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Nets and Drawings for
Visualizing Geometry
1-1
Vocabulary
Review
Identify each figure as two-dimensional or three-dimensional.
1.
2.
3.
Vocabulary Builder
polygon
polygon (noun)
PAHL
ih gahn
Main Idea: A polygon is a closed figure, so all sides meet. No sides cross each
ch other
other.
Examples: Triangles, rectangles, pentagons, hexagons, and octagons are polygons.
Use Your Vocabulary
Underline the correct word(s) to complete each sentence.
4. A polygon is formed by two / three or more straight sides.
5. A circle is / is not a polygon.
6. A triangle / rectangle is a polygon with three sides.
7. The sides of a polygon are curved / straight .
8. Two / Three sides of polygon meet at the same point.
Cross out the figure(s) that are NOT polygons.
9.
10.
A
E
B
X
W
D
Chapter 1
S
N
L
C
R
11.
M
P
Q
2
V
T
U
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Definition A polygon is a two-dimensional figure with three or more
sides, where each side meets exactly two other sides at their endpoints.
Underline the correct word(s) to complete the sentence.
12. A net is a two-dimensional / three-dimensional diagram that you can fold to form
a two-dimensional / three-dimensional figure.
13. Circle the net that you can NOT fold into a cube.
Use the net of a cube at the right for Exercises 14 and 15.
14. Suppose you fold the net into a cube. What color will be opposite each face?
red
blue
green
15. Suppose you fold the net into a cube. What color is missing from each view?
?
?
?
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Problem 1 Identifying a Solid From a Net
Got It? The net at the right folds into the cube shown.
E
Which letters will be on the top and right side of the cube?
16. Four of the five other letters will touch some side of Face B when
the net is folded into a cube. Cross out the letter of the side that
will NOT touch some side of Face B.
A
C
D
E
A
B
C
D
B
F
F
17. Which side of the cube will that letter be on? Circle your answer.
Top
Bottom
Right
Left
Back
18. Use the net. Which face is to the right of Face B? How do you know?
_______________________________________________________________________
_______________________________________________________________________
19. Use the net. Which face is on the top of the cube? How do you know?
_______________________________________________________________________
_______________________________________________________________________
3
Lesson 1-1
Problem 2 Drawing a Net From a Solid
Got It? What is a net for the figure at the right? Label the net with
its dimensions.
10 cm
10 cm
Write T for true or F for false.
7 cm
20. Three of the faces are rectangles.
4 cm
21. Four of the faces are triangles.
22. The figure has five faces in all.
23. Now write a description of the net.
_______________________________________________________________________
_______________________________________________________________________
24. Circle the net that represents the figure above.
10 cm
10 cm
7 cm
4 cm
10 cm
4 cm
7 cm
10 cm
7 cm
10 cm
Problem 3 Isometric Drawing
Got It? What is an isometric drawing of this cube structure?
25. The cube structure has
edges that you can see and
vertices that you can see.
26. The isometric dot paper shows 2 vertices and
1 edge of the cube structure. Complete the
isometric drawing.
Chapter 1
7 cm
10 cm
4
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10 cm
7 cm
Problem 4 Orthographic Drawing
Got It? What is the orthographic drawing for this isometric drawing?
27. Underline the correct word to complete the sentence.
If you built the figure out of cubes, you would use seven / eight cubes
28. Cross out the drawing below that is NOT part of the orthographic drawing.
Then label each remaining drawing. Write Front, Right, or Top.
Fro
nt
ht
Rig
Lesson Check • Do you UNDERSTAND?
Vocabulary Tell whether each drawing is isometric, orthographic, a net, or none.
29. Write dot paper, one view, three views or none. Then label each figure.
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Top
Front
Fro
nt
Right
ht
Rig
Math Success
Check off the vocabulary words that you understand.
net
isometric drawing
orthographic drawing
Rate how well you can use nets, isometric drawings, and orthographic drawings.
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Lesson 1-1
Name
1-1
Class
Date
Practice
Form K
Nets and Drawings for Visualizing Geometry
Match each three-dimensional figure with its net.
1.
2.
A.
3.
B.
C.
Draw a net for each figure. Label the net with its dimensions. To start, visualize
opening the end flaps of the prism.
4.
5.
Make an isometric drawing of each cube structure on isometric
dot paper. To start, draw the front edge.
6.
7.
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Practice (continued)
Form K
Nets and Drawings for Visualizing Geometry
8. Visualization If the net shown at the right is
folded so that side A is the front of the cube, what
letters will be on the top, bottom, right, left, and
back?
9. Multiple Representations How many different
nets can you make for a cube? Draw at least five
nets.
10. Reasoning Are there more, fewer, or the same
number of nets possible for a rectangular prism
than for a cube? Explain.
11. Open-Ended Make an isometric drawing of a
structure you can build using six cubes.
11. Error Analysis A classmate drew the net of a
triangular prism shown at the right. Explain the
error in your classmate’s drawing. Draw the net
correctly.
Match the package with its net.
13.
A.
14.
15.
B.
C.
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Additional Problems
1-1
Net and Drawings for Visualizing Geometry
Problem 1
The net below folds into a cube. Which letters will be on the top
and front of the cube?
B
F
E
D
C
A
Problem 2
What is a net for the cereal box below? Label the net with its
dimensions.
12 in.
7 in.
2 in.
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Additional Problems (continued)
Net and Drawings for Visualizing Geometry
Problem 3
What is the isometric drawing of the cube structure below?
Problem 4
What is the front orthographic drawing for the isometric
drawing below?
Fro
nt
ht
Rig
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Form G
Practice
Nets and Drawings for Visualizing Geometry
Match each three-dimensional figure with its net.
1.
2.
A.
B.
3.
C.
Make an isometric drawing of each cube structure on isometric dot paper.
4.
5.
6.
7. Error Analysis Two students draw nets for the solid shown below. Who is
correct, Student A or Student B? Explain.
8. You want to make a one-piece cardboard cutout for a child to fold and tape
together to make a dollhouse. It includes a floor, a complete roof, and four
walls. Draw a net for the dollhouse.
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(continued)
Form G
Nets and Drawings for Visualizing Geometry
For each isometric drawing, make an orthographic drawing. Assume there are
no hidden cubes.
9.
10.
11.
12. Visualization Look at the orthographic
drawing at the right. Make an isometric drawing
of the structure.
13. Choose the nets that will fold to make a cube.
A.
B.
C.
14. Writing To make a net from a container, you start by cutting one of the seams along
an edge where two sides meet. If you wanted to make a different net for the container,
what would you do differently?
15. Multiple Representations Draw two different
nets for the solid shown at the right.
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1-2
Points, Lines, and Planes
Vocabulary
Review
Draw a line from each net in Column A to the three-dimensional figure it
represents in Column B.
Column A
Column B
1.
2.
Vocabulary Builder
conjecture (noun, verb) kun JEK chur
Main Idea: A conjecture is a guess or a prediction.
Definition: A conjecture is a conclusion reached by using inductive reasoning.
Use Your Vocabulary
Write noun or verb to identify how the word conjecture is used in each sentence.
4. You make a conjecture that your volleyball team will win.
5. Assuming that your sister ate the last cookie is a conjecture.
6. You conjecture that your town will build a swimming pool.
Chapter 1
6
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3.
Key Concept Undefined and Defined Terms
Write the correct word from the list on the right. Use each word only once.
Undefined or Defined Term
Diagram
Name
A
A
point
7.
line
opposite rays
plane
point
B
8.
line
9.
plane
ray
AB
segment
A
10.
segment
11.
ray
12.
opposite rays
A
B
P
P
C
A
B
A
B
A
C
AB
AB
B
CA, CB
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Column A
Column B
z
B
Draw a line from each item in Column A to its description in Column B.
A
C
13. plane HGE
intersection of AB and line z
14. BF
plane AEH
15. plane DAE
line through points F and E
16. line y
intersection of planes ABF and CGF
17. point A
plane containing points E, F, and G
D
F
G
x
E
y
H
Postulates 1–1, 1–2, 1–3, and 1–4
18. Complete each postulate with line, plane, or point.
Postulate 1-1 Through any two points there is exactly one 9.
line
Postulate 1-2 If two distinct lines intersect, then they intersect in exactly one 9.
point
Postulate 1-3 If two distinct planes intersect, then they intersect in exactly one 9.
line
Postulate 1-4 Through any three noncollinear points there is exactly one 9.
plane
7
Lesson 1-2
Write P if the statement describes a postulate or U if it describes an undefined term.
19. A point indicates a location and has no size.
20. Through any two points there is exactly one line.
21. A line is represented by a straight path that has no thickness and extends in
two opposite directions without end.
22. If two distinct planes intersect, then they intersect in exactly one line.
23. If two distinct lines intersect, then they intersect in exactly one point.
24. Through any three nontcollinear points there is exactly one plane.
Naming Segments and Rays
Problem 2
)
)
Got It? Reasoning EF and FE form a line. Are they opposite rays? Explain.
For Exercises 25–29, use the line below.
)
)
)
)
)
)
26. Do EF and FE share an endpoint?
Yes / No
27. Do EF and FE form a line?
Yes / No
28. Are EF and FE opposite rays?
Yes / No
)
)
29. Explain your answer to Exercise 28.
_______________________________________________________________________
_______________________________________________________________________
Problem 3 Finding the Intersection of Two Planes
D
Got It? Each surface of the box at the right represents
* ) part of a plane.
A
What are the names of two planes that intersect in BF ?
* )
E
30. Circle the points that are on BF or in one of the two planes.
A
B
C
E
D
F
H
G
BCD
BCG
CDH
* )
32. Now name two planes that intersect in BF .
Chapter 1
8
B
F
H
31. Circle another name for plane BFG. Underline another name for plane BFE.
ABF
C
FGH
G
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25. Draw and label points E and F. Then draw EF in one color and FE in another color.
Problem 4 Using Postulate 1–4
Got It? What plane contains points L, M, and N? Shade the plane.
M
J
33. Use the figure below. Draw LM , LN , and MN as dashed segments.
Then shade plane LMN.
M
L
J
L
K
R
N
Q
P
K
R
Q
P
N
Underline the correct word to complete the sentence.
34. LM , LN , and MN form a triangle / rectangle .
35. Name the plane.
_______________________________________________________________________
Lesson Check • Do you UNDERSTAND?
