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19 41. What is the value of x? Find the value of s A and B are the remote interior angles of BCD in ABC. Which of these equations must be true? m A = m B m A = 90 - m B m BCD = m BCA - m A m B = m BCD - m A 44. Extended Response The measures of the angles in a triangle are in the ratio 2 : 3 : 4. Describe how to use algebra to find the measures of these angles. Then find the measure of each angle and classify the triangle. CHALLENGE AND EXTEND 45. An exterior angle of a triangle measures 117. Its remote interior angles measure (2 y 2 + 7) and (61 - y 2 ). Find the value of y. 46. Two parallel lines are intersected by a transversal. What type of triangle is formed by the intersection of the angle bisectors of two same-side interior angles? Explain. (Hint: Use geometry software or construct a diagram of the angle bisectors of two same-side interior angles.) 47. Critical Thinking Explain why an exterior angle of a triangle cannot be congruent to a remote interior angle. 48. Probability The measure of each angle in a triangle is a multiple of 30. What is the probability that the triangle has at least two congruent angles? 49. In ABC, m B is 5 less than What is m A in degrees? times m A. m C is 5 less than 2 1 times m A. 2 SPIRAL REVIEW Make a table to show the value of each function when x is -2, 0, 1, and 4. (Previous course) 50. f (x) = 3x f (x) = x f (x) = (x - 3) Find the length of NQ. Name the theorem or postulate that justifies your answer. (Lesson 1-2) Classify each triangle by its side lengths. (Lesson 4-1) 54. ACD 55. BCD 56. ABD 57. What if? If CA = 8, What is the effect on the classification of ACD? 230 Chapter 4 Triangle Congruence
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21 EXAMPLE 2 Using Corresponding Parts of Congruent Triangles When you write a statement such as ABC DEF, you are also stating which parts are congruent. Given: EFH GFH A Find the value of x. FHE and FHG are rt.. FHE FHG m FHE = m FHG (6x - 12) = 90 6x = 102 x = 17 B Find m GFH. m EFH + m FHE + m E = 180 m EFH = 180 Def. of lines Rt. Thm. Def. of m EFH = 180 m EFH = 68.4 GFH EFH m GFH = m EFH m GFH = 68.4 Substitute values for m FHE and m FHG. Add 12 to both sides. Divide both sides by 6. Sum Thm. Substitute values for m FHE and m E. Simplify. Subtract from both sides. Corr. of are. Def. of Trans. Prop. of = Given: ABC DEF 2a. Find the value of x. 2b. Find m F. EXAMPLE 3 Proving Triangles Congruent Given: P and M are right angles. R is the midpoint of PM. PQ MN, QR NR Prove: PQR MNR Proof: Statements Reasons 1. P and M are rt. 2. P M 3. PRQ MRN 4. Q N 5. R is the mdpt. of PM. 6. PR MR 7. PQ MN ; QR NR 8. PQR MNR 1. Given 2. Rt. Thm. 3. Vert. Thm. 4. Third Thm. 5. Given 6. Def. of mdpt. 7. Given 8. Def. of 3. Given: AD bisects BE. BE bisects AD. AB DE, A D Prove: ABC DEC 232 Chapter 4 Triangle Congruence
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28 Quiz for Lessons 4-1 Through 4-3 SECTION 4A 4-1 Classifying Triangles Classify each triangle by its angle measures. 1. ACD 2. ABD 3. ADE Classify each triangle by its side lengths. 4. PQR 5. PRS 6. PQS 4-2 Angle Relationships in Triangles Find each angle measure. 7. m M 8. m ABC 9. A carpenter built a triangular support structure for a roof. Two of the angles of the structure measure 37 and 55. Find the measure of RTP, the angle formed by the roof of the house and the roof of the patio. 4-3 Congruent Triangles Given: JKL DEF. Identify the congruent corresponding parts. 10. KL? 11. DF? 12. K? 13. F? Given: PQR STU. Find each value. 14. PQ 15. y 16. Given: AB CD, AB CD, AC BD, AC CD, DB AB Prove: ACD DBA Proof: Statements Reasons 1. AB CD 2. BAD CDA 3. AC CD, DB AB 4. ACD and DBA are rt. 5. e.? 6. f.? 7. AB CD, AC BD 8. h.? 9. ACD DBA 1. a.? 2. b.? 3. c.? 4. d.? 5. Rt. Thm. 6. Third Thm. 7. g.? 8. Reflex Prop. of 9. i.? Ready to Go On? 239
