1.

2 Measuring and Constructing Segments

Essential Question How can you measure and construct a
line segment?
Measuring Line Segments Using
Nonstandard Units
12
Work with a partner. 10
11
3
2
1 CM

4
9
5

a. Draw a line segment that has a length

6
7
8 9
8

7
10
11

of 6 inches.
12
6 14
13

5
15
16
17
4 18

b. Use a standard-sized paper clip to

19
3
20
21
22
2 24
23

measure the length of the line segment. 1

25
26
H
27
INC 28
29

Explain how you measured the line

30

MAKING SENSE segment in “paper clips.”

OF PROBLEMS
To be proficient in math,
you need to explain to c. Write conversion factors from paper clips
ps
yourself the meaning of to inches and vice versa.
a problem and look for
entry points to its solution. 1 paper clip = in.
1 in. = paper clip
d. A straightedge is a tool that you can use to draw a straight line. An example of a
straightedge is a ruler. Use only a pencil, straightedge, paper clip, and paper to draw
another line segment that is 6 inches long. Explain your process.

Measuring Line Segments Using

Nonstandard Units

Work with a partner.

a. Fold a 3-inch by 5-inch index card
fo
on one of its diagonals. ld
3 in.
b. Use the Pythagorean Theorem
to algebraically determine the
length of the diagonal in inches.
Use a ruler to check your answer. 5 in.
c. Measure the length and width of the index card in paper clips.
d. Use the Pythagorean Theorem to algebraically determine the length of the diagonal
in paper clips. Then check your answer by measuring the length of the diagonal in
paper clips. Does the Pythagorean Theorem work for any unit of measure? Justify
your answer.

Measuring Heights Using Nonstandard Units

Work with a partner. Consider a unit of length that is equal to the length of the
diagonal you found in Exploration 2. Call this length “1 diag.” How tall are you in
diags? Explain how you obtained your answer.

Communicate Your Answer

4. How can you measure and construct a line segment?

Section 1.2 Measuring and Constructing Segments 11

1.2 Lesson What You Will Learn
Use the Ruler Postulate.
Copy segments and compare segments for congruence.
Core Vocabul
Vocabulary
larry Use the Segment Addition Postulate.
postulate, p. 12
axiom, p. 12
Using the Ruler Postulate
coordinate, p. 12
distance, p. 12 In geometry, a rule that is accepted without proof is called a postulate or an axiom.
A rule that can be proved is called a theorem, as you will see later. Postulate 1.1 shows
construction, p. 13
how to find the distance between two points on a line.
congruent segments, p. 13
between, p. 14
Postulate names of points
Postulate 1.1 Ruler Postulate
A B
The points on a line can be matched one to
one with the real numbers. The real number x1 x2
that corresponds to a point is the coordinate
of the point. coordinates of points

The distance between points A and B, written

A AB B
as AB, is the absolute value of the difference
of the coordinates of A and B. x1 x2
AB = #x2 − x1#

Using the Ruler Postulate

—
Measure the length of ST to the nearest tenth of a centimeter.

S T

SOLUTION
Align one mark of a metric ruler with S. Then estimate the coordinate of T. For
example, when you align S with 2, T appears to align with 5.4.

S T

cm 1 2 3 4 5 6

ST = ∣ 5.4 − 2 ∣ = 3.4 Ruler Postulate

— is about 3.4 centimeters.

So, the length of ST

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

Use a ruler to measure the length of the segment to the nearest —18 inch.
1. 2.
M N P Q

3. 4.
U V W X

12 Chapter 1 Basics of Geometry

 Constructing and Comparing Congruent Segments
A construction is a geometric drawing that uses a limited set of tools, usually a
compass and straightedge.

Copying a Segment
Use a compass and straightedge to construct a line segment
—.
that has the same length as AB
A B

SOLUTION
Step 1 Step 2 Step 3

A B A B A B

C C C D

Draw a segment Use a straightedge Measure length Set your Copy length Place the compass at
—.
to draw a segment longer than AB —.
compass at the length of AB C. Mark point D on the new segment.
Label point C on the new segment. — has the same length as AB
So, CD —.

Core Concept
Congruent Segments
READING Line segments that have the same length are called congruent segments. You
In the diagram, the — is equal to the length of CD
can say “the length of AB —,” or you can say “AB
— is
red tick marks indicate —
congruent to CD .” The symbol ≅ means “is congruent to.”
—
AB ≅ — CD . When there
Lengths are equal. Segments are congruent.
is more than one pair of
A B AB = CD — ≅ CD
AB —
congruent segments, use
multiple tick marks.
C D “is equal to” “is congruent to”

Comparing Segments for Congruence

Plot J(−3, 4), K(2, 4), L(1, 3), and M(1, −2) in a coordinate plane. Then determine
— and LM
whether JK — are congruent.
SOLUTION
y Plot the points, as shown. To find the length of a horizontal segment, find the absolute
J(−3, 4) K(2, 4)
value of the difference of the x-coordinates of the endpoints.