)
)
Are AB and BA the same ray? Explain.
Underline the correct symbol to complete each sentence.
)
)
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36. The endpoint of AB is A / B .
37. The endpoint of BA is A / B .
)
)
38. Use the line. Draw and label points A and B. Then draw AB and BA .
)
)
39. Are AB and BA the same ray? Explain.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
point
line
plane
segment
ray
postulate
axiom
Rate how well you understand points, lines, and planes.
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Lesson 1-2
1-2
Points, Lines, and Planes
PART 2
Vocabulary
Review
a
Use the diagram at the right. Complete each statement with the correct
word from the list below.
intersect
intersecting
P
b
intersection
1. Line a and line b 9 at point P.
2. The 9 of lines a and b is point P.
3. The diagram shows two 9 lines.
Vocabulary Builder
plane P
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plane (noun) playn
A
Related Word: coplanar
P
C
B
Definition: A plane is represented by a flat surface that
extends without end.
Word Origin: The word plane comes from the Latin planos, which means “flat.”
Math Usage: A plane contains infinitely many points. You can name a plane by
one capital letter or by at least three noncollinear points in the plane.
Use Your Vocabulary
Write T for true or F for false.
4. A plane has length, width, and height.
5. A plane extends without end.
6. All points in a line are coplanar.
7. All points in a plane are collinear.
13
Lesson 1-2, Part 2
Postulates 1–1, 1–2, 1–3, and 1–4
8. Complete each postulate with line, plane, or point.
Postulate 1-1 Through any two points there is exactly one 9.
Postulate 1-2 If two distinct lines intersect, then they intersect in exactly one 9.
Postulate 1-3 If two distinct planes intersect, then they intersect in exactly one 9.
Postulate 1-4 Through any three noncollinear points there is exactly one 9.
Write P if the statement describes a postulate or U if it describes an undefined term.
9. A point indicates a location and has no size.
10. Through any two points there is exactly one line.
11. If two distinct planes intersect, then they intersect in exactly one line.
12. If two distinct lines intersect, then they intersect in exactly one point.
D
Got It? Each surface of the box at the right represents
* ) part of a plane.
A
What are the names of two planes that intersect in BF ?
* )
E
13. Circle the points that are on BF or in one of the two planes.
A
B
C
D
E
F
H
G
C
B
G
F
H
14. Circle another name for plane BFG. Underline another name for plane BFE.
ABF
BCD
BCG
CDH
* )
FGH
15. Now name two planes that intersect in BF .
Use the diagram at the right for Exercises 16–19.
B
16. The intersection of plane DFC and plane ACB is
.
17. The intersection of plane CBF and plane ACF is
.
18. The intersection of plane DFE and plane BEF is
.
19. Name two planes that do NOT intersect.
Chapter 1
C
A
14
E
F
D
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Problem 3 Finding the Intersection of Two Planes
Problem 4 Using Postulate 1–4
Got It? What plane contains points L, M, and N? Shade the plane.
M
J
20. Use the figure below. Draw LM , LN , and MN as dashed segments.
Then shade plane LMN.
M
L
J
L
K
R
N
Q
P
K
R
Q
P
N
Underline the correct word to complete the sentence.
21. LM , LN , and MN form a triangle / rectangle .
22. Name the plane.
_______________________________________________________________________
Lesson Check • Do you UNDERSTAND?
Reasoning Can two planes intersect at a ray or a segment? Explain.
23. Complete Postulate 1-3.
If two distinct planes intersect, then they intersect in exactly one 9.
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24. Circle the figures that extend without end.
line
plane
ray
segment
25. Now answer the question.
_______________________________________________________________________
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
point
line
plane
segment
ray
postulate
axiom
Rate how well you understand points, lines, and planes.
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Lesson 1-2, Part 2
Name:
WORKSHEET: Points, Lines and Planes
For 1 – 5, use the diagram to determine if each statement is TRUE or FALSE. JUSTIFY
your answer.
1.
Point A lies on line m.
2.
B, C and D are collinear.
3.
A, B and F are coplanar.
4.
A, B, C, and D are collinear.
5.
CD and CE are coplanar.
For 6 – 9, name a point that is COLLINEAR with the given points.
6.
E and D
7.
C and A
8.
D and B
9.
B and G
For 10 – 13, name a point that is COPLANAR with the given points.
10.
J, K and L
11.
J, K, and E
12.
E, K, and M
13.
J, L, and G
For 14 – 17, use the diagram.
14.
Name THREE (3) points that are COLLINEAR.
15.
Name TWO (2) lines that are COPLANAR.
16.
Name THREE (3) points that are NOT COLLINEAR.
17.
Name FOUR (4) points that are NOT COPLANAR.
For 18 – 24, decide whether the statement is TRUE or FALSE. JUSTIFY your answer.
18.
Planes Q and R intersect at line n.
19.
Planes P and Q intersect at line m.
20.
Planes R and S do not appear to intersect.
21.
Planes S and P do not appear to intersect.
22.
Lines n and
23.
Planes Q and S intersect at line m.
24.
Lines
appear to intersect.
and m do not appear to intersect.
For 25 – 28, sketch each figure described.
25.
Two lines that lie in a plane and intersect at a point.
26.
Two planes that intersect in a line.
27.
Two planes that do not intersect.
28.
A line that intersects a plane at a point.
Name
1-2
Class
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ELL Support
Points, Lines, and Planes
Complete the vocabulary chart by filling in the missing information.
Word or Word
Phrase
Definition
Picture or Example
intersection
An intersection is the set of
points two or more figures have
in common.
Point E is the intersection
of the lines.
ray
A ray is part of a line that consists
of one endpoint and all the points
of the line on one side of the
endpoint.
1.
opposite rays
2.
Point Q is the endpoint shared
by these two rays.
segment
A segment is part of a line that
consists of two endpoints and all
points between them.
3.
collinear
Points that lie on the same line
are collinear.
4.
postulate
5.
Example: Through any two
points there is exactly one line.
coplanar
6.
Points G, H, and Z are coplanar.
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Additional Problems
Points, Lines, and Planes
Problem 1
* )
a. What are two other ways to name AB ?
B
b. What are two ways to name plane Q?
C
c. What are the names of three collinear
points?
m
E A D
Q
F
d. What are the names of four coplanar
points?
Problem 2
a. What are the names of the segments in the figure below?
b. What are the names of the rays in the figure?
c. Which of the rays in part (b) are opposite rays?
P
N
M
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Additional Problems (continued)
1-2
Points, Lines, and Planes
Problem 3
Each surface of the box represents part of
a plane. What is the intersection of plane
AEH and plane EGH?
Problem 4
Use the figure below.
a. Which plane contains points J, M, and L?
b. Which plane contains points L, P, and Q?
M
J
N
R
L
K
Q
P
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E
H
G
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Date
Reteaching
Points, Lines, and Planes
Review these important geometric terms.
Term
Examples of Labels
Point
Italicized capital letter: D
Line
Two capital letters with a line drawn
Diagram
over them:
One italicized lowercase letter: m
Line Segment Two capital letters (called endpoints)
with a segment drawn over them:
AB orBA
Ray
Two capital letters with a ray symbol
drawn over them:
Plane
Three capital letters: ABF, AFB, BAF,
BFA, FAB, or FBA
One italicized capital letter: W
Remember:
1. When you name a ray, an arrowhead is not drawn over the beginning point.
2. When you name a plane with three points, choose no more than two collinear
points.
3. An arrow indicates the direction of a path that extends without end.
4. A plane is represented by a parallelogram. However, the plane actually has no
edges. It is flat and extends forever in all directions.
Exercises
Identify each figure as a point, segment, ray, line, or plane, and name each.
1.
2.
3.
4.
5.
6.
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Reteaching (continued)
Points, Lines, and Planes
A postulate is a statement that is accepted as true.
Postulate 1–4 states that through any three
noncollinear points, there is only one plane.
Noncollinear points are points that do not all lie on
the same line.
In the figure at the right, points D, E, and F are
noncollinear. These points all lie in one plane.
Three noncollinear points lie in only one plane. Three
points that are collinear can be contained by more than
one plane. In the figure at the right, points P, Q, and R
are collinear, and lie in both plane O and plane N.
Exercises
Identify the plane containing the given points as front, back, left side, right side, top,
or bottom.
7. F, G, and X
8. F, G, and H
9. H, I, and Z
10. F, W, and X
11. I, W, and Z
12. Z, X, and Y
13. H, G, and X
14. W, Y, and Z
Use the figure at the right to determine how
many planes contain the given group of points.
Note that
pierces the plane at R,
coplanar with X, and
is not
does not intersect
.
15. C, D, and E
16. D, E, and F
17. C, G, E, and F
18. C and F
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Practice
Form K
Points, Lines, and Planes
Use the figure at the right for Exercises 1–4. Note that line r
pierces the plane at X. It is not coplanar with V.
1. What are two other ways to name
?
To start, remember you can name a line by any
?
point(s) on the line or by
Two other ways to name
?
lowercase letter(s).
are line
?
and
?.
2. What are two other ways to name plane V?
3. Name three collinear points.
4. Name four coplanar points.
Use the figure at the right for Exercises 5–7.
5. Name six segments in the figure. To start,
remember that a segment is part of a line that
consists of
?
endpoints.
Six segments are AB, BC,
?,?
,
?
, and
?
.
6. Name the rays in the figure.
7. a. Name the pairs of opposite rays with endpoint C.
b. Name another pair of opposite rays.
For Exercises 8–12, determine whether each statement is always,
sometimes, or never true.
8. Plane ABC and plane DEF are the same plane.
9.
and
are the same line.
10. Plane XYZ does not contain point Z.
11. All the points of a line are coplanar.
12. Two rays that share an endpoint form a line.
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Practice (continued)
Form K
Points, Lines, and Planes
Use the figure at the right for Exercises 13–21.
Name the intersection of each pair of planes. To start,
identify the points that both planes contain.
13. planes DCG and EFG
14. planes EFG and ADH
15. planes BCG and ABF
Name two planes that intersect in the given line. To start,
identify the planes that contain the given line.
16.
17.
18.
Copy the figure. Shade the plane that contains the given points.
19. A, B, C
20. C, D, H
21. E, H, B
Postulate 1-4 states that any three noncollinear points lie in one
plane. Find the plane that contains the first three points listed. Then
determine whether the fourth point is in that plane. Write coplanar
or noncoplanar to describe the points.
