped off the ramp.
100.In the statements below, the ° symbol is used in two different ways. Explain the difference.
m( A) 85° and 85°F
101.Can two angles that are complementary be equal? Explain.
102.Explain why the angles highlighted below are not vertical angles.
9.2 Parallel and Perpendicular Lines |
725 |
REVIEW
103.Add: 12 23 34
104.Subtract: 34 18 12
105.Multiply: 58 152 65
106.Divide: 1217 344
S E C T I O N 9.2
Parallel and Perpendicular Lines
In this section, we will consider parallel and perpendicular lines. Since parallel lines are always the same distance apart, the railroad tracks shown in figure (a) illustrate one application of parallel lines. Figure (b) shows one of the events of men’s gymnastics, the parallel bars. Since perpendicular lines meet and form right angles, the monument and the ground shown in figure (c) illustrate one application of perpendicular lines.
Objectives
1Identify and define parallel and perpendicular lines.
2Identify corresponding angles, interior angles, and alternate interior angles.
3Use properties of parallel lines cut by a transversal to find unknown angle measures.
The symbol 
indicates a right angle.
(a) |
(b) |
(c) |
1 Identify and define parallel and perpendicular lines.
If two lines lie in the same plane, they are called coplanar. Two coplanar lines that do not intersect are called parallel lines. See figure (a) on the next page. If two lines do not lie in the same plane, they are called noncoplanar. Two noncoplanar lines that do not intersect are called skew lines.
726 |
Chapter 9 An Introduction to Geometry |
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Parallel lines |
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Perpendicular lines |
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Parallel Lines
Parallel lines are coplanar lines that do not intersect.
Some lines that intersect are perpendicular. See figure (b) above.
Perpendicular Lines
Perpendicular lines are lines that intersect and form right angles.
The Language of Mathematics If lines l1 (read as “l sub 1”) and l2
(read as “l sub 2”) are parallel, we can write l1 l2, where the symbol is read as “is parallel to.”
If lines l1 and l2 are perpendicular, we can write l1 l2, where the symbol is read as “is perpendicular to.”
2Identify corresponding angles, interior angles, and alternate interior angles.
A line that intersects two coplanar lines in two distinct |
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intersecting lines l2 and l3. |
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When two lines are cut by a transversal, all eight |
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angles that are formed are important in the study of parallel lines. Descriptive names are given to several pairs of these angles.
In the figure below, four pairs of corresponding angles are formed.
Corresponding angles |
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Corresponding Angles
If two lines are cut by a transversal, then the angles on the same side of the transversal and in corresponding positions with respect to the lines are called corresponding angles.
9.2 Parallel and Perpendicular Lines |
727 |
In the figure below, four interior angles are formed.
Interior angles |
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In the figure below, two pairs of alternate interior angles are formed.
Alternate interior angles |
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Alternate Interior Angles
If two lines are cut by a transversal, then the nonadjacent angles on opposite sides of the transversal and on the interior of the two lines are called alternate interior angles.
Success Tip Alternate interior angles are easily spotted because they form a Z-shape or a backward Z-shape, as shown below.
EXAMPLE 1 Refer to the figure. Identify:
a.all pairs of corresponding angles
b.all interior angles
c.all pairs of alternate interior angles
Strategy When two lines are cut by a transversal, eight angles are formed. We will consider the relative position of the angles with respect to the two lines and the transversal.
Transversal
7
8
6
5
3 4
2 1
WHY There are four pairs of corresponding angles, four interior angles, and two pairs of alternate interior angles.
Self Check 1
Refer to the figure below. Identify:
a.all pairs of corresponding angles
b.all interior angles
c.all pairs of alternate interior angles
8
1
7
2
6
3
5
4
Now Try Problem 21
728 |
Chapter 9 An Introduction to Geometry |
Solution
a.To identify corresponding angles, we examine the angles to the right of the transversal and the angles to the left of the transversal. The pairs of corresponding angles in the figure are
• 1 and 5 |
• 4 and 8 |
• 2 and 6 |
• 3 and 7 |
b.To identify the interior angles, we determine the angles inside the two lines cut by the transversal. The interior angles in the figure are
3, 4, 5, and 6
c.Alternate interior angles are nonadjacent angles on opposite sides of the transversal inside the two lines. Thus, the pairs of alternate interior angles in the figure are
• 3 and 5 |
• 4 and 6 |
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3Use properties of parallel lines cut by a transversal to find unknown angle measures.
