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Lesson 10.1 Assignment
Name_________________________________________________________ Date__________________________
Pulling a One-Eighty!
Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems
1. Use the figure shown to answer the questions.
P
21°
35°
B
M
62°
U
a. Explain how you can use the Exterior Angle Theorem to calculate the measure of /PMU.
The Exterior Angle Theorem states that the measure of the exterior angle of a triangle is
equal to the sum of the measures of the two remote interior angles of the triangle. Angle
PMU is an exterior angle to nPBM and its corresponding remote interior angles are /PBM
and /BPM. So, I can calculate the sum of those two angle measures to find the measure
of /PMU.
b. Calculate the measure of /PMU. Show your work.
© 2011 Carnegie Learning
m/PMU 5 m/PBM 1 m/BPM
5 35º 1 21º
5 56º
The measure of /PMU is 56°.
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Lesson 10.1 Assignment
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c. Explain how you can use the Triangle Sum Theorem to calculate the measure of /UPM.
The Triangle Sum Theorem states that the sum of the measures of the interior angles of a
triangle is 180°. I know that the sum of the three angles in nUPM must equal 180°. I can
add the measures of /PUM and /PMU and then subtract that total from 180° to find the
measure of /UPM.
d. Calculate the measure of /UPM. Show your work.
m/PMU 1 m/MUP 1 m/UPM 5 180º
56º 1 62º 1 m/UPM 5 180º
5 62º
The measure of /UPM is 62°.
e. Which side of nPUB is the longest? Explain your reasoning.
The measure of /UPB is 62° 1 21° or 83°, which makes it the largest angle in the triangle.
___
So, ​UB ​ is the longest side because it is across from the largest angle.
Side PU is the shortest side because it is across from the smallest angle.
g. List the sides of nPMB in order from shortest to longest. Explain how you determined your
answer.
____ ____ ___
, PM ​
​  
, PB ​
​  
​MB​ 
Side MB is across from the smallest angle, which makes it the shortest side. Side PB is
across from the largest angle, which makes it the longest side.
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f. Which side of nPUB is the shortest? Explain your reasoning.
Lesson 10.2 Assignment
Name_________________________________________________________ Date__________________________
Triangle Construction I
Constructing Triangles
In each exercise, do the following:
a.Use the given information to construct a triangle.
b.Determine whether it is possible to use the given information to construct another triangle that
is not congruent to the first triangle.
c.If it is possible to construct another triangle that is not congruent to the first triangle, construct
it. If it is not possible, explain why not.
1. Use the two line segments and the included angle to construct n XYZ.
X
Y
Y
Z
Y
a.
Z
X
Y
b. It is not possible to construct another triangle that is not congruent to the first triangle.
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c. It is not possible because two segments and an included angle determine a unique triangle.
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Lesson 10.2 Assignment
page 2
2. Use the three angles to construct n JKL.
J
a.
K
L
K
L
J
b. It is possible to construct another triangle that is not congruent to the first triangle.
c. Triangle JKL below has the same angle measures as triangle JKL, but the lengths of the
sides are different.
K
L
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J
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Lesson 10.2 Assignment
page 3
Name_________________________________________________________ Date__________________________
3. Use the two angles and the included side to construct n MNP.
N
M
a.
M
N
P
M
N
b. It is not possible to construct another triangle that is not congruent to the first triangle.
© 2011 Carnegie Learning
c. It is not possible because two angles and an included side determine a unique triangle.
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200 • Chapter 10 Assignments
Lesson 10.3 Assignment
Name_________________________________________________________ Date__________________________
Triangle Construction II
Congruent Figures and Constructing Congruent Triangles
1. Consider nPQR and nSTU.
Q
T
S
P
U
R
a.Explain how to determine whether nPQR is congruent to nSTU.
Measure all of the corresponding angles and corresponding sides of triangles PQR
and STU. If corresponding angles are congruent and corresponding sides are congruent,
then the triangles are congruent.
b.Is nPQR congruent to nSTU? Explain your reasoning.
No. nPQR is not congruent to nSTU because not all of the corresponding sides
and corresponding angles are congruent.