22. P, T, R, N
23. P, O, S, N
24. T, R, N, U
25. P, O, R, S
Use the diagram at the right. How many planes
contain each line and point?
26.
28.
and G
and G
27.
and F
29.
and M
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Date
Practice
Form G
Points, Lines, and Planes
Use the figure below for Exercises 1–8. Note that
not coplanar with V.
pierces the plane at N. It is
1. Name two segments shown in the figure.
2. What is the intersection of
and
?
3. Name three collinear points.
4. What are two other ways to name plane V?
5. Are points R, N, M, and X coplanar?
6. Name two rays shown in the figure.
7. Name the pair of opposite rays with endpoint N.
8. How many lines are shown in the drawing?
For Exercises 9–14, determine whether each statement is always, sometimes, or
never true.
9.
and
10.
and
are the same ray.
are opposite rays.
11. A plane contains only three points.
12. Three noncollinear points are contained in only one plane.
13. If
lies in plane X, point G lies in plane X.
14. If three points are coplanar, they are collinear.
15. Reasoning Is it possible for one ray to be shorter in length than
another? Explain.
16. Open-Ended Draw a figure of two planes that intersect in
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13
Name
Class
1-2
Date
Practice (continued)
Form G
Points, Lines, and Planes
17. Draw a figure to f t each description.
a. Through any two points there is exactly one line.
b. Two distinct lines can intersect in only one point.
18. Reasoning Point F lies on
and point M lies on
collinear, what must be true of these rays?
. If F, E, and M are
19. Writing What other terms or phrases mean the same as postulate?
20. How many segments can be named
from the figure at the right?
Use the figure at the right for Exercises 21–29.
Name the intersection of each pair of planes or lines.
21. planes ABP and BCD
22.
and
23. planes ADR and DCQ
24. planes BCD and BCQ
25.
and
Name two planes that intersect in the given line.
26.
27.
28.
29.
Coordinate Geometry Graph the points and state whether they are collinear.
30. (0, 0), (4, 2), (6, 3)
31. (0, 0), (6, 0), (9, 0)
32. (−1, 1), (2, −2), (4, −3)
33. (1, 2), (2, 3), (4, 5)
34. (−2, 0), (0, 4), (2, 0)
35. (−4, −1), (−1, −2), (2, −3)
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14
Measuring Segments
1-3
Vocabulary
Review
Draw an example of each.
1. point
* )
2. AB
3. DF
)
Vocabulary Builder
segment HJ
segment (noun)
SEG
munt
H
J
Definition: A segment is part of a line that consists of two endpoints
and all points between them.
Use Your Vocabulary
Complete each sentence with endpoint, endpoints, line, or points.
4. A ray has one 9.
5. A line contains infinitely many 9.
6. A segment has two 9.
7. A segment is part of a 9.
Place a check ✓ if the phrase describes a segment. Place an ✗ if it does not.
8. Earth’s equator
9. the right edge of a book’s cover
10. one side of a triangle
Postulate 1–5 Ruler Postulate
Every point on a line can be paired with a real number, called the coordinate of the point.
Chapter 1
10
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Main Idea: You name a segment by its endpoints.
Problem 1 Measuring Segment Lengths
Got It? What are UV and SV on the number line?
11. Label each point on the number line with its coordinate.
S
U
Ľ6
Ľ2
0
2
4
6
8
V
12
16
12. Find UV and SV. Write a justification for each statement.
UV 5 P
UV 5 P
UV 5
2
P
P
SV 5 P
SV 5 P
SV 5
2
P
P
Postulate 1–6 Segment Addition Postulate
If three points A, B, and C are collinear and B is between A and C, then AB 1 BC 5 AC.
Given points A, B, and C are collinear and B is between A and C, complete each equation.
13. AB 5 5 and BC 5 4, so AB 1 BC 5
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14. AC 5 12 and BC 5 7, so AC 2 BC 5
and AC 5
1
.
and AB 5
2
.
Problem 2 Using the Segment Addition Postulate
Got It? In the diagram, JL 5 120. What are JK and KL?
4x 6
J
15. Write a justification for each statement.
7x 15
K
L
JK 1 KL 5 JL
(4x 1 6) 1 (7x 1 15) 5 120
11x 1 21 5 120
11x 5 99
x59
16. You know that JK 5 4x 1 6 and KL 5 7x 1 15. Use the value of x from Exercise 15
to to find JK and KL. find JK and KL.
17. JK 5
and KL 5
11
Lesson 1-3
Problem 3 Comparing Segment Lengths
Got It? Use the diagram below. Is AB congruent to DE?
A
6 4 2
B
0
2
C
4
D
6
8
E
10 12 14 16
In Exercises 18 and 19, circle the expression that completes the equation.
18. AB 5 j
22 2 2
u 22 2 2 u
u 22 2 3 u
u 22 2 4 u
10 1 14
u 5 2 14 u
u 10 2 14 u
19. DE 5 j
3 214
20. After simplifying, AB 5
and DE 5
.
21. Is AB congruent to DE? Explain.
_______________________________________________________________________
The midpoint of a segment is the point that divides the segment into two
congruent segments.
A
B
C
D
E
F
G
H
I
J
K
Ľ5
Ľ4
Ľ3
Ľ2
Ľ1
0
1
2
3
4
5
22. Point
is halfway between points B and J.
23. The midpoint of AE is point
24. Point
divides EK into two congruent segments.
25. Find the midpoint of each segment. Then write the coordinate of the midpoint.
AG
DH
AK
Midpoint
Coordinate
26. Find the coordinate of the midpoint of each segment.
segment with
endpoints at 24 and 2
segment with
endpoints at 22 and 4
Coordinate of midpoint
27. Circle the expression that relates the coordinate of the midpoint to the coordinates
of the endpoints.
x1 1 x2
Chapter 1
(x1 1 x2)
(x1 2 x2)
2
2
12
.
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Use the number line below for Exercises 22–25.
Problem 4 Using the Midpoint
Got It? U is the midpoint of TV . What are TU, UV, and TV?
8x 11
T
28. Use the justifications at the right to complete the steps below.
12x 1
U
V
Step 1 Find x.
TU 5 UV
Definition of midpoint
8x 1 11 5
8x 1 11 1
Substitute.
5
Add 1 to each side.
1
5
Subtract 8x from each side.
5x
Divide each side by 4.
Step 2 Find TU and UV.
TU 5 8 ?
1 11 5
Substitute
UV 5 12 ?
215
Substitute.
for x.
Step 3 Find TV.
TV 5 TU 1 UV
5
Definition of midpoint
Substitute.
1
Simplify.
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5
Lesson Check • Do you UNDERSTAND?
Vocabulary Name two segment bisectors of PR.
Underline the correct word or symbol to complete each sentence.
P Q R S T
2
29. A bisector / midpoint may be a point, line, ray, or segment.
3
4
5
6
30. The midpoint of PR is point P / Q / R .
31. Line ℓ passes through point P / Q / R .
32. Two bisectors of PR are
and
.
Math Success
Check off the vocabulary words that you understand.
congruent segments
coordinate
midpoint
segment bisector
Rate how well you can find lengths of segments.
Need to
review
0
2
4
6
8
Now I
get it!
10
13
Lesson 1-3
Name
Class
Date
Form G
Practice
1-3
Measuring Segments
In Exercises 1–6, use the figure below. Find the length of each segment.
1. AB
2. BC
3. AC
4. AD
5. BD
6. CD
For Exercises 7–11, use the figure at the right.
7. If PQ = 7 and QR = 10, then PR =
.
8. If PQ = 20 and QR = 22, then PR =
.
9. If PR = 25 and PQ = 12, then QR =
.
10. If PR = 19 and QR = 12, then PQ =
.
11. If PR = 10 and PQ = 4, then QR =
.
Use the number line below for Exercises 12–16. Tell whether the segments
are congruent.
12. GH and HI
13. GH and IK
14. HJ and IK
15. IJ and JK
16. HJ and GI
17. HK and GI
18. Reasoning Points A, Q, and O are collinear. AO = 10, AQ = 15, and OQ
= 5. What must be true about their positions on the line?
Algebra Use the figure at the right for
Exercises 19 and 20.
19. Given: ST = 3x + 3 and TU = 2x + 9.
a. What is the value of ST?
b. What is the value of TU?
20. Given: ST = x + 3 and TU = 4x − 6.
a. What is the value of ST?
b. What is the value of SU?
21. Algebra On a number line, suppose point E has a coordinate of 3, EG = 6, and
EX = 12. Is point G the midpoint of EX ? What are possible coordinates for G
and X?
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Name
1-3
Class
Date
Practice (continued)
Form G
Measuring Segments
On a number line, the coordinates of P, Q, R, and S are −12, −5, 0,
and 7, respectively.
22. Draw a sketch of this number line. Use this sketch to answer Exercises 23–26.
23. Which line segment is the shortest?
24. Which line segment is the longest?
25. Which line segments are congruent?
26. What is the coordinate of the midpoint of PR ?
27. You plan to drive north from city A to town B and then continue north to city
C. The distance between city A and town B is 39 mi, and the distance
between town B and city C is 99 mi.
a. Assuming you follow a straight driving path, after how many
miles of driving will you reach the midpoint between city A and
city C?
b. If you drive an average of 46 mi/h, how long will it take you to drive
from city A to city C?
28. Algebra Point O lies between points M and P on a line. OM = 34z and OP = 36z − 7.
If point N is the midpoint of MP , what algebraic equation can you use to find MN?
Algebra Use the diagram at the right for Exercises 29–32.
29. If AD = 20 and AC = 3x + 4, find the value of x. Then find AC and DC.
30. If ED = 5y + 6 and DB = y + 30, find the value of y. Then find ED, DB,
and EB.
31. If DC = 6x and DA = 4x + 18, find the value of x. Then find AD, DC, and AC.
32. If EB = 4y − 12 and ED = y + 17, find the value of y. Then find ED, DB, and EB.
33. Writing Is it possible that PQ + QR < PR? Explain.
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Name
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Date
Additional Problems
1-3
Measuring Segments
Problem 1
What is CD?
B
4
C
0
D
4
8
E
12 16
Problem 2
If LN 5 32, what are LM and MN?
