Congruent Triangles. The flag of the United Kingdom is shown below. Consider the four large triangles appearing on the top and the bottom of the flag.

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1 Congruent Triangles Why? Then You identified triangles with congruent sides. (Lesson 9-3) The flag of the United Kingdom is shown below. Consider the four large triangles appearing on the top and the bottom of the flag. Now Identify corresponding parts of congruent triangles. Identify congruent triangles. New Vocabulary congruent corresponding parts Math Online glencoe.com Extra Examples Personal Tutor Self-Check Quiz Homework Help a. Measure the sides of the four triangles. What is true about the lengths? b. Use a protractor to measure the angles of the four triangles. What is true about the measures? c. MAKE A CONJECTURE Suppose the triangles were cut out and laid on top of one another so that the parts with the same measures were matched up. What is true about the triangles? Corresponding Parts of Congruent Triangles Figures that have the same size and shape are congruent. In the figure below, triangle ABC is congruent to triangle DEF. Arcs are used to indicate which angles are congruent. Tick marks are used to indicate which sides are congruent. The parts of congruent triangles that match or correspond, are called corresponding parts. Words Model Key Concept Corresponding Parts of Congruent Triangles If two triangles are congruent, their corresponding sides are congruent and their corresponding angles are congruent. For Your Symbols ABC DEF Congruent Angles: A D, B E, C F Congruent Sides: AB DE, BC EF, CA FD 598 Chapter 11 Distance and Angle

2 StudyTip Congruence Statements You can write the following congruence statements for the congruent triangles at the right. RSQ UVT SQR VTU SRQ VUT RQS UTV QSR TVU QRS TUV In a congruence statement, the letters are written so that corresponding vertices appear in the same order. QRS TUV Vertex Q corresponds to vertex T. Vertex R corresponds to vertex U. Vertex S corresponds to vertex V. EXAMPLE 1 Name Corresponding Parts a. Name the corresponding parts in the congruent triangles below. Then complete the congruence statement JKL?. Use the matching arcs and tick marks to identify the corresponding parts. Corresponding angles: J M, K N, L P Corresponding sides: JK MN, KL NP, LJ PM The congruence statement is JKL MNP. StudyTip Draw a Figure When given only a congruence statement, it is helpful to draw the two triangles in the same position. Then it will be easier to visualize and confirm the corresponding angles and sides. b. If ABC HGF, name the corresponding parts. Then complete the congruence statement BAC?. Use the order of the vertices in the congruence statement ABC HGF to identify the corresponding parts. Corresponding angles: A H, B G, C F Corresponding sides: AB HG, BC GF, CA FH The congruence statement is BAC GHF. Check Your Progress Name the corresponding parts in each pair of congruent triangles. Then complete the congruence statement. 1A. 1B. TUV CDE VTU? DEF? Personal Tutor glencoe.com Lesson 11- Congruent Triangles 599

3 You can use corresponding parts to find the measures of angles and sides in a figure that is congruent to a figure with known measures. EXAMPLE SIGNS In the signs below, RST VWX. Find Missing Measures S W R T V X a. If m W = 60, what is the measure of S? W and S are corresponding angles, so they are congruent. m S = 60 Congruent angles have equal measures. b. The length of ST is 30 inches. What is the length of WX? ST and WX are corresponding sides, so they are congruent. WX = 30 inches Congruent sides have equal lengths. Check Your Progress. QUILTING In the quilt design 18 cm shown, ABC ADE. What is the measure of BCA? What is 3 cm the length of AD? 59 Personal Tutor glencoe.com Identify Congruent Triangles In congruent triangles, all corresponding angles are congruent and all corresponding sides are congruent. StudyTip Identify Congruent Parts To write a congruence statement, note which angles are congruent and which segments have the same length. Then write corresponding vertices in the same order. EXAMPLE 3 Identify Congruent Triangles Determine whether the triangles shown at the right are congruent. If so, name the 6 cm corresponding parts and write a 8 cm 9.5 cm 9.5 cm congruence statement. 8 cm The arcs indicate that G P, H Q, 6 cm and F R. The side measures indicate that GH PQ, HF QR, and FG RP. Since all pairs of corresponding angles and sides are congruent, the triangles are congruent. One congruence statement is GHF PQR. Check Your Progress 3. Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement. 11 in. 3 in. 6 in. 8 in. 4 in. 14 in. Personal Tutor glencoe.com 600 Chapter 11 Distance and Angle

4 Check Your Understanding Example 1 p. 599 Name the corresponding parts in each pair of congruent triangles. Then complete the congruence statement. 1.. RST GJK STR? YZX? Example 3. KITES In the umbrella kite shown at the right, JLK NLM. a. If m JKL = 67.5, what is m NML? b. If NL = 14 inches and MN = 1 inches, what is JK? Example 3 Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement ft 15 ft 18 ft 15 ft Practice and Problem Solving = Step-by-Step Solutions begin on page R11. Extra Practice begins on page 810. Example 1 p. 599 Name the corresponding parts in each pair of congruent triangles. Then complete the congruence statement GJF? STZ? 8. KLM WXY 9. NPQ DEF LKM? EFD? Lesson 11- Congruent Triangles 601

