Chapter 4: Congruent Triangles

1 Chapter 4 Congruent TrianglesChapter 5 Relationships in TrianglesChapter 6 Proportions and SimilarityChapter 7 Right Triangles and TrigonometryChapter 4 Congruent TrianglesChapter 5 Relationships in TrianglesChapter 6 Proportions and SimilarityChapter 7 Right Triangles and TrigonometryTrianglesTrianglesYou can use trianglesand their properties tomodel and analyzemany real-worldsituations. In thisunit, you will learnabout relationships inand among Triangles ,including congruenceand similarity.(l)A. Ramey/Woodfin Camp & Associates, (r)Dennis MacDonald/PhotoEdit174 Unit 2 TrianglesContinue working onyour WebQuest as youstudy Unit on to your WebQuest by reading the Is Behind This Geometry Concept Anyway?Unit 2 Triangles1754-65-16-67-1216241325347 LessonPage Have you ever wondered who first developedsome of the ideas you are learning in your geometryclass? Today, many students use the Internet forlearning and research. In this project, you will beusing the Internet to research a topic in geometry.

2 You will then prepare a portfolio or poster to displayyour children use the InternetPercentage of children 3-17using the Internet at home,by region:USA TODAY Snapshots By Sam Ward, USA TODAYS ource: Census exterior angle (p. 186) flow proof (p. 187) corollary (p. 188) Congruent Triangles (p. 192) coordinate proof (p. 222)Key Vocabulary Lesson 4-1 Classify Triangles . Lesson 4-2 Apply the Angle Sum Theoremand the Exterior Angle Theorem. Lesson 4-3 Identify corresponding parts ofcongruent Triangles . Lessons 4-4 and 4-5 Test for trianglecongruence using SSS, SAS, ASA, and AAS. Lesson 4-6 Use properties of isosceles andequilateral Triangles . Lesson 4-7 Write coordinate are found everywhere you look. Triangles with thesame size and shape can even be found on the tail of a will learn more about orca whales in Lesson Chapter 4 Congruent Triangles176 Chapter 4 Congruent TrianglesDaniel J. Cox/Getty Images Prerequisite SkillsTo be successful in this Chapter , you ll need to masterthese skills and be able to apply them in problem-solving situations.

3 Reviewthese skills before beginning Chapter Lesson 4-1 Solve EquationsSolve each equation.(For review , see pages 737 and 738.) 18 16 12 8 2a 12 6. 23 b 9 15 For Lessons 4-2, 4-4, and 4-5 Congruent AnglesName the indicated angles or pairs of angles if p qand m .(For review , see Lesson 3-1.) Congruent to Congruent to supplementary to supplementary to 12 For Lessons 4-3 and 4-7 Distance FormulaFind the distance between each pair of points. Round to the nearest tenth.(For review , see Lesson 1-3.)11.(6, 8), ( 4, 3)12.( 15, 12), (6, 18)13.(11, 8), ( 3, 4)14.( 10, 4), (8, 7)56711131415161234891210mpq Chapter 4 Congruent Triangles177 Chapter 4 Congruent Triangles177 TrianglesMake this Foldable to help you organize your notes. Begin with twosheets of grid paper and one sheet of construction and WritingAs you read and study the Chapter , use your journal for sketches and examples ofterms associated with Triangles and sample and CutStaple and LabelStack the gridpaper on theconstructionpaper.

4 Folddiagonally asshown and cutoff the the edge to form a label eachpage with alesson numberand Triangles BY ANGLESR ecall that a triangle is a three-sidedpolygon. Triangle ABC, written ABC, has parts that are named using the letters A, B, and C. The sides of ABCare A B , B C , and C A . The vertices are A, B, and C. The angles are ABCor B, BCAor C, and BACor A. There are two ways to classify Triangles . One way is by their angles. All triangleshave at least two acute angles, but the third angle is used to classify the triangle. An acute triangle with all angles Congruent is an .equiangular triangleBCAV ocabulary acute triangle obtuse triangle right triangle equiangular triangle scalene triangle isosceles triangle equilateral triangleClassifying Triangles178 Chapter 4 Congruent Triangles Identify and classify Triangles by angles. Identify and classify Triangles by structures use triangular shapes asbraces for construction.

5 The roof sections ofhouses are made of triangular trusses thatsupport the roof and the house. Classify Triangles by AnglesARCHITECTUREThe roof of this house ismade up of three different Triangles . Use aprotractor to classify DFH, DFG, and HFGas acute, equiangular, obtuse, or right. DFHhas all angles with measures less than 90, so it is an acute triangle. DFGand HFGboth have one angle withmeasure equal to 90. Both of these are (t)Martin Jones/CORBIS, (b)David Scott/Index StockCommonMisconceptionsThese classifications aredistinct groups. Forexample, a triangle cannotbe right and acute. Study TipClassifying Triangles by AnglesIn an , allof the angles are angle measures 90In an ,one angle is angle measure 90In a , oneangle is angle measure 9042 90 48 right triangle13 142 25 obtuse triangle37 76 67 acute triangleare Triangles importantin construction?are Triangles importantin construction?Equilateral TrianglesModel Align three pieces of patty paper as indicated.