L
M
3x 8
N
2x 4
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Name
Class
1-3
Date
Additional Problems (continued)
Measuring Segments
Problem 3
Are AD and BE congruent?
A
4
B
C
0
4
D
8
E
12 16
Problem 4
S is the midpoint of RT . What are RS, ST, and RT ?
R
S
7x 3
T
3x 1
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Date
Reteaching
Measuring Segments
The Segment Addition Postulate allows you to use known segment lengths to find
unknown segment lengths. If three points, A, B, and C, are on the same line, and point
B is between points A and C, then the distance AC is the sum of the distances AB and
BC.
AC = AB + BC
If QS = 7 and QR = 3, what is RS?
QS = QR + RS
Segment Addition Postulate
QS − QR = RS
Subtract QR from each side.
7 − 3 = RS
4 = RS
Substitute.
Simplify.
Exercises
For Exercises 1–5, use the figure at the right.
1. If PN = 29 cm and MN = 13 cm, then PM =
.
2. If PN = 34 cm and MN = 19 cm, then PM =
.
3. If PM = 19 and MN = 23, then PN =
4. If MN = 82 and PN = 105, then PM =
5. If PM = 100 and MN = 100, then PN =.
For Exercises 6-8, use the figure at the right.
6. If UW = 13 cm and UX = 46 cm, then WX =
7. UW = 2 and UX = y. Write an expression for WX.
8. UW = m and WX = y + 14. Write an expression for UX.
On a number line, the coordinates of A, B, C, and D are −6, −2, 3, and 7, respectively. Find
the lengths of the two segments. Then tell whether they are congruent.
9. AB and CD
10. AC and BD
11. BC and AD
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Reteaching (continued)
Measuring Segments
The midpoint of a line segment divides the segment into two segments that are equal
in length. If you know the distance between the midpoint and an endpoint of a
segment, you can find the length of the segment. If you know the length of a
segment, you can find the distance between its endpoint and midpoint.
X is the midpoint of WY . XW = WY, so XW and WY are congruent.
C is the midpoint of BE . If BC = t + 1, and CE = 15 − t, what is BE?
Definition of midpoint
BC = CE
Substitute.
t + 1 = 15 − t
Add t to each side.
t + t + 1 = 15 − t + t
Simplify.
2t + 1 = 15
Subtract 1 from each side.
2t + 1 − 1 = 15 − 1
Simplify.
2t = 14
Divide each side by 2.
t=7
BC = t + 1
Given.
BC = 7 + 1
Substitute.
BC = 8
Simplify.
BE = 2(BC)
Definition of midpoint.
BE = 2(8)
Substitute.
BE = 16
Simplify.
Exercises
12. W is the midpoint of UV . If UW = x + 23, and WV = 2x + 8, what is x?
13. W is the midpoint of UV . If UW = x + 23, and WV = 2x + 8, what is WU?
14. W is the midpoint of UV . If UW = x + 23, and WV = 2x + 8, what is UV?
15. Z is the midpoint of YA . If YZ = x + 12, and ZA = 6x − 13, what is YA?
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Name
1-3
Class
Date
Practice
Form K
Measuring Segments
Find the length of each segment. To start, find the coordinate of
each endpoint.
1. PR
2. QT
3. QS
Use the number line at the right for Exercises 4–6.
4. If GH = 31 and HI = 11, then GI =
.
5. If GH = 45 and GI = 61, then HI =
.
6. Algebra GH = 7y + 3, HI = 3y − 5, and GI = 9y + 7.
a. What is the value of y?
b. Find GH, HI, and GI.
Use the number line below for Exercises 7–9. Tell whether the segments are
congruent. To start, use the definition of distance. Use the coordinates of the
points to write an equation for each distance.
7. CE and FD
8. CD and FG
9. GE and BD
For Exercises 10–12, use the figure below. Find the value of KL.
10. KL = 3x + 2 and LM = 5x − 10
11. KL = 8x − 5 and LM = 6x + 3
12. KL = 4x + 7 and LM = 5x − 4
On a number line, the coordinates of D, E, F, G, and H are −9, −2, 0, 3, and 5,
respectively. Find the lengths of the two segments. Then tell whether they are
congruent.
13. DG and DH
14. DE and EH
15. EG and GH
16. EG and FH
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1-3
Class
Date
Practice (continued)
Form K
Measuring Segments
Suppose the coordinate of P is 2, PQ = 8, and PR = 12. What are the possible
coordinates of the midpoint of the given segment?
18. PR
17. PQ
19. QR
Visualization Without using your ruler, sketch a segment with the
given length. Use your ruler to see how well your sketch approximates
the length provided.
20. 5 cm
21. 8 in.
22. 8 cm
23. 12 cm
24. 85 mm
25. 5 in.
26. Suppose point J has a coordinate of −2 and JK = 4. What are the possible
coordinates of point K?
27. Suppose point X has a coordinate of 5 and XY = 10. What are the possible
coordinates of point Y?
Algebra Use the diagram at the right for Exercises 28–32.
28. If NO = 17 and NP = 5x − 6, find the value of x.
Then find NP and OP.
29. If RO = 6 + x and OQ = 2x + 1, find the value of x.
Then find RO, OQ, and RQ.
30. If NO = 3x + 4 and NP = 10x − 10, find the value of x.
Then find NO, NP, and OP.
31. If RO = 5x and RQ = 12x − 20, find the value of x.
Then find RO, OQ, and RQ.
32. Vocabulary What term describes the relationship between NP and RQ
33. Reasoning If KL = 5 and KJ = 10, is it possible that LJ = 5? Explain.
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Measuring Angles
1-4
Vocabulary
Review
Write T for true or F for false.
)
1. AB names a ray with endpoints A and B.
2. You name a ray by its endpoint and another point on the ray.
Vocabulary Builder
angle (noun, verb)
ANG
gul
Other Word Forms: angular (adjective), angle (verb), angled (adjective)
Use Your Vocabulary
Name the rays that form each angle.
A
3.
B
B
4.
C
and
A
C
and
Key Concept Angle
Definition
How to Name It
An angle is formed by
two rays with the same
endpoint.
You can name an angle by
The rays are the sides of the
angle. The endpoint is the
vertex of the angle.
Chapter 1
Diagram
• its vertex
B
• a point on each ray and the
vertex
• a number
14
A
1
C
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Definition: An angle is formed by two rays with the same endpoint.
For Exercises 5–8, use the diagram in the Take Note on page 14. Name each part of the angle.
5. the vertex
6. two points that are NOT the vertex
7. the sides
)
and
)
and
8. Name the angle three ways.
by its vertex
by a point on each side and the vertex
by a number
Problem 1 Naming Angles
Got It? What are two other names for lKML?
)
MK
MJ
)
)
1 2
ML
/2
/JKL
L
M
10. Circle all the possible names of /KML.
/1
K
J
9. Cross out the ray that is NOT a ray of /KML.
/JMK
/JML
/KMJ
/LMK
Key Concept Types of Angles
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
11. Draw your own example of each type of angle.
acute
right
0,x,
x5
obtuse
straight
,x,
x5
In the diagram, mlABC 5 70 and mlBFE 5 90. Describe each angle as acute,
right, obtuse or straight. Give an angle measure to support your description.
12. /ABC
C
13. /CBD
70í B
A
D
14. /CFG
G
90í
15. /CFH
F
E
H
15
Lesson 1-4
Problem 2 Measuring and Classifying Angles
Got It? What are the measures of /LKH , /HKN , and /MKH in the art below?
Classify each angle as acute, right, obtuse, or straight.
J
L
30
15
0 1 40
40
0 10
20
180 170 1
60
1
inches
1
2
K
M
170 180
160
0
10 0
15
20
0
0
14 0 3
4
H
80 90 100 11
0 1
70
90 80
20
70
60 110 100
60 13
0
2
0
1
50
30
50
3
4
5
6
N
16. Write the measure of each angle. Then classify each angle.
/LKH
/HKN
8
/MKH
8
Problem 3
8
Using Congruent Angles
Got It? Use the photo at the right. If m/ABC 5 49,
17. /ABC has
angle mark(s).
D
18. The other angle with the same number of
marks is /
.
B
C
19. Underline the correct word to complete the sentence.
F
E
A
in Exercise 18 are equal / unequal .
Postulate 1–8 Angle Addition Postulate
If point B is in the interior of /AOC, then m/AOB 1 m/BOC 5 m/AOC.
21. Draw /ABT with point L in the interior and /ABL and /LBT .
22. Complete: m/ABL 1 m/
Chapter 1
J
G
20. m/DEF 5
5 m/
16
M
K
The measure of /ABC and the measure of the angle
L
H
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
what is m/DEF ?
Problem 4
Using the Angle Addition Postulate
C
Got It? /DEF is a straight angle. What are m/DEC and m/CEF ?
(11x 12)
(2x 10)
E
F
D
23. Write a justification for each statement.
m/DEF 5 180
m/DEC 1 m/CEF 5 180
(11x 2 12) 1 (2x 1 10) 5 180
13x 2 2 5 180
13x 5 182
x 5 14
24. Use the value of x to find m/DEC and m/CEF .
m/DEC 5 11x 2 12 5 11(
) 2 12 5
m/CEF 5
Lesson Check • Do you know How?
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Algebra If mlABD 5 85, what is an expression to represent mlABC?
25. Use the justifications at the right to complete the statements below.
m/ABC 1 m/CBD 5 m/ABD
m/ABC 1
m/ABC 1
Angle Addition Postulate
C
1 x
D
B
Substitute.
5
2
A
5
Subtract
2
m/ABC 5
from each side.
Simplify.
Math Success
Check off the vocabulary words that you understand.
acute angle
obtuse angle
right angle
straight angle
Rate how well you can classify angles.
Need to
review
0
2
4
6
8
Now I
get it!
10
17
Lesson 1-4
Name
1-4
Class
Date
Practice
Form G
Measuring Angles
Use the diagram below for Exercises 1–11. Find the measure of each angle.
1. ∠MLN
2. ∠NLP
3. ∠NLQ
4. ∠OLP
5. ∠MLQ
Classify each angle as acute, right, obtuse, or straight.
6. ∠MLN
9. ∠OLP
7. ∠NLO
8. ∠MLP
10. ∠OLQ
11. ∠ MLQ
Use the figure at the right for Exercises 12 and 13.