5 Example 10. STAINED GLASS In the stained glass window at the right, ABC ADC. " a. If BC = 6 centimeters, what is DC? b. If m ADC = 90, and m DCA = 45, what is m BAC? # % 11. ALGEBRA Triangle STU is congruent to triangle MNP. In the figures, TU = 15.4 millimeters, US = 13 millimeters, and ST = 18.1 millimeters. If MN = x + 1, what is the value of x? Example 3 $ Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement ; 1 4 / 4 8 m ' 1 6 cm 9 cm 3 ( 4 cm cm 6 cm 3 ) m 13 m 37 1 m - 1 m 63 0, 75 9m cm ART The structure shown at the right, Moondog, created by Tony Smith, is located at the Hirshhorn Museum and Sculpture Garden in Washington, D.C. If ABC DFG, name all corresponding sides. O M P N The average weight of an individual block of stone that makes up the Khafre pyramid in Egypt is.5 tons. Some of the outside stones weigh up to 7 tons. A B D C F G 17 SOCIAL STUDIES The Khafre pyramid shown at the left is composed of triangles. If MNO NPO, B m M = 60, and m OPN = 50, find m O. 18. GEOMETRY Hexagon ABCDEF has six congruent sides. Segments CA, CF, and CE are drawn on the hexagon, forming four triangles. Make a conjecture about which triangles are congruent. Test your conjecture by measuring the sides and angles of the triangles. Source: PBS 19. In the figure below, FGI LKM. Find the value of x. ', 5 cm x + 4 cm * 60 Chapter 11 Distance and Angle. 10 cm ( - # $ % " ' &

6 0. CONSTRUCTION The roof at the right contains two congruent triangles. a. Write a congruence statement involvingthe triangles. Then write a congruence statement involving 14 ft a pair of corresponding sides. b. What is the distance from the right side of the roof at the base to the center? Name the side. c. If the length of XW is 4 feet longer than WY, what is the length of WY? 1. BUTTERFLIES Butterfly wings are triangular in shape. Using the art of the butterfly as a model, draw two sets of congruent triangles, label the vertices, and write a congruence statement for each. Biological Scientists Biological scientists study the behavior, genetics, and diseases of living things. Since the job requires collecting and analyzing data, knowledge of arithmetic, algebra and statistics are important. A four-year degree in a biological field is the minimum level of education required for this career. C ALGEBRA Find the value of x for each pair of congruent triangles x - 9 3x + 5 H.O.T. Problems Use Higher-Order Thinking Skills 4. OPEN ENDED Find a real-world example of congruent triangles. Explain how you know the triangles are congruent. 5. FIND THE ERROR Jade and Fernando are writing a congruence statement for the congruent triangles at the right. Is either of them correct? Justify your reasoning. Jade YXZ STR Fernando ST ZY 6. CHALLENGE Determine whether each statement is true or false. If true, explain your reasoning. If false, give a counterexample. a. Triangles EFG and KLM are congruent. So, their perimeters are equal. b. Triangles WXY and HJK are congruent. So, their areas are equal. c. The perimeter of ABC is 4 millimeters and the perimeter of RST is 4 millimeters. So, triangles ABC and RST are congruent. 7. CHALLENGE Describe the scale factor of two congruent triangles. 8. WRITING IN MATH Describe structures that are built using congruent triangles. Explain why you think congruent triangles are used. Lesson 11- Congruent Triangles 603

7 Standardized Test Practice 9. In the road sign below, ABC DCB, AC is.5 meters long, BC is 1 meter long, and AB is.7 meters long. What is the length of BD? 5 North 31. Which of the following statement is not true if FGH LMN? A FH LN B GH MN C G M D F N A 1 meter C.7 meters B.5 meters D meters 30. Guy wires connected to a telephone pole create two congruent triangles PQR and SQR. Find the length of QS. F 1 ft G 4 ft H 48 ft J 65 ft 4 ft 3. EXTENDED RESPONSE Determine whether the triangles shown below are congruent. If so, name the corresponding parts and write a congruence statement Spiral Review 33. Angles R and S are complementary. Find m R if m S = 65.7 (Lesson 11-1) 34. In a triangle, a leg is 4 centimeters long. Find the length of the hypotenuse to the nearest tenth. (Lesson 10-6) 35. BIRDS A mockingbird uses about 5-4 Joules of energy to sing a song. Write the amount of energy the bird uses as an expression using a positive exponent. (Lesson 9-4) 36. CAR RENTAL The costs for renting a car from Able Car Rental and from Baker Car Rental are shown in the table. For what mileage does Baker have the better deal? Use the inequality x > x. Explain why this inequality works. (Lesson 5-5) Able Car Rental Baker Car Rental Rental Car Costs Cost per Day Cost per Mile $30 $0.05 $0 $0.10 Skills Review Graph each ordered pair on a coordinate plane. (Lesson -6) 37. F (4, 5) 38. G (, -) 39. C (-1.4) 40. A (-3, -) 604 Chapter 11 Distance and Angle

Geometry Regular Midterm Exam Review (Chapter 1, 2, 3, 4, 7, 9)

Geometry Regular Midterm Exam Review (Chapter 1, 2, 3, 4, 7, 9) Name: Date: Mod: Use the figure at the right for #1-4 1. What is another name for plane P? A. plane AE B. plane A C. plane BAD D. plane BAC