Lines that are cut by a transversal may or may not be parallel. When a pair of parallel lines are cut by a transversal, we can make several important observations about the angles that are formed.
1.Corresponding angles property: If two parallel lines are cut by a transversal,
each pair of corresponding angles are congruent. In the figure below, if l1 l2, then 1 5, 3 7, 2 6, and 4 8.
2.Alternate interior angles property: If two parallel lines are cut by a transversal,
alternate interior angles are congruent. In the figure below, if l1 l2, then3 6 and 4 5.
3.Interior angles property: If two parallel lines are cut by a transversal, interior
angles on the same side of the transversal are supplementary. In the figure below, if l1 l2, then 3 is supplementary to 5 and 4 is supplementary to 6.
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4.If a transversal is perpendicular to one of two parallel lines, it is also perpendicular to the other line. In figure (a) below, if l1 l2 and l3 l1, then l3 l2.
5.If two lines are parallel to a third line, they are parallel to each other. In figure
(b) below, if l1 l2 and l1 l3, then l2 l3.
l3
l1 
l1 
l2 
l2 
l3 
(a) |
(b) |
Refer to the figure. If l1 l2 and m( 3) 120°, find the measures of the other seven angles that are labeled.
Strategy We will look for vertical angles, supplementary angles, and alternate interior angles in the figure.
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9.2 Parallel and Perpendicular Lines |
729 |
Self Check 2
Refer to the figure for Example 2. If l1 l2 and m( 8) 50°, find the measures of the other seven angles that are labeled.
Now Try Problem 23
WHY The facts that we have studied about
vertical angles, supplementary angles, and alternate interior angles enable us to use known angle measures to find unknown angle measures.
Solution
m( 1) 60° 3 and 1 are supplementary: m( 3) m( 1) 180°. m( 2) 120° Vertical angles are congruent: m( 2) m( 3). m( 4) 60° Vertical angles are congruent: m( 4) m( 1).
m( 5) 60° If two parallel lines are cut by a transversal, alternate interior angles are congruent: m( 5) m( 4).
m( 6) 120° If two parallel lines are cut by a transversal, alternate interior angles are congruent: m( 6) m( 3).
m( 7) 120° Vertical angles are congruent: m( 7) m( 6). m( 8) 60° Vertical angles are congruent: m( 8) m( 5).
Some geometric figures contain two transversals.
Refer to the figure. If AB DE, which pairs of angles are congruent?
Strategy We will use the corresponding angles property twice to find two pairs of congruent angles.
· ·
WHY Both AC and BC are transversals cutting the
parallel line segments AB and DE.
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Solution Since AB DE, and AC is a transversal cutting them, corresponding angles are congruent. So we have
A 1
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Since AB DE and BC is a transversal cutting them, corresponding angles must be congruent. So we have
B 2
EXAMPLE 4 |
In the figure, l1 |
l2. Find x. |
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Strategy We will use the corresponding angles property to write an equation that mathematically models the situation.
WHY We can then solve the equation to find x.
9x − 15°
l1 
6x + 30°
l2 
Solution In the figure, two corresponding angles have degree measures that are represented by the algebraic expressions 9x 15° and 6x 30°. Since l1 l2, this pair of corresponding angles are congruent.
Self Check 3
See the figure below. If YZ MN, which pairs of angles are congruent?
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Self Check 4
In the figure below, l1 l2. Find y.
l1
7y − 14°
l2 
4y + 10°
Now Try Problem 27
730 |
Chapter 9 An Introduction to Geometry |
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9x 15° 6x 30° Since the angles are congruent, their measures are equal. |
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3x 15° 30° |
To eliminate 6x from the right side, subtract 6x from both |
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sides. |
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3x 45° |
To isolate the variable term 3x, undo the subtraction of |
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15° by adding 15° to both sides: 30° 15° 45°. |
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To isolate x, undo the multiplication by 3 by dividing both |
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Thus, x is 15°.
Self Check 5
In the figure below, l1 l2.
a.Find x.
b.Find the measures of both angles labeled in the figure.
l1
2x + 50°
x + 40°
l2
Now Try Problem 29
EXAMPLE 5 |
In the figure, l1 |
l2. |
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a.Find x.
b.Find the measures of both angles labeled in the figure.