12. What is another name for ∠XYW?
13. What is another name for ∠WYZ?
Use a protractor. Measure and classify each angle.
14.
15.
16.
17.
18.
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Name
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Practice (continued)
Date
Form G
Measuring Angles
19. ∠JKL and ∠CDE are congruent. If m∠JKL = 137, what is m∠CDE?
Use the figure at the right for Exercises 20–23.
m∠FXH = 130 and m∠FXG = 49.
20. ∠FXG ≅
21. m∠GXH =
22. Name a straight angle in the figure.
23. ∠IXJ ≅
24. Algebra If m∠RZT = 110, m∠RZS = 3s, and
m∠TZS = 8s, what are m∠RZS and m∠TZS?
25. Algebra m∠OZP = 4r + 2, m∠PZQ = 5r − 12, and
m∠OZQ = 125. What are m∠OZP and m∠PZQ?
26. Reasoning Elsa draws an angle that measures 56. Tristan draws a congruent angle.
Tristan says his angle is obtuse. Is he correct? Why or why not?
27. Lisa makes a cherry pie and an apple pie. She cuts the cherry pie into six equal wedges
and she cuts the apple pie into eight equal wedges. How many degrees greater is the
measure of a cherry pie wedge than the measure of an apple pie wedge?
28. Reasoning ∠JNR and ∠RNX are congruent. If the sum of the measures of the two
angles is 180, what type of angle are they?
29. A new pizza place in town cuts their circular pizzas into 12 equal slices. What is the
measure of the angle of each slice?
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Additional Problems
Measuring Angles
Problem 1
What are two other names for /1?
L
1
M
N
Problem 2
What are the measures of
/LKN , /NKM , and /JKN ?
Classify each angle as acute,
right, obtuse, or straight.
J
K
Prentice Hall Geometry • Additional Problems
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7
M
170 180
160
0
10 0
15
20
0
0
14 0 3
4
H
80 90 100 11
0
70
00 90 80 70 120
1
0
6
10
60 13
01
0
2
0
1
5
50
0
3
1
0 10
20
180 170 1
30
60
15
0 1 40
40
L
N
Name
Class
Date
Additional Problems (continued)
1-4
Measuring Angles
Problem 3
Use the diagram below. Which angle is congruent to /WBM ?
M
B
W
Y
A
DC
E
X
N Z
Problem 4
If m/ABC 5 175, what are m/ABD and m/CBD?
(6x 5) D
(4x 10)
A
B
C
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8
Name
1-4
Class
Date
ELL Support
Measuring Angles
Tamsin wants to find the measures of
∠ SOP and ∠ POT, when m∠ SOT = 100.
She wrote the following steps on note
cards, but they got mixed up.
Use the note cards to complete the steps below. Follow the instructions on each note
card to solve the problem.
1.
First,
2.
Second,
3.
Next,
4.
Finally,
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31
Name
Class
1-4
Date
Reteaching
Measuring Angles
The vertex of an angle is the common endpoint of the rays
that form the angle. An angle may be named by its vertex. It
may also be named by a number or by a point on each ray
and the vertex (in the middle).
This is ∠Z, ∠XZY , ∠YZX, or ∠1.
It is not ∠ZYX, ∠XYZ, ∠YXZ, or ∠ZXY .
Angles are measured in degrees, and the measure of an angle is used to classify it.
The measure of
an acute angle
is between 0 and 90.
The measure of a
right angle is 90.
The measure of an
obtuse angle is between
90 and 180.
The measure of a
straight angle is 180.
Exercises
Use the figure at the right for Exercises 1 and 2.
1. What are three other names for ∠S?
2. What type of angle is ∠S?
3. Name the vertex of each angle.
a. ∠ LGH
b. ∠ MBX
Classify the following angles as acute, right, obtuse, or straight.
4. m∠LGH = 14
5. m∠SRT = 114
6. m∠SLI = 90
7. m∠1 = 139
8. m∠L = 179
9. m∠P = 73
Use the diagram below for Exercises 10–18.
Find the measure of each angle.
10. ∠ADB
11. ∠FDE
12. ∠BDC
13. ∠CDE
14. ∠ADC
15. ∠FDC
16. ∠BDE
17. ∠ADE
18. ∠BDF
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Name
1-4
Class
Date
Reteaching (continued)
Measuring Angles
The Angle Addition Postulate allows you to use a known angle measure to
find an unknown angle measure. If point B is in the interior of ∠AXC, the
sum of m∠AXB and m∠BXC is equal to m∠AXC.
m∠AXB + m∠BXC = m∠AXC
If m∠LYN = 125, what are
m∠LYM and m∠MYN?
Step 1 Solve for p.
m∠LYN = m∠LYM + m∠MYN
Angle Addition Postulate
125 = (4p + 7) + (2p − 2)
Substitute.
125 = 6p + 5
Simplify
120 = 6p
Subtract 5 from each side.
Divide each side by 6.
20 = p
Step 2 Use the value of p to find the measures of the angles.
m∠LYM = 4p + 7
Given
m∠LYM = 4(20) + 7
Substitute.
m∠LYM = 87
Simplify.
m∠MYN = 2p − 2
Given
m∠MYN = 2(20) − 2
Substitute.
m∠MYN = 38
Simplify.
Exercises
19. X is in the interior of ∠LIN. m∠LIN = 100, m∠LIX = 14t, and
m∠XIN = t + 10.
a. What is the value of x?
b. What are m∠LIX and m∠XIN?
20. Z is in the interior of ∠GHI. m∠GHI = 170, m∠GHZ = 3s − 5, and
m∠ZHI = 2s + 25.
a. What is the value of s?
b. What are m∠GHZ and m∠ZHI?
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40
Name
Class
Date
Practice
1-4
Form K
Measuring Angles
Name each shaded angle in three different ways. To start, identify the rays that
form each angle.
1.
2.
3.
Use the diagram below. Find the measure of each angle. Then classify the angle
as acute, right, obtuse, or straight.
4. ∠AFB
To start, identify ∠AFB. Then use the definition of the measure of an angle to
find m∠AFB.
m∠AFB =
This angle is a(n)
?
angle.
5. ∠AFD
6. ∠ CFD
7. ∠BFD
8. ∠AFE
9. ∠BFE
10. ∠AFC
Use the diagram at the right. Complete each statement.
11. ∠MIG
≅
12. ∠ PMJ
≅
13. If m∠KJL = 30, then m∠
= 30.
14. If m∠LMP = 100, then m∠QHG =
.
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Name
1-4
Class
Date
Practice (continued)
Form K
Measuring Angles
15. If m∠FHI = 142, what are
16. ∠JKL is a right angle. What are
m∠FHG and m∠GHI?
m∠JKM and m∠MKL?
Use a protractor. Measure and classify each angle.
17.
18.
19.
20.
Algebra Use the diagram at the right for
Exercises 21–23. Solve for x. Find the angle
measures to check your work.
21. m∠CGD = 4x + 2, m∠DGE = 3x − 5,
m∠EGF = 2x + 10
22. m∠CGD = 2x − 2, m∠EGF = 37, m∠CGF = 7x + 2
23. If m∠DGF = 72, what equation can you use to find m∠EGF?
24. The flag of the United Kingdom is shown
at the right. Copy the flag on a separate
piece of paper. Label at least two of each
type of angle:
a. acute
b. obtuse
c. right
d. straight
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36
1-5
Exploring Angle Pairs
Vocabulary
Review
Use a word from the list below to complete each sentence. Use each word just once.
interior
rays
vertex
1. The 9 of an angle is the region containing all of the points
between the two sides of the angle.
2. When you use three points to name an angle, the 9 must go in the middle.
3. The sides of /QRS are 9 RS and RQ.
Use the figure below for Exercises 4–7. Identify each
angle as acute, right, obtuse, or straight.
5. /TRS
T
S
6. /TRQ
Q
Vocabulary Builder
conclusion (noun) kun KLOO zhun
Other Word Forms: conclude (verb)
Definition: A conclusion is the end of an event or the last step in a
reasoning process.
Use Your Vocabulary
Complete each sentence with conclude or conclusion.
8. If it rains, you can 9 that soccer practice will be canceled.
9. The last step of the proof is the 9.
Chapter 1
U
7. /VRQ
18
R
V
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
4. /SRV
Key Concept Types of Angle Pairs
Definition
Angle Pair
Adjacent angles
Two coplanar angles with a common side, a common
vertex, and no common interior points
Vertical angles
Two angles whose sides are opposite rays
Complementary angles
Two angles whose measures have a sum of 90
Supplementary angles
Two angles whose measures have a sum of 180
Draw a line from each word in Column A to the angles it describes in Column B.
Column A
Column B
10. supplementary
/1 and /2
11. adjacent
/2 and /3
12. vertical
/2 and /5
13. complementary
/3 and /6
1
2
6
3
4
5
Problem 1 Identifying Angle Pairs
Got It? Use the diagram at the right. Are lAFE and lCFD vertical angles?
Explain.
)
)
B
)
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)
)
28
A
14. The rays of /AFE are FE and FC / FA .
)
62
15. The rays of /CFD are FC and FD / FA .
E
Complete each statement.
)
)
16. FE and
are opposite rays.
17. FA and
are opposite rays.
18. Are /AFE and /AFE vertical angles?
T
Got It? Can you conclude that TW O WV from the diagram? Explain.
19. Circle the items marked as congruent in the diagram.
/TWQ and /PWT
D
118
Yes / No
Problem 2 Making Conclusions From a Diagram
PW and WQ
C
F
TW and WV
P
/TWQ and /VWQ
W
Q
V
20. Can you conclude that TW > WV ? Why or why not?
_______________________________________________________________________
19
Lesson 1-5
Postulate 1–9 Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
21. If /A and /B form a linear pair, then m/A 1 m/B 5
.
Problem 3 Finding Missing Angle Measures
L
Got It? Reasoning lKPL and lJPL are a linear pair, mlKPL 5 2x 1 24,
and mlJPL 5 4x 1 36. How can you check that mlKPL 5 64 and
mlJPL 5 116?
(2x 24) (4x 36)
K
P
J
22. What is one way to check solutions? Place a ✓ in the box if the response
is correct. Place an ✗ in the box if it is incorrect.
Draw a diagram. If it looks good, the solutions are correct.
Substitute the solutions in the original problem statement.