L(1, 3) JK = ∣ 2 − (−3) ∣ = 5 Ruler Postulate

2 To find the length of a vertical segment, find the absolute value of the difference of the
y-coordinates of the endpoints.
−4 −2 2 4 x LM = ∣ −2 − 3 ∣ = 5 Ruler Postulate
−2
M(1, −2) — — have the same length. So, JK
JK and LM — ≅ LM
—.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

5. Plot A(−2, 4), B(3, 4), C(0, 2), and D(0, −2) in a coordinate plane. Then
— and CD
determine whether AB — are congruent.

Section 1.2 Measuring and Constructing Segments 13

 Using the Segment Addition Postulate
When three points are collinear, you can say that one point is between the other two.
A D E
B
C F

Point B is between Point E is not between

points A and C. points D and F.

Postulate
Postulate 1.2 Segment Addition Postulate
If B is between A and C, then AB + BC = AC. AC
If AB + BC = AC, then B is between A and C. A B C
AB BC

Using the Segment Addition Postulate

a. Find DF.
D 23 E 35 F

b. Find GH. 36

F 21 G H

SOLUTION
a. Use the Segment Addition Postulate to write an equation. Then solve the equation
to find DF.
DF = DE + EF Segment Addition Postulate
DF = 23 + 35 Substitute 23 for DE and 35 for EF.
DF = 58 Add.
b. Use the Segment Addition Postulate to write an equation. Then solve the
equation to find GH.
FH = FG + GH Segment Addition Postulate
36 = 21 + GH Substitute 36 for FH and 21 for FG.
15 = GH Subtract 21 from each side.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

Use the diagram at the right. 23

50
6. Use the Segment Addition Postulate to find XZ. X Y
Z
7. In the diagram, WY = 30. Can you use the W
Segment Addition Postulate to find the distance
between points W and Z ? Explain your reasoning.
144
8. Use the diagram at the left to find KL.
J 37 K L

14 Chapter 1 Basics of Geometry

 Using the Segment Addition Postulate

The cities shown on the map lie approximately in a straight line. Find the distance
from Tulsa, Oklahoma, to St. Louis, Missouri.

St. Louis
738 mi

T Tulsa

377 mi

L Lubbock

SOLUTION
1. Understand the Problem You are given the distance from Lubbock to St. Louis
and the distance from Lubbock to Tulsa. You need to find the distance from Tulsa
to St. Louis.
2. Make a Plan Use the Segment Addition Postulate to find the distance from
Tulsa to St. Louis.
3. Solve the Problem Use the Segment Addition Postulate to write an equation.
Then solve the equation to find TS.
LS = LT + TS Segment Addition Postulate
738 = 377 + TS Substitute 738 for LS and 377 for LT.
361 = TS Subtract 377 from each side.
So, the distance from Tulsa to St. Louis is about 361 miles.
4. Look Back Does the answer make sense in the context of the problem? The
distance from Lubbock to St. Louis is 738 miles. By the Segment Addition
Postulate, the distance from Lubbock to Tulsa plus the distance from Tulsa to
St. Louis should equal 738 miles.
377 + 361 = 738 ✓
Monitoring Progress Help in English and Spanish at BigIdeasMath.com

9. The cities shown on the map lie approximately in a straight line. Find the distance
from Albuquerque, New Mexico, to Provo, Utah.

Provo
P

680 mi Albuquerque
A
231 mi

Carlsbad
C

Section 1.2 Measuring and Constructing Segments 15

 1.2 Exercises Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check

—
1. WRITING Explain how XY and XY are different.

2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

7 B
3
C
A

Find AC + CB. Find BC − AC.

Find AB. Find CA + BC.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 –6, use a ruler to measure the length of In Exercises 15–22, find FH. (See Example 3.)
the segment to the nearest tenth of a centimeter.
15. 8
(See Example 1.) F
3. G 14

4. H

5. 16. 7
H
19 G
6.

F
CONSTRUCTION In Exercises 7 and 8, use a compass and
straightedge to construct a copy of the segment. 17. 12
H
7. Copy the segment in Exercise 3. 11
G
8. Copy the segment in Exercise 4. F

In Exercises 9–14, plot the points in a coordinate 18.

plane. Then determine whether — AB and —
4
CD are F
congruent. (See Example 2.) G
15
9. A(−4, 5), B(−4, 8), C(2, −3), D(2, 0)
H
10. A(6, −1), B(1, −1), C(2, −3), D(4, −3)
19.
11. A(8, 3), B(−1, 3), C(5, 10), D(5, 3)
37 G
H 13
12. A(6, −8), B(6, 1), C(7, −2), D(−2, −2)
F
13. A(−5, 6), B(−5, −1), C(−4, 3), D(3, 3)