Strategy We will use the interior angles property to write an equation that mathematically models the situation.
WHY We can then solve the equation to find x.
Solution
l1 
3x + 20° l2 
3x − 80°
a.Because the angles are interior angles on the same side of the transversal, they are supplementary.
3x 80° 3x 20° 180° |
The sum of the measures of two supplementary |
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6x 60° 180° |
Combine like terms: 3x 3x 6x. |
6x 240° |
To undo the subtraction of 60°, add 60° to both |
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sides: 180° 60° 240°. |
x 40° |
To isolate x, undo the multiplication by 6 by |
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Thus, x is 40°.
This problem may be solved using a different approach. In the figure below, we see that 1 and the angle with measure 3x 80° are corresponding angles.
Because l1 and l2 are parallel, all pairs of corresponding angles are congruent. Therefore,
m( 1) 3x 80°
1
l1
3x + 20° l2 
3x − 80°
In the figure, we also see that 1 and the angle with measure 3x 20° are supplementary. That means that the sum of their measures must be 180°. We have
m( 1) |
3x 20° |
180° |
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3x 80° |
3x 20° |
180° |
Replace m( 1) with 3x 80°. |
This is the same equation that we obtained in the previous solution. When it is solved, we find that x is 40°.
9.2 Parallel and Perpendicular Lines |
731 |
b.To find the measures of the angles in the figure, we evaluate the expressions 3x 20° and 3x 80° for x 40°.
3x 20° 3(40°) 20° |
3x 80° 3(40°) 80° |
120° 20° |
120° 80° |
140° |
40° |
The measures of the angles labeled in the figure are 140° and 40°.
ANSWERS TO SELF CHECKS
1. |
a. 1 and 3, 2 and 4, 8 and 6, 7 and 5 b. 2, 7, 3, and 6 c. 2 and |
6, 7 and 3 2. m( 5) 50°, m( 7) 130°, m( 6) 130°, m( 3) 130°, |
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m( 4) 50°, m( 1) 50°, and m( 2) 130° 3. 1 Y, 3 Z 4. 8° |
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a. 30° b. 110°, 70° |
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S E C T I O N |
9.2 |
STUDY SET |
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VOCABULARY |
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lines are lines that intersect and form |
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6.In the figure above, 2 and 6 are angles.
CONCEPTS
7.a. Draw two parallel lines. Label them l1 and l2.
b.Draw two lines that are not parallel. Label them l1 and l2.
8.a. Draw two perpendicular lines. Label them l1 and l2.
b.Draw two lines that are not perpendicular. Label them l1 and l2.
9.a. Draw two parallel lines cut by a transversal. Label the lines l1 and l2 and label the transversal l3.
b.Draw two lines that are not parallel and cut by a transversal. Label the lines l1 and l2 and label the transversal l3.
10.Draw three parallel lines. Label them l1, l2, and l3.
In Problems 11–14, two parallel lines are cut by a transversal. Fill in the blanks.
11.In the figure below, on the left, ABC BEF. When two parallel lines are cut by a transversal, angles are congruent.
12.In the figure below, on the right, 1 2. When two parallel lines are cut by a transversal,
angles are congruent.
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732Chapter 9 An Introduction to Geometry
13.In the figure below, on the left, m( ABC) m( BCD) 180°. When two parallel lines are
cut by a transversal, angles on the same
side of the transversal are supplementary.
14.In the figure below, on the right, 8 6. When two parallel lines are cut by a transversal,
angles are congruent.
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16.In the figure below, on the right, l1 l2 and l2 l3. What can you conclude about l1 and l3?
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NOTATION
Fill in the blanks.
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The symbol indicates a |
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The symbol is read as “is |
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The symbol is read as “is |
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GUIDED PRACTICE
21.Refer to the figure below and identify each of the following. See Example 1.
a.corresponding angles
b.interior angles
c.alternate interior angles
2
1 3
4
6
5 7
22.Refer to the figure below and identify each of the following. See Example 1.
a.corresponding angles
b.interior angles
c.alternate interior angles
C
A B D
E F G
H
23.In the figure below, l1 l2 and m( 4) 130°. Find the measures of the other seven angles that are labeled.