24. How does your check show that you found the correct angle measurements?
_______________________________________________________________________
_______________________________________________________________________
Problem 4
Using an Angle Bisector to Find Angle Measures
)
Got It? KM bisects lJKL. If mlJKL 5 72, what is mlJKM ?
25. Write a justification for each step.
m/JKM 5 m/MKL
m/JKM 1 m/MKL 5 m/JKL
Overmatter
2m/JKM 5 m/JKL
m/JKM 5 12 m/JKL
Chapter 1
20
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23. Use your answer(s) to Exercise 22 to check the solutions.
26. Complete.
m/JKL 5
, so m/JKM 5
.
27. Now complete the diagram below.
Lesson Check • Do you UNDERSTAND?
Error Analysis Your friend calculated the value of x below. What is her error?
2x
4x + 2x = 180
6x = 180
x = 30
4x
28. Circle the best description of the largest angle in the figure.
acute
obtuse
right
straight
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29. Complete: 4x 1 2x 5
30. What is your friend’s error? Explain.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
angle
complementary
supplementary
angle bisector
vertical
Rate how well you can find missing angle measures.
Need to
review
0
2
4
6
8
Now I
get it!
10
21
Lesson 1-5
Name
1-5
Class
Date
Additional Problems
Exploring Angle Pairs
Problem 1
Use the diagram below. Is each statement true? Explain.
a. /PAL and /LAM are adjacent angles.
b. /PAO and /NAM are vertical angles.
c. /PAO and /NAO are supplementary.
L
A
P 74
106
O
M
N
Problem 2
What can you conclude from the
information in the diagram?
4
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9
5
3
1
2
Name
1-5
Class
Date
Additional Problems (continued)
Exploring Angle Pairs
Problem 3
/ABC and /DBC are a linear pair, m/ABC 5 3x 1 19,
and m/DBC 5 7x 2 9. What are the measures of /ABC
and /DBC?
Problem 4
)
LM bisects /JLN . If m/JLM 5 42, what is m/JLN ?
A. 21
B. 42
C. 60
D. 84
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10
Name
Class
1-5
Date
ELL Support
Exploring Angle Pairs
adjacent angles
angle bisector
complementary angles
supplementary angle
vertical angles
Choose the word from the list above that is defined by each statement.
1. two angles whose measures have a sum of 180
2. two angles whose measures have a sum of 90
3. two angles whose sides are opposite rays
4. two angles that share a side but do not overlap
5. a ray that divides one angle into two angles with
the same measure
Use a word from the list to complete each sentence.
6. If the measure of one angle in a pair of
measure of the other angle is 180 – n.
7.
never share a side.
8.
always share a side.
Draw the angle bisector for each angle.
Multiple Choice
11. Which of the following are a pair of vertical angles?
∠ GBA and ∠ CBJ
∠ ABM and ∠ MBC
∠ GBA and ∠ GBM
∠ ABJ and ∠ MBC
12. Which of the following are supplementary angles?
∠ GBA and ∠ CBJ
∠ ABM and ∠ MBC
∠ GBA and ∠ GBM
∠ ABM and ∠ MBC
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is n, the
Name
Class
1-5
Date
Practice
Form K
Exploring Angle Pairs
Use the diagram at the right. Is each statement true? Explain.
1. ∠5 and ∠4 are supplementary angles.
2. ∠6 and ∠5 are adjacent angles.
3. ∠1 and ∠2 are a linear pair.
Name an angle or angles in the diagram described by each
of the following.
4. a pair of vertical angles
5. supplementary to ∠RPS
To start, remember that supplementary angles are two angles whose measures
have a sum of
6. a pair of complementary angles
To start, remember that complementary angles are two angles whose
measures have a sum of
.
7. adjacent to ∠TPU
For Exercises 8–11, can you make each conclusion from the
information in the diagram? Explain.
8. ∠CEG ≅ ∠FED
10. ∠BCE ≅ ∠BAD
9. DE ≅ EF
11. ∠ADB and ∠FDE are vertical angles.
Use the diagram at the right for Exercises 12 and 13.
12. Name two pairs of angles that form a linear pair.
13. Name two pairs of angles that are complementary.
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Name
Class
Practice (continued)
1-5
Exploring Angle Pairs
14. Algebra In the diagram,
bisects ∠WXZ.
a. Solve for x and find m∠WXY.
b. Find m∠YXZ.
c. Find m∠WXZ.
Algebra
bisects ∠PQS. Solve for x and find m∠PQS.
15. m∠PQR = 3x, m∠RQS = 4x − 9
16. m∠PQS = 4x − 6, m∠PQR = x + 11
17. m∠PQR = 5x − 4, m∠SQR = 3x + 10
18. m∠PQR = 8x + 1, m∠SQR = 6x + 7
Algebra Find the measure of each angle in the angle pair described.
19. The measure of one angle is 5 times the measure of its complement.
20. The measure of an angle is 30 less than twice its supplement.
21. Draw a Diagram Make a diagram that matches the
following description.
•
∠1 is adjacent to ∠2.
•
∠2 and ∠3 are a linear pair.
•
∠2 and ∠4 are vertical angles.
•
∠4 and ∠5 are complementary.
In the diagram at the right, m∠HKI = 48.
Find each of the following.
22. m∠HKJ
23. m∠IKJ
24. m∠FKG
25. m∠FKH
26. m∠FKJ
27. m∠GKI
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Date
Form K
Name
Class
Date
Reteaching
1-5
Exploring Angle Pairs
Adjacent Angles and Vertical Angles
Adjacent means “next to.” Angles are adjacent if they lie next to each other. In other
words, the angles have the same vertex and they share a side without overlapping.
Adjacent Angles
Overlapping Angles
Vertical means “related to the vertex.” So, angles are vertical if they share a vertex, but
not just any vertex. They share a vertex formed by the intersection of two straight lines.
Vertical angles are always congruent.
Vertical Angles
Non-Vertical Angles
Exercises
1. Use the diagram at the right.
a. Name an angle that is adjacent to ∠ABE.
b. Name an angle that overlaps ∠ABE.
2. Use the diagram at the right.
a. Mark ∠DOE and its vertical angle as congruent angles.
b. Mark ∠AOE and its vertical angle as congruent angles.
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Name
Class
1-5
Date
Reteaching (continued)
Exploring Angle Pairs
Supplementary Angles and Complementary Angles
Two angles that form a line are supplementary angles. Another term for these angles is
a linear pair. However, any two angles with measures that sum to 180 are also
considered supplementary angles. In both figures below, m∠1 = 120 and m∠2 = 60, so
∠1 and ∠2 are supplementary.
Two angles that form a right angle are complementary angles. However, any two
angles with measures that sum to 90 are also considered complementary angles. In both
figures below, m∠1 = 60 and m∠2 = 30, so ∠1 and ∠2 are complementary.
Exercises
3. Copy the diagram at the right.
a. Label ∠ABD as ∠1.
b. Label an angle that is supplementary to
∠ABD as ∠2.
c. Label as ∠3 an angle that is adjacent
and complementary to ∠ABD.
d. Label as ∠4 a second angle that is
complementary to ∠ABD.
e. Name an angle that is supplementary to ∠ABE.
f. Name an angle that is complementary to ∠EBF.
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Name
1-5
Class
Date
Practice
Form G
Exploring Angle Pairs
Use the diagram at the right. Is each statement true? Explain.
1. ∠ 2 and ∠5 are adjacent angles.
2. ∠1 and ∠4 are vertical angles.
3. ∠4 and ∠5 are complementary.
Name an angle or angles in the diagram described by each of the following.
4. complementary to ∠BOC
5. supplementary to ∠DOB
6. adjacent and supplementary to ∠AOC
Use the diagram below for Exercises 7 and 8. Solve for x.
Find the angle measures.
7. m∠AOB = 4x − 1; m∠BOC = 2x + 15; m∠AOC = 8x + 8
8. m∠COD = 8x + 13; m∠BOC = 3x − 10; m∠BOD = 12x − 6
9. ∠ABC and ∠EBF are a pair of vertical angles; m∠ABC = 3x + 8 and m∠EBF
= 2x + 48. What are m∠ABC and m∠EBF?
10. ∠JKL and ∠MNP are complementary; m∠JKL = 2x − 3 and
m∠MNP = 5x + 2. What are m∠JKL and m∠MNP?
For Exercises 11–14, can you make each conclusion from
the information in the diagram? Explain.
11. ∠3 ≅ ∠4
12. ∠2 ≅ ∠4
13. m∠1 + m∠5 = m∠3
14. m∠3 = 90
uuuur
15. KM bisects ∠JKL. If m∠JKM = 86, what is m∠JKL?
uuur
16. SV bisects ∠RST. If m∠RST = 62, what is m∠RSV?
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Name
1-5
Class
Date
Practice (continued)
Form G
Exploring Angle Pairs
bisects ∠PQR. Solve for x and find m∠PQR.
17. m∠PQS = 3x; m∠SQR = 5x − 20
18. m∠PQS = 2x + 1; m∠RQS = 4x − 15
19. m∠PQR = 3x − 12; m∠PQS = 30
20. m∠PQS = 2x + 10; m∠SQR = 5x − 17
For Exercises 21–24, can you make each conclusion from the information in the
diagram below? Explain.
21. ∠DAB and ∠CDB are congruent.
22. ∠ADB and ∠CDB are complementary.
23. ∠ADB and ∠CDB are congruent.
24. ∠ADB and ∠BCD are congruent.
25. Algebra ∠MLN and ∠JLK are complementary, m∠MLN = 7x − 1, and m∠JLK =
4x + 3.
a. Solve for x.
b. Find m∠MLN and m∠JKL.
c. Show how you can check your answer.
26. Reasoning Describe all the situations in which the following statements are
true.
a. Two vertical angles are also complementary.
b. A linear pair is also supplementary.
c. Two supplementary angles are also a linear pair.
d. Two vertical angles are also a linear pair.
27. Open-Ended Write and solve an equation using an angle bisector to find the
measure of an angle.
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Basic Constructions
1-6
Vocabulary
Review
Draw a line from each word in Column A to its symbol or picture in Column B.
Column A
Column B
1. congruent
S
W
2. point
3. ray
G
4. vertex
P
W
O
Vocabulary Builder
perpendicular (adjective) pur pun DIK yoo lur
Definition: Perpendicular means at right angles to a given line or plane.
