LESSON

132 Domain 4: Geometry

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24 Angles in Triangles

UNDERSTAND Atrianglehas3sidesand3interior angles .Ifonesideofatriangleisextended,anexterior angleisformed .

Thesumofthemeasuresoftheinterioranglesinanytriangleis180° .Thisdiagramcanbeusedtoshowthatthisistruefortheshadedtriangle .Inthediagram,linel||linemandtheyarecutbytwotransversals,linesnandp .

n p

l

m

54 2

1 3

Adjacentangles4,2,and5formastraightline .

m/41m/21m/55180°

Lookathowparallellineslandmarecutbytransversallinesnandp .

n p

l

m

54 2

1 3

Whenlinencutsparallellineslandm,itformsalternateinteriorangles5and3 .

Alternateinterioranglesarecongruent,som/55m/3 .

Whenlinepcutsparallellineslandm,itformsalternateinteriorangles4and1 .

Alternateinterioranglesarecongruent,som/45m/1 .

m/45m/1andm/55m/3 .Substitutem/1form/4andm/3form/5intheequationfromabove:m/41m/21m/55180° .

m/11m/21m/35180°

exteriorangle

interiorangle

Connect

Lesson 24: Angles in Triangles 133

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Findthevalueofx .

x°a°40°

79°

Whatdoesxrepresent?

Itrepresentsthedegreemeasureofanexteriorangleofthetriangle .

Usethefactthatthe61°interiorangleanditsexterioranglearesupplementarytofindthevalueofx .

61°1x°5180°

  x°5119°

▸x5119

3

1

First,finda°,themeasureofthethirdinteriorangleofthetriangle .

Thesumofthemeasuresoftheinterioranglesofatriangleis180° .

40°179°1a°5180°

119°1a°5180°

a°561°

2

Istheexteriorangleintheexampletheonlyexterioranglethatcanbedrawnforthistriangle?Ifnot,drawtwoothersandlabeltheirmeasuresy°andz° .

x°a°40°

79°

MODEl

Practice

134 Domain 4: Geometry

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Write an equation that could be used to find the value of x for each triangle.

1.

x°60°

70°2.

61°

x°

REMEMBER  A right angle is 90°.

3.

110° 38°

x°

State whether or not it is possible to draw a triangle with the given interior angle measures.

4. 45°,55°,80° 5. 30°,60°,100° 6. 12°,84°,84°

Find the value of z for each triangle. Show your work.

7.

60°

48°

z°

8.

42°

34°

z°

9.

40°z°

Find the value of x for each triangle. Show your work.

10.

x°28°

32°

11.

x°

30° 82°

Lesson 24: Angles in Triangles 135

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12.

x°

(2x)°13.

x°110°

41°

Use the diagram below for questions 14–15.

41 3

2

14. Whyisthisstatementtrue?

m/11m/21m/35180°

15. Whyisthisstatementtrue?

m/31m/45180°

Solve.

16. EXPLAIN Iftwotriangleshavethreepairsofcorrespondinganglesthatarecongruent,thetrianglesmustbesimilar .WouldtheinformationgiveninthediagrambelowbesufficienttoallowyoutoprovethatnABCandnDEFaresimilar?Explain .

A C D F

EB

60° 60°40° 40°

17. CONCLUDE Thetriangleonthelefthasinteriorangles1,2,and3 .

1 13

2

23

Martacopiedthistriangle3times,rotatedthosetriangles,andplacedthemtogethersothatangles1,2,and3werealigned .Whatdoesthisallowyoutoconcludeabouttheinterioranglemeasuresofatriangle?Explain .