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c.Use your ruler and protractor to draw nXYZ such that it is congruent to nPQR.
Y
X
Z
d.Explain how you know that nXYZ is congruent to nPQR.
Triangle XYZ is congruent to nPQR because all of the corresponding sides
and corresponding angles are congruent.
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Lesson 10.3 Assignment
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Name_________________________________________________________ Date__________________________
e.Use the congruence symbol to show the corresponding sides that are congruent in
nPQR and nXYZ.
___
___
____
___
___
___
PQ ​
​  
​  > XY ​
​QR ​ > YZ ​
​  
​PR ​ > XZ ​
​  
f.Use the congruence symbol to show the corresponding angles that are congruent in nPQR
and nXYZ.
/P > /X
/Q > /Y
/R > /Z
g.Construct nABC so that it is congruent to nPQR by copying all three sides.
A
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C
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B
Lesson 10.3 Assignment
page 3
Name_________________________________________________________ Date__________________________
h.Construct DEF so that it is congruent to PQR by copying two sides and their
included angle.
E
D
F
i.Construct GHI so that it is congruent to PQR by copying two angles and their included side.
H
I
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G
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204 • Chapter 10 Assignments
Lesson 10.4 Assignment
Name_________________________________________________________ Date__________________________
Pasta Anyone?
Triangle Inequality Theorem
1. You are building a triangular pen for baby ducks. The sides of the pen will be made from lumber
you have left from other projects. You have two 12-foot boards, one 14-foot board, one 8-foot
board, one 4-foot board, one 3-foot board, and one 2-foot board. Use this information to answer
parts (a) through (f ).
a.Suppose you choose the 14-foot board and the 4-foot board. Of the boards you have left over,
what is the longest board that can be used for the third side of the pen? Explain.
The length of any side of a triangle must be less than the sum of the measures of the two
other sides. So, the third side must be less than 14 1 4 5 18 feet long. The longest board
you have left that is less than 18 feet is one of the 12-foot boards.
b.Suppose you choose a 12-foot board and the 8-foot board. Of the boards you have left over,
what is the shortest board that can be used for the third side of the pen? Explain.
The length of any side of a triangle must be less than the sum of the measures of the two
other sides. The longest side of the triangle is 12 feet and one of the other sides is 8 feet.
For the sum of the other two sides to be greater than 12 feet, the third side must be greater than 4 feet. The shortest board you have left that is greater than 4 feet is one of © 2011 Carnegie Learning
the 12-foot boards.
c.Suppose you choose a 12-foot board and the 4-foot board. Of the boards you have left over,
which board(s) can be used for the third side of the pen? Explain.
The length of any side of a triangle must be less than the sum of the measures of the two
other sides. So, the longest possible board must be less than 12 1 4 5 16 feet, and the
shortest possible board must be greater than 12 2 4 5 8 feet. Of the boards you have left,
the other 12-foot board or the 14-foot board can be used for the third side of the pen.
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Lesson 10.4 Assignment
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d. How many different triangular pens can be formed using the 4-foot board?
List the side lengths of each possible triangular pen.
4 feet, 2 feet, 3 feet
4 feet, 12 feet, 12 feet
4 feet, 12 feet, 14 feet
Three different triangular pens can be formed using the 4-foot board.
e. If you only have three boards and their lengths are 5 feet, 8 feet, and 4 feet, can you form a
triangular pen? Explain.
Yes. The two shortest sides are 4 feet and 5 feet, and their sum, 9 feet, is greater than the length of the other side, which is 8 feet. So, these board lengths can be used to form a triangular pen.
f.Suppose you decide to build a pen with side lengths of 14 feet, 12 feet, and 8 feet as shown in
the figure. Which angle has the greatest measure? Which angle has the least measure? Explain.
14 ft
In any triangle, the angle with the greatest measure is opposite the longest side, and the angle with the least measure is opposite the shortest side. So, the angle that is opposite the
14-foot side has the greatest measure, and the angle that is opposite the 8-foot side has the least measure.
206 • Chapter 10 Assignments
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12 ft
8 ft