Example: Each corner of this paper is formed by perpendicular edges of the page.
Non-Examples: Acute, obtuse, and straight angles do not have perpendicular rays.
Use Your Vocabulary
6. Circle the figure that shows perpendicular segments.
s
s
Chapter 1
22
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5. intersection of segments
Problem 1 Constructing Congruent Segments
Got It? Use a straightedge to draw XY . Then construct RS so that RS 5 2XY.
7. A student did the construction at the right. Describe each
Y
step of the construction.
X
Step 1
Step 2
Step 3
R
S
Step 4
Step 5
Problem 2 Constructing Congruent Angles
Got It? Construct lF so that mlF 5 2mlB at the right.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
8. Use arc or compass to complete the sentence(s) in each step.
In the large box, construct /F .
Step 1 Use a straightedge to construct
a ray with endpoint F.
Step 2 With your ? point on vertex B,
draw a(n) ? that intersects both sides of
ƋB. Label the points of intersection A and C.
Step 3 Use the same compass
setting. Put the ? point on point
F. Draw a long ? and label its
intersection with the ray as S.
A
B
Step 6 Draw FR.
B
C
Step 5 Use the same compass setting.
Put the ? point on point T. Draw an
? and label its intersection with
the first ? as point R.
23
Step 4 Open the ? to the length of
AC. With the compass point on point
S, draw an ? . Label where this arc
intersects the other arc as point T.
Lesson 1-6
A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular
to the segment at its midpoint.
9. Circle the drawing that shows the perpendicular bisector of a segment.
E
A
F
Problem 3
E
E
B
A
A
F B
F
B
Constructing the Perpendicular Bisector
Got It? Draw ST . Construct its perpendicular bisector.
10. Error Analysis A student’s construction of the perpendicular bisector of ST is
shown below. Describe the student’s error.
T
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
11. Do the construction correctly in the box below.
X
S
T
Y
Chapter 1
24
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S
Problem 4 Constructing the Angle Bisector
)
Got It? Draw obtuse lXYZ. Then construct its bisector YP .
12. Obtuse /XYZ is drawn in the box at the right. Complete
the flowchart and do each step of the construction.
X
Step 1 Put the compass point on vertex
. Draw an arc . Label the points of
that intersects the sides of
intersection A and B.
Y
Z
Step 2 Put the compass point on point A and draw an arc. With the same / a different
compass setting, draw an arc using point B. Be sure the arcs intersect. Label
the point where the two arcs intersect P.
Step 3 Draw
.
Lesson Check • Do you UNDERSTAND?
Vocabulary What two tools do you use to make constructions?
Draw a line from each task in Column A to the tool used in Column B.
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Column A
Column B
13. measure lines
compass
14. measure angles
protractor
15. construct arcs
ruler
16. construct lines
straightedge
Math Success
Check off the vocabulary words that you understand.
straightedge
compass
construction
perpendicular bisector
Rate how well you can construct angles and bisectors.
Need to
review
0
2
4
6
8
Now I
get it!
10
25
Lesson 1-6
Name
Class
1-6
Date
Additional Problems
Basic Constructions
Problem 1
Construct EF so that EF > CD.
C
D
Problem 2
Construct /R so that /R > /T .
T
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11
Name
Class
1-6
Date
Additional Problems (continued)
Basic Constructions
Problem 3
Construct line LM so that LM is the perpendicular
bisector of QR.
Q
R
Problem 4
Construct DE, the bisector of /D.
D
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Name
Class
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Date
ELL Support
Basic Constructions
Concept List
angle bisector
is greater than
parallel lines
arc
is perpendicular to
perpendicular bisector
congruent
segments midpoint
perpendicular lines
Choose the concept from the list above that best represents the item in
each box.
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51
1-7
Midpoint and Distance
in the Coordinate Plane
Vocabulary
Review
Use the figure at the right for Exercises 1–6. Write T for true or F for false.
1. Points A and B are both at the origin.
y
8
2. If AB 5 BC, then B is the midpoint of AC.
C
3. The midpoint of AE is F.
D
B
4. The Pythagorean Theorem can be used for any triangle.
x
Ľ10
5. Point C is at (6, 0).
Ľ5
E
A
F
Vocabulary Builder
midpoint (noun)
MID
poynt
Definition: A midpoint of a segment is a point that divides the segment into two
congruent segments.
Use Your Vocabulary
Use the figure at the right for Exercises 7–9.
7. The midpoint of EF is G(
8. The midpoint of AB is (
,
,
y
4
).
G B
), or the origin.
C
Ľ4
9. The midpoint of CD is (
,
).
E
Ľ2
A
26
D
O
Ľ4
Chapter 1
F
x
4
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
6. Point E has a y-coordinate of 28.
Key Concept Midpoint Formulas
On a Number Line
In the Coordinate Plane
The coordinate of the midpoint M of AB
a àb
.
with endpoints at a and b is
2
Given A(x1, y1) and B(x2, y2), the coordinates of the
x1 àx2 y1 ày2
,
midpoint of AB are M
2
2
(
)
Find the coordinate of the midpoint M of each segment with the given endpoints
on a number line.
10. endpoints 5 and 9
11. endpoints 23 and 5
12. endpoints 210 and 23
13. endpoints 28 and 21
14. Complete the diagram below.
4
2
y
(à4) ó2
(1 à) ó2
(,)
(1, 2)
x
2
O
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(17, 4)
4
6
8
10
12
14
16
Problem 2 Finding an Endpoint
Got It? The midpoint of AB has coordinates (4, 29). Endpoint A has coordinates
(23, 25). What are the coordinates of B?
15. Complete the equations below.
(
,
)
Midpoint Coordinates
Midpoint Formula
Endpoint A Coordinates
(
x1 1
,
2
y1 1
2
)
x1 1
2
x1 1
(
,
)
y1 1
5
← Solve two equations. →
y1 1
5
x1 5
16. The coordinates of endpoint B are (
2
5
5
y1 5
).
27
Lesson 1-7
Formula The Distance Formula
The distance between two points A(x1, y1) and B(x2, y2) is d 5 "(x2 2 x1) 2 1 (y2 2 y1) 2 .
The Distance Formula is based on the Pythagorean Theorem.
y
B
y2
d
A
y1
O
c
a
x2 x1
x1
b
y2 y1
a2
x
b2 c2
x2
Use the diagrams above. Draw a line from each triangle side in Column A to the
corresponding triangle side in Column B.
Column A
Column B
17. y2 2 y1
a
18. x2 2 x1
b
19. distance, d
c
Problem 3 Finding Distance
y
S(–2, 14)
Got It? SR has endpoints S(22, 14) and
20. Complete the diagram at the right.
8
21 2 14 5
21. Let S(22, 14) be (x1, y1) and let
R(3, 21) be (x2, y2) . Use the
justifications and complete the
steps below to find SR.
4
x
Ľ8
Ľ4
3 2 (22) 5
d5
SR 5
5
Ä
Ä
5
5
<
Chapter 1
Ä
Ä
Ä
Q
Q
Q
2 x1 R 2 1 Q
2 (22) R 2 1 Q
R2 1 Q
1
R2
2 y1 R 2
2 14 R 2
Use the Distance Formula.
Substitute.
Subtract.
Simplify powers.
Add.
Use a calculator.
28
6
R(3, –1)
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R(3, 21). What is SR to the nearest tenth?
Problem 4 Finding Distance
Got It? On a zip-line course, you are harnessed to a cable that travels through the
treetops. You start at Platform A and zip to each of the other platforms. How far do
you travel from Platform D to Platform E? Each grid unit represents 5 m.
2200
A
C
D
F
10
10
10
10
10 O
30
30
30
50
y
10
10
20
20
30
30
40
40
x
50
10
110
0
E
20
220
0
B
22. The equation is solved below. Write a justification for each step.
d 5 "(x2 2 x1) 2 1 (y2 2 y1) 2
DE 5 "(30 2 20) 2 1 (215 2 20) 2
5 "102 1 (235) 2 5 "100 1 1225 5 "1325
23. To the nearest tenth, you travel about
m.
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Lesson Check • Do you UNDERSTAND?
Reasoning How does the Distance Formula ensure that the distance between two
different points is positive?
24. A radical symbol with no sign in front of it indicates a positive / negative
square root.
25. Now answer the question.
__________________________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
midpoint
distance
coordinate plane
Rate how well you can use the Midpoint and Distance Formulas.
Need to
review
0
2
4
6
8
Now I
get it!
10
29
Lesson 1-7
Name
Class
1-7
Date
Additional Problems
Midpoint and Distance in the Coordinate Plane
Problem 1
FG has endpoints at 23 and 7. What is the coordinate
of its midpoint?
F
6 4 2
G
0
2
4
6
8 10 12 14 16
Problem 2
y
The midpoint of LM is A(2, 21).
One endpoint is L(23, 25). What
are the coordinates of the
other endpoint?
6
4
2
6 4 2
2
4
L
6
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13
A
4
6 x
Name
1-7
Class
Date
Additional Problems (continued)
Midpoint and Distance in the Coordinate Plane
Problem 3
What is the distance between (6, 22) and (25, 3)? Round to the
nearest tenth.
Problem 4
y
On a zip-line course, you
40
are harnessed to a cable that
travels through the treetops.
20
C
You start at Platform A and
zip to each of the other
60 40 20
20 40
platforms. How far do you
20
travel from Platform B to
B
40
Platform C?
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14
x
60
1-8
Perimeter, Circumference,
and Area
Vocabulary
Review
1. Cross out the shapes that are NOT polygons.
2. Write the name of each figure. Use each word once.
triangle
square
rectangle
circle
consecutive (adjective) kun SEK yoo tiv
Definition: Consecutive means following in order without interruption.
Related Word: sequence
Example: The numbers 2, 4, 6, 8, . . . are consecutive even numbers.
Non-Example: The numbers 1, 3, 2, 5, 4, . . . are NOT consecutive numbers.
Use Your Vocabulary
Draw a line from each sequence of letters in Column A to the next consecutive
letter in Column B.