14. A(10, −4), B(3, −4), C(−1, 2), D(−1, 5)

16 Chapter 1 Basics of Geometry

20. 22 26. MODELING WITH MATHEMATICS In 2003, a remote-
controlled model airplane became the first ever to fly
F H 15 G nonstop across the Atlantic Ocean. The map shows the
airplane’s position at three different points during its
21. flight. Point A represents Cape Spear, Newfoundland,
F point B represents the approximate position after
42 1 day, and point C represents Mannin Bay, Ireland.
The airplane left from Cape Spear and landed in
H
Mannin Bay. (See Example 4.)
22
G

22. Europe
North C
G America B
A 601 mi
1282 mi
53
40
Atlantic Ocean

H
F a. Find the total distance the model airplane flew.
b. The model airplane’s flight lasted nearly 38 hours.
ERROR ANALYSIS In Exercises 23 and 24, describe and Estimate the airplane’s average speed in
correct the error in finding the length of —
AB . miles per hour.

A B 27. USING STRUCTURE Determine whether the statements

are true or false. Explain your reasoning.
cm 1 2 3 4 5 6
C
B
23.

✗ AB = 1 − 4.5 = −3.5 A D E H

24.

✗ AB = ∣ 1 + 4.5 ∣ = 5.5
a. B is between A and C.
b. C is between B and E.
c. D is between A and H.

25. ATTENDING TO PRECISION The diagram shows an d. E is between C and F.

insect called a walking stick. Use the ruler to estimate
the length of the abdomen and the length of the 28. MATHEMATICAL CONNECTIONS Write an expression
thorax to the nearest —14 inch. How much longer is the for the length of the segment.
walking stick’s abdomen than its thorax? How many —
a. AC
times longer is its abdomen than its thorax?

abdomen thorax A x+2 B 7x − 3 C

—
b. QR

13y + 25

P 8y + 5 Q R
INCH
1 2 3 4 5 6 7
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Section 1.2 Measuring and Constructing Segments 17

29. MATHEMATICAL CONNECTIONS Point S is between 33. REASONING You travel from City X to City Y. You
—. Use the information to write an
points R and T on RT know that the round-trip distance is 647 miles. City Z, a
equation in terms of x. Then solve the equation and city you pass on the way, is 27 miles from City X. Find
find RS, ST, and RT. the distance from City Z to City Y. Justify your answer.
a. RS = 2x + 10 b. RS = 3x − 16
ST = x − 4 ST = 4x − 8 34. HOW DO YOU SEE IT? The bar graph shows the
win-loss record for a lacrosse team over a period of
RT = 21 RT = 60
three years. Explain how you can apply the Ruler
c. RS = 2x − 8 d. RS = 4x − 9 Postulate (Post. 1.1) and the Segment Addition
ST = 11 ST = 19 Postulate (Post. 1.2) when interpreting a stacked
bar graph like the one shown.
RT = x + 10 RT = 8x − 14
Win-Loss Record
30. THOUGHT PROVOKING Is it possible to design a table
where no two legs have the same length? Assume Wins Losses
that the endpoints of the legs must all lie in the same
Year 1
plane. Include a diagram as part of your answer.
Year 2

Year 3
31. MODELING WITH MATHEMATICS You have to walk
0 2 4 6 8 10 12
from Room 103 to Room 117.
Number of games

103 107 113 117

86 ft 35. ABSTRACT REASONING The points (a, b) and (c, b)

22 ft form a segment, and the points (d, e) and (d, f ) form
a segment. Create an equation assuming the segments
101 105 109 111 115 119 121 are congruent. Are there any letters not used in the
equation? Explain.

a. How many feet do you travel from Room 103 36. MATHEMATICAL CONNECTIONS In the diagram,
to Room 117? — ≅ BC
AB —, AC— ≅ CD —, and AD = 12. Find the
lengths of all segments in the diagram. Suppose you
b. You can walk 4.4 feet per second. How many
choose one of the segments at random. What is the
minutes will it take you to get to Room 117?
probability that the measure of the segment is greater
c. Why might it take you longer than the time in than 3? Explain your reasoning.
part (b)?
D
32. MAKING AN ARGUMENT Your friend and your cousin C
B
discuss measuring with a ruler. Your friend says that A
you must always line up objects at the zero on a ruler.
Your cousin says it does not matter. Decide who is 37. CRITICAL THINKING Is it possible to use the Segment
correct and explain your reasoning. Addition Postulate (Post. 1.2) to show FB > CB or
that AC > DB? Explain your reasoning.

A D F C B

Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons

Simplify. (Skills Review Handbook)

−4 + 6 — — 7+6
38. — 39. √ 20 + 5 40. √ 25 + 9 41. —
2 2

Solve the equation. (Skills Review Handbook)

3+y −5 + x
42. 5x + 7 = 9x − 17 43. — = 6 44. — = −9 45. −6x − 13 = −x − 23
2 2

18 Chapter 1 Basics of Geometry