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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.

U

M

B

P

35°

21°

62°

a. Explain how you can use the Exterior Angle Theorem to calculate the measure of /PMU.

TheExteriorAngleTheoremstatesthatthemeasureoftheexteriorangleofatriangleis

equaltothesumofthemeasuresofthetworemoteinterioranglesofthetriangle.Angle

PMUisanexteriorangletonPBManditscorrespondingremoteinterioranglesare/PBM

and/BPM.So,Icancalculatethesumofthosetwoanglemeasurestofindthemeasure

of/PMU.

b. Calculate the measure of /PMU. Show your work.

m/PMU5m/PBM1m/BPM

5 35º1 21º

5 56º

Themeasureof/PMU is56°.

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Lesson 10.1 Assignment page 2

c. Explain how you can use the Triangle Sum Theorem to calculate the measure of /UPM.

TheTriangleSumTheoremstatesthatthesumofthemeasuresoftheinterioranglesofa

triangleis180°.IknowthatthesumofthethreeanglesinnUPMmustequal180°.Ican

addthemeasuresof/PUM and/PMU andthensubtractthattotalfrom180°tofindthe

measureof/UPM.

d. Calculate the measure of /UPM. Show your work.

m/PMU1m/MUP1m/UPM 5 180º

56º1 62º1m/UPM 5 180º

5 62º

Themeasureof/UPMis62°.

e. Which side of nPUB is the longest? Explain your reasoning.

Themeasureof/UPB is62°121°or83°,whichmakesitthelargestangleinthetriangle.

So,___

UB isthelongestsidebecauseitisacrossfromthelargestangle.

f. Which side of nPUB is the shortest? Explain your reasoning.

SidePUistheshortestsidebecauseitisacrossfromthesmallestangle.

g. List the sides of nPMB in order from shortest to longest. Explain how you determined your

answer.

____

MB , ____

PM , ___

PB

SideMBisacrossfromthesmallestangle,whichmakesittheshortestside.SidePBis

acrossfromthelargestangle,whichmakesitthelongestside.

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Triangle Construction IConstructing 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.

X Y

Z

b. Itisnotpossibletoconstructanothertrianglethatisnotcongruenttothefirsttriangle.

c. Itisnotpossiblebecausetwosegmentsandanincludedangledetermineauniquetriangle.

Lesson 10.2 Assignment

Name ________________________________________________________ Date _________________________

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Lesson 10.2 Assignment page 2

2. Use the three angles to construct n JKL.

J K L

a.K

J L

b. Itispossibletoconstructanothertrianglethatisnotcongruenttothefirsttriangle.

c. TriangleJKLbelowhasthesameanglemeasuresastriangleJKL,butthelengthsofthe

sidesaredifferent.

K

J L

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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.

M

N

NM

a.

NM

P

b. Itisnotpossibletoconstructanothertrianglethatisnotcongruenttothefirsttriangle.

c. Itisnotpossiblebecausetwoanglesandanincludedsidedetermineauniquetriangle.

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Triangle Construction IICongruent Figures and Constructing Congruent Triangles

1. Consider nPQR and nSTU.

P R

Q T

S

U

a. Explain how to determine whether nPQR is congruent to nSTU.

MeasureallofthecorrespondinganglesandcorrespondingsidesoftrianglesPQR

andSTU.Ifcorrespondinganglesarecongruentandcorrespondingsidesarecongruent,

thenthetrianglesarecongruent.

b. Is nPQR congruent to nSTU? Explain your reasoning.

No.nPQRisnotcongruenttonSTUbecausenotallofthecorrespondingsides

andcorrespondinganglesarecongruent.

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.

TriangleXYZiscongruenttonPQRbecauseallofthecorrespondingsides

andcorrespondinganglesarecongruent.

Lesson 10.3 Assignment

Name ________________________________________________________ Date _________________________

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Lesson 10.3 Assignment page 2

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.

B

A C

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h. Construct nDEF so that it is congruent to nPQR by copying two sides and their

included angle.

E

D F

i. Construct nGHI so that it is congruent to nPQR by copying two angles and their included side.

H

G I

Lesson 10.3 Assignment page 3

Name ________________________________________________________ Date _________________________

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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.

Thelengthofanysideofatrianglemustbelessthanthesumofthemeasuresofthetwo

othersides.So,thethirdsidemustbelessthan1414518feetlong.Thelongestboard

youhaveleftthatislessthan18feetisoneofthe12-footboards.

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.

Thelengthofanysideofatrianglemustbelessthanthesumofthemeasuresofthetwo

othersides.Thelongestsideofthetriangleis12feetandoneoftheothersidesis8feet.

Forthesumoftheothertwosidestobegreaterthan12feet,thethirdsidemustbe

greaterthan4feet.Theshortestboardyouhaveleftthatisgreaterthan4feetisoneof

the12-footboards.

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.

Thelengthofanysideofatrianglemustbelessthanthesumofthemeasuresofthetwo

othersides.So,thelongestpossibleboardmustbelessthan1214516feet,andthe

shortestpossibleboardmustbegreaterthan122458feet.Oftheboardsyouhaveleft,

theother12-footboardorthe14-footboardcanbeusedforthethirdsideofthepen.

Lesson 10.4 Assignment

Name ________________________________________________________ Date _________________________

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Lesson 10.4 Assignment page 2

d. How many different triangular pens can be formed using the 4-foot board?

List the side lengths of each possible triangular pen.

4feet,2feet,3feet

4feet,12feet,12feet

4feet,12feet,14feet

Threedifferenttriangularpenscanbeformedusingthe4-footboard.

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.Thetwoshortestsidesare4feetand5feet,andtheirsum,9feet,isgreaterthan

thelengthoftheotherside,whichis8feet.So,theseboardlengthscanbeusedtoform

atriangularpen.

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.

8 ft12 ft

14 ft

Inanytriangle,theanglewiththegreatestmeasureisoppositethelongestside,andthe

anglewiththeleastmeasureisoppositetheshortestside.So,theanglethatisoppositethe

14-footsidehasthegreatestmeasure,andtheanglethatisoppositethe8-footsidehas

theleastmeasure.