See Example 2.
l1
7 8
5
6
l2
3
4
1
2
24.In the figure below, l1 l2 and m( 2) 40°. Find the measures of the other angles. See Example 2.
l1
1 2
6
5
l2
3 7
4 8
25.In the figure below, YM XN. Which pairs of angles are congruent? See Example 3.
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26.In the figure below, AE BD. Which pairs of angles are congruent? See Example 3.
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In Problems 27 and 28, l1 l2. First find x. Then determine the measure of each angle that is labeled in the figure.
See Example 4.
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4x – 8°
2x + 16°
28.
2x + 10°
l1 
4x – 10°
l2
In Problems 29 and 30, l1 l2. First find x. Then determine the measure of each angle that is labeled in the figure.
See Example 5.
29.
l1
5x
6x + 70°
l2 
30.
l1 
5x + 5°
l2 
2x
9.2 Parallel and Perpendicular Lines |
733 |
TRY IT YOURSELF
31.In the figure below, l1 AB. Find:
a.m( 1), m( 2), m( 3), and m( 4)
b.m( 3) m( 4) m( ACD)
c.m( 1) m( ABC) m( 4)
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32.In the figure below, AB DE. Find m( B), m( E), and m( 1).
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33.In the figure below, AB DE. What pairs of angles are congruent? Explain your reasoning.
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15x + 36° |
734 |
Chapter 9 An Introduction to Geometry |
In Problems 35–38, first find x. Then determine the measure of each angle that is labeled in the figure.
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9x – 38°
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6x – 2° |
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38. |
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7x – 2° |
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APPLICATIONS
39.CONSTRUCTING PYRAMIDS The Egyptians used a device called a plummet to tell whether stones were properly leveled. A plummet (shown below) is made up of an A-frame and a plumb bob suspended from the peak of the frame. How could a builder use a plummet to tell that the two stones on the left are not level and that the three stones on the right are level?
Plummet
Plumb bob
40.DIAGRAMMING SENTENCES English instructors have their students diagram sentences to help teach proper sentence structure. A diagram of the sentence The cave was rather dark and damp is shown below. Point out pairs of parallel and perpendicular lines used in the diagram.
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dark |
cave |
was |
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and |
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rather |
damp |
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41.BEAUTY TIPS The figure to the right shows how one website
illustrated the “geometry” of the ideal eyebrow. If l1 l2 and m( DCF) 130°, find m( ABE).
l1 E
A l2
D
B
C
F
42.PAINTING SIGNS For many sign painters, the most difficult letter to paint is a capital E because of all the right angles involved. How many right angles are there?
E
43.HANGING WALLPAPER Explain why the concepts of perpendicular and parallel are both important when hanging wallpaper.
44.TOOLS What geometric concepts are seen in the design of the rake shown here?
45.SEISMOLOGY The figure shows how an earthquake fault occurs when two blocks of earth move apart and one part drops down. Determine the measures of 1, 2, and 3.
3
2
1
105°
46.CARPENTRY A carpenter cross braced three 2 4’s as shown below and then used a tool to
measure the three highlighted angles in red. Are the 2 4’s parallel? Explain your answer.
Cross brace |
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2 × 4 |
45° |
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2 × 4 |
43° |
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WRITING
47.PARKING DESIGN Using terms from this section, write a paragraph describing the parking layout shown below.
North side of street
West 
East
Planter
South side of street
48.In the figure below, l1 l2. Explain why m( BDE) 91°.
l1 |
l2 |
A |
C |
F 89° |
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91° B |
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E |
9.2 Parallel and Perpendicular Lines |
735 |
49.In the figure below, l1 l2. Explain why m( FEH) 100°.
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100° B
E
D H
50.In the figure below, l1 l2. Explain why the figure must be mislabeled.
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59° |
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118° |
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51.Are pairs of alternate interior angles always congruent? Explain.
52.Are pairs of interior angles on the same side of a transversal always supplementary? Explain.
REVIEW
53.Find 60% of 120.
54.80% of what number is 400?
55.What percent of 500 is 225?
56.Simplify: 3.45 7.37 2.98
57.Is every whole number an integer?
58.Multiply: 2 15 4 37
59.Express the phrase as a ratio in lowest terms: 4 ounces to 12 ounces
60.Convert 5,400 milligrams to kilograms.