6 Draw a dot at X. Fold the patty paperthrough Xand Yandthrough Xand XYZequilateral? Explain. three pieces of patty paper to make atriangle that is isosceles, but not three pieces of patty paper to make ascalene 4-1 Classifying Triangles179 Classify Triangles by SidesIdentify the indicated type of triangle in the isosceles trianglesb. scalene trianglesIsosceles Triangles have Scalene Triangles have at least two sides no Congruent sides. Congruent . So, ABD AEB, AED, ACB,and EBDare isosceles. ACD, BCE, and DCEare Triangles by SidesNo two sides of least two sides of of the sides of triangleisosceles trianglescalene triangleFind Missing ValuesALGEBRAFind xand the measure of each side of equilateral triangle RSTif RS x 9, ST 2x, and RT 3x RSTis equilateral, RS ST. x 9 2xSubstitution9 xSubtract xfrom each , substitute to find the length of each side. RS x 9ST 2xRT 3x 9 9 9 or 18 2(9) or 18 3(9) 9 or 18 For RST, x 9, and the measure of each side is Triangles BY SIDEST riangles can also be classified accordingto the number of Congruent sides they have.

7 To indicate that sides of a triangle arecongruent, an equal number of hash marks are drawn on the corresponding equilateral triangle is a special kind of isosceles + 93x Chapter 4 Congruent TrianglesConcept CheckGuided PracticeUse the Distance FormulaCOORDINATE GEOMETRYFind the measures of the sides of DEC. Classify the triangle by the Distance Formula to find the lengths of each ( 5 2)2 (3 2 )2 ED ( 5 3)2 (3 9 )2 49 1 64 3 6 50 100 DC (3 2 )2 (9 2)2 1 49 50 Since E C and D C have the same length, DECis Explainhow a triangle can be classified in two ENDEDDraw a triangle that is isosceles and whether each of the following statements is always, sometimes, ornever true. Triangles are also Triangles are a protractor to classify each triangle as acute, equiangular, obtuse, or the obtuse Triangles the right Triangles if MJK KLM, m MJK 126, I J G H , G H D F , and G I E F .and m JNM x, JM, MN, and x, QR, RS, and QSJN if JMNis an isosceles triangle if QRSis an equilateral triangle.

8 With J M M N .QRS4x2x + 16x 1 JMN3x 92x 5x 2 DHJGEIFMLNJKE xample4 Example4 Look BackTo review the DistanceFormula, see Lesson TipyxOECDP ractice and ApplyPractice and ApplyLesson 4-1 Classifying the measures of the sides of TWZwith vertices at T(2, 6), W(4, 5), andZ( 3, 0). Classify the star-shaped composite quilting square is made up of four different Triangles . Use a ruler to classify the four Triangles by a protractor to classify each triangle as acute, equiangular, obtuse, or right. May 5, 2002, Venus, Saturn, and Mars were aligned in a triangular formation. Use a protractoror ruler to classify the triangle formed by sides and the Internet or other resource to find out how astronomers canpredict planetary restored and decorated Victorian houses in San Franciscoare called the Painted Ladies. Use a protractor to classify the trianglesindicated in the photo by sides and the indicated type of Triangles in the figureif A B B D D C C A and B C A D.

9 Xand the measure of each side of the GHJis isosceles, with H G J G , GH x 7, GJ 3x 5,and HJ x MPNis equilateral with MN 3x 6, MP x 4, and NP 2x QRSis equilateral. QRis two less than two times a number, RSis six morethan the number, and QSis ten less than three times the JKLis isosceles with K J L J . JLis five less than two times a number. JKisthree more than the number. KLis one less than the number. Find the measureof each Sohm/Stock Boston ForExercises13 1819, 21 2526 2930, 3132 37,40, 41 SeeExamples11, 2324 Extra Practice See page Practice See page Painted Ladies arelocated in Alamo area is one of 11designated historic districts in San : Chapter 4 Congruent top of the crystal bowl shown is circular. The diameter at the top of the bowl is MN. Pis the midpoint of M N , andO P M N . If MN 24 and OP 12, determine whether MPOand NPOare total distance from Nashville, Tennessee, to Cairo, Illinois, to Lexington,Kentucky, and back to Nashville, Tennessee,is 593 miles.

10 The distance from Cairo toLexington is 81 more miles than the distance from Lexington to Nashville. Thedistance from Cairo to Nashville is 40 milesless than the distance from Nashville toLexington. Classify the triangle formed by its the measures of the sides of ABCand classify each triangle by its sides. (5, 4), B(3, 1), C(7, 1) ( 4, 1), B(5, 6), C( 3, 7) ( 7, 9), B( 7, 1), C(4, 1) ( 3, 1), B(2, 1), C(2, 3) (0, 5), B(5 3 , 2), C(0, 1) ( 9, 0), B 5, 6 3 , C( 1, 0) a two-column a paragraph proofproof to prove that EQLis to prove that RPMis an if m NPM 33. GEOMETRYShow that Sis the midpoint of Show that ADCis T and Uis the midpoint of T V . THINKINGK L is a segment representing one side of isosceles right triangle KLM, with K(2, 6), and L(4, 2). KLMis a right angle, and K L L M .Describe how to find the coordinates of vertex Mand name these (0, 0) (a, 0) a2 ( ), byxOR( 10, 2) U(0, 8) S( 7, 8) T( 4, 14) V(4, 2) yxO 444128 8 MNPR33 EQLUIPROOFPROOFN ashvilleCairoLexingtonMNOPM aintain Your SkillsMaintain Your SkillsLesson 4-1 Classifying the question that was posed at the beginning of the are Triangles important in construction?