Column A
Column B
3. L, M, N, O, . . .
R
4. V, U, T, S, . . .
I
5. A, C, E, G….
P
Chapter 1
30
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Vocabulary Builder
Key Concept Perimeter, Circumference, and Area
6. Label the parts of each of the figures below.
Square
P 5 4s
A5
s2
Triangle
Rectangle
Circle
P5a1b1c
P 5 2b 1 2h
C 5 pd or C 5 2pr
A 5 12bh
A 5 bh
A 5 pr2
Problem 1 Finding the Perimeter of a Rectangle
Got It? You want to frame a picture that is 5 in. by 7 in. with a 1-in.-wide frame.
What is the perimeter of the picture?
7. The picture is
in. by
in.
8. Circle the formula that gives the perimeter of the picture.
P 5 4s
P 5 2b 1 2h
P5a1b1c
C 5 pd
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9. Solve using substitution.
10. The perimeter of the picture is
in.
Problem 2 Finding Circumference
Got It? What is the circumference of a circle with radius 24 m in terms of π?
11. Error Analysis At the right is one student’s solution. What error did the
student make?
_________________________________________________________
_________________________________________________________
24 m
c = πd
c = π(24)
c = 24π
12. Find the correct circumference.
31
Lesson 1-8
Problem 3
Finding Perimeter in the Coordinate Plane
Got It? Graph quadrilateral JKLM with vertices J(23, 23),
5
K(1, 23), L(1, 4), and M(23, 1). What is the perimeter of JLKM?
4
3
13. Graph the quadrilateral on the coordinate plane at the right.
2
14. Use the justifications at the right to find the length of each side.
JK 5 P 23 2 1 P
Use the Ruler Postulate.
KL 5 P 42
P
5
JM 5 P 232
P
5
ML 5
5
5
5
5
Ä
Ä
Ä
Ä
1
Ľ5 Ľ4 Ľ3 Ľ2 Ľ1 O
Simplify.
5
)2 1 32
(
)1(
(
)
x
1
2
3
4
5
Ľ2
Use the Ruler Postulate.
Ľ3
Simplify.
Ľ5
Ľ4
Use the Ruler Postulate.
Simplify.
(1 2 (23))2 1 (4 2
(
y
)2
Use the Distance Formula.
Simplify within parentheses.
Simplify powers.
)
Add.
Take the square root.
JK 1 KL 1 JM 1 ML 5
16. The perimeter of JKLM is
1
1
1
5
units.
Problem 5 Finding Area of a Circle
Got It? The diameter of a circle is 14 ft. What is its area in terms of p?
17. Label the diameter and radius of the circle at the right.
18. Use the formula A 5
pr2
19. The area of the circle is
to find the area of the circle in terms of p.
p ft 2 .
Key Concept Postulate 1–10 Area Addition Postulate
20. The area of a region is the sum / difference of the areas of its nonoverlapping parts.
Chapter 1
32
ft
ft
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15. Add the side lengths to find the perimeter.
Problem 6 Finding Area of an Irregular Shape
Got It? Reasoning The figure below shows one way to separate the figure at the
left. What is another way to separate the figure?
3 cm
A1
3 cm
3 cm
A2
9 cm
3 cm
6 cm
A3
3 cm
9 cm
21. Draw segments to show two different ways to separate the figure. Separate the
left-hand figure into three squares.
3 cm
3 cm
9 cm
9 cm
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Lesson Check • Do you UNDERSTAND?
Compare and Contrast Your friend can’t remember whether 2pr computes the
circumference or the area of a circle. How would you help your friend? Explain.
22. Underline the correct word(s) to complete each sentence.
Area involves units / square units .
Circumference involves units / square units .
The formula 2pr relates to area / circumference because it involves units / square units .
Math Success
Check off the vocabulary words that you understand.
perimeter
area
Rate how well you can find the area of irregular shapes.
Need to
review
0
2
4
6
8
Now I
get it!
10
33
Lesson 1-8
Name
1-8
Class
Date
Additional Problems
Perimeter, Circumference, and Area
Problem 1
To place a fence on the outside of the garden,
how much material will you need?
5 ft
12 ft
Problem 2
What is the circumference of the circle in terms
of p? What is the circumference of each circle to
the nearest tenth?
5 in.
U
Problem 3
What is the perimeter of triangle LMN?
y
6
L
4
2
4
2
6 4 2
2
4
M
6
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15
N
6 x
Name
1-8
Class
Date
Additional Problems (continued)
Perimeter, Circumference, and Area
Problem 4
You are designing a rectangular flag for your city’s museum.
The flag will be 15 feet wide and 2 yards high. How many square
yards of material do you need?
Problem 5
The diameter of (L is 10 cm. What is its
area in terms of p?
10 cm
L
Problem 6
What is the area of the figure below?
20 m
16 m
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16
Name
1-8
Class
Date
Practice
Form G
Perimeter, Circumference, and Area
Find the perimeter of each figure.
1.
2.
3. An 8-ft-by-10-ft rug leaves 1 ft of the bedroom floor exposed on all four sides.
Find the perimeter of the bedroom floor.
Find the circumference of each circle in terms of π.
4.
5.
6.
Graph each figure in the coordinate plane. Find the perimeter.
7. X(−4, 2), Y(2, 10), Z(2, 2)
8. R(1, 2), S(1, −2), T(4, −2)
9. A(0, 0), B(0, 5), C(6, 5), D(6, 0)
10. L(−3, 2), M(2, 14), N(2, 20), P(−3, 20)
Find the area of the rectangle with the given base and height.
11. 4 ft, 15 in.
12. 90 in., 3 yd
13. 3 m, 130 cm
Find the area of each circle in terms of π.
14.
15.
16.
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Name
Class
1-8
Date
Practice (continued)
Form G
Perimeter, Circumference, and Area
Find the area of each shaded region. All angles are right angles.
17.
18.
19.
Find the circumference and area of each circle, using π = 3.14. If necessary, round to
the nearest tenth.
20. r = 5 m
21. d = 2.1 in.
22. d = 2 m
23. r = 4.7 ft
2
24. The area of a circle is 25 π in. . What is its radius?
25. A rectangle has twice the area of a square. The rectangle is 18 in. by 4 in. What is the
perimeter of the square?
26. Reasoning If two circles have the same circumference, what do you know about
their areas? Explain.
27. Coordinate Geometry The center of a circle is A(−3, 3), and B(1, 6) is on the
circle. Find the area of the circle in terms of π.
28. Algebra Use the formula for the circumference of a circle to write a formula for the
area of a circle in terms of its circumference.
29. Coordinate Geometry On graph paper, draw polygon ABCDEF with vertices A(0,
0), B(0, 10), C(5, 10), D(5, 7), E(9, 7), and F(9, 0). Find the perimeter and the area of
the polygon.
30. The units of the floor plan at the right are in feet.
Find the perimeter and area of each room.
a. the kitchen
b. the bedroom
c. the bathroom
d. the closet
e. What is the area of the main hallway? Explain how you
could find this area using the area of each room.
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Name
1-8
Class
Date
ELL Support
Perimeter, Circumference, and Area
For Exercises 1-7, draw a line from each word in Column A to its definition in
Column B. The first one is done for you.
Column A
Column B
1. perimeter
a polygon with three sides
2. radius
3. diameter
the sum of the lengths of the sides of a polygon, or the
distance around a polygon
the distance from the center to a point on a circle
4. area
a polygon with four sides
5. circumference
the distance across a circle, through the center
6. triangle
the number of square units a figure encloses
7. quadrilateral
the distance around a circle
For Exercises 8-13, draw a line from each phrase in Column A to its formula
in Column B.
Column A
Column B
8. perimeter of a square
4s
9. circumference of a circle
bh
10. area of a rectangle
1
bh
2
11. area of a triangle
πd or 2πr
12. perimeter of a rectangle
2b + 2h
13. area of a circle
πr2
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Name
1-8
Class
Date
Reteaching
Perimeter, Circumference, and Area
The perimeter of a rectangle is the sum of the lengths of its sides. So, the perimeter
is the distance around its outside. The formula for the perimeter of a rectangle is
P = 2(b + h).
The area of a rectangle is the number of square units contained within the
rectangle. The formula for the area of a rectangle is A = bh.
Exercises
1. Fill in the missing information for each rectangle in the table below.
Dimensions Perimeter, P = 2(b + h)
1 ft × 9ft
2(1 ft + 9 ft) = 20 ft
Area, A = bh
1 ft × 9 ft = 9 ft
2
2 ft × 8ft
3 ft × 7ft
4ft × 6 ft
2. How does the perimeter vary as you move down the table? How does the area vary
as you move down the table?
3. What pattern in the dimensions of the rectangles explains your answer to
Exercise 2?
4. Fill in the missing information for each rectangle in the table below.
Dimensions Perimeter, P = 2(b + h)
Area, A = bh
1 ft × 24 ft
2 ft × 12 ft
3 ft × 8 ft
4 ft × 6 ft
5. How does the perimeter vary as you move down the table? How does the area vary
as you move down the table?
6. What pattern in the dimensions of the rectangles explains your answer to
Exercise 5?
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Name
Class
1-8
Date
Reteaching (continued)
Perimeter, Circumference, and Area
A square is a rectangle that has four sides of the same length. Because the perimeter is
s + s + s + s, the formula for the perimeter of a square is P = 4s. The formula for the
area of a square is A = s2.
The circumference of a circle is the distance around the circle. The formula for the
circumference of a circle is C = πd or C = 2πr. The area of a circle is the number of
square units contained within the circle. The formula for the area of a circle is A = πr2.
Exercises
7. Fill in the missing information for each square in the table below.
Side
Perimeter, P = 4s
3 cm
4 × 3 cm = 12 cm
Area, A = s
2
2
(3 cm) = 9 cm
2
4 cm
4 × 5 cm = 20 cm
2
(10 cm) = 100 cm
2
8. Fill in the missing information for each circle in the table below.
Radius
2 in.
Diameter, D = 2r
2 × 2 = 4 in.
Circumference, C = 2π r
Area, A = π r
2
π × 2 × 2 = 4π in.2
2π × 2 = 4π in.
3 in.
2 × 5 = 10 in.
2π × 8 = 16π in.
π × 10 × 10 = 100π in.2
2
9. A rectangle has a length of 5 cm and an area of 20 cm . What is its width?
2
10. What is the perimeter of a square whose area is 81 ft ?
11. Can you find the perimeter of a rectangle if you only know its area? What
about a square? Explain.
12. Can you find the area of a circle if you only know its circumference? Explain.
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