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31 4-4 Triangle Congruence: SSS and SAS Objectives Apply SSS and SAS to construct triangles and to solve problems. Prove triangles congruent by using SSS and SAS. Vocabulary triangle rigidity included angle Who uses this? Engineers used the property of triangle rigidity to design the internal support for the Statue of Liberty and to build bridges, towers, and other structures. (See Example 2.) In Lesson 4-3, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent. The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape. For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate. Postulate Side-Side-Side (SSS) Congruence POSTULATE HYPOTHESIS CONCLUSION If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. ABC FDE EXAMPLE 1 Using SSS to Prove Triangle Congruence Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts. Use SSS to explain why PQR PSR. It is given that PQ PS and that QR SR. By the Reflexive Property of Congruence, PR PR. Therefore PQR PSR by SSS. 1. Use SSS to explain why ABC CDA. An included angle is an angle formed by two adjacent sides of a polygon. B is the included angle between sides AB and BC. 242 Chapter 4 Triangle Congruence
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33 EXAMPLE 3 Verifying Triangle Congruence Show that the triangles are congruent for the given value of the variable. A UVW YXW, x = 3 ZY = x - 1 = 3-1 = 2 XZ = x = 3 XY = 3x - 5 = 3 (3) - 5 = 4 UV YX. VW XZ, and UW YZ. So UVW YXZ by SSS. B DEF JGH, y = 7 JG = 2y + 1 = 2 (7) + 1 = 15 GH = y 2-4y + 3 = (7) 2-4 (7) + 3 = 24 m G = 12y + 42 = 12 (7) + 42 = 126 DE JG. EF GH, and E G. So DEF JGH by SAS. 3. Show that ADB CDB when t = 4. EXAMPLE 4 Proving Triangles Congruent Given: l m, EG HF Prove: EGF HFG Proof: Statements 1. EG HF 2. l m 3. EGF HFG 4. FG GF 5. EGF HFG 1. Given 2. Given Reasons 3. Alt. Int. Thm. 4. Reflex Prop. of 5. SAS Steps 1, 3, 4 4. Given: QP bisects RQS. QR QS Prove: RQP SQP 244 Chapter 4 Triangle Congruence
34 THINK AND DISCUSS 1. Describe three ways you could prove that ABC DEF. 2. Explain why the SSS and SAS Postulates are shortcuts for proving triangles congruent. 3. GET ORGANIZED Copy and complete the graphic organizer. Use it to compare the SSS and SAS postulates. 4-4 Exercises KEYWORD: MG7 4-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In RST which angle is the included angle of sides ST and TR? SEE EXAMPLE 1 p. 242 Use SSS to explain why the triangles in each pair are congruent. 2. ABD CDB 3. MNP MQP SEE EXAMPLE 2 p Sailing Signal flags are used to communicate messages when radio silence is required. The Zulu signal flag means, I require a tug. GJ = GH = GL = GK = 20 in. Use SAS to explain why JGK LGH. J G H K L SEE EXAMPLE 3 p. 244 Show that the triangles are congruent for the given value of the variable. 5. GHJ IHJ, x = 4 6. RST TUR, x = Triangle Congruence: SSS and SAS 245
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36 Which postulate, if any, can be used to prove the triangles congruent? Explain what additional information, if any, you would need to prove ABC DEC by each postulate. a. SSS b. SAS Multi-Step Graph each triangle. Then use the Distance Formula and the SSS Postulate to determine whether the triangles are congruent. 19. QRS and TUV 20. ABC and DEF Q (-2, 0), R (1, -2), S (-3, -2) A (2, 3), B (3, -1), C (7, 2) T (5, 1), U (3, -2), V (3, 2) D (-3, 1), E (1, 2), F (-3, 5) 21. Given: ZVY WYV, ZVW WYZ, VW YZ Prove: ZVY WYV Proof: Statements Reasons 1. ZVY WYV, ZVW WYZ 2. m ZVY = m WYV, m ZVW = m WYZ 3. m ZVY + m ZVW = m WYV + m WYZ 4. c.? 5. WVY ZYV 6. VW YZ 7. f.? 8. ZVY WYV 1. a.? 2. b.? 3. Add. Prop. of = 4. Add. Post. 5. d.? 6. e.? 7. Reflex. Prop. of 8. g.? 22. This problem will prepare you for the Multi-Step Test Prep on page 280. The diagram shows two triangular trusses that were built for the roof of a doghouse. a. You can use a protractor to check that A and D are right angles. Explain how you could make just two additional measurements on each truss to ensure that the trusses are congruent. b. You verify that the trusses are congruent and find that AB = AC = 2.5 ft. Find the length of EF to the nearest tenth. Explain. 4-4 Triangle Congruence: SSS and SAS 247
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40 Activity 2 1 Construct YZ with a length of 6.5 cm. 2 Using YZ as a side, construct XYZ measuring Draw a circle at Z with a radius of 5 cm. Construct ZW, a radius of circle Z. 4 Move W around circle Z. Observe the possible shapes of YZW. Try This 7. In Step 4 of the activity, how many different triangles were possible? Does Side-Side-Angle make only one triangle? 8. Repeat Activity 2 using an angle measure of 90 in Step 2 and a circle with a radius of 7 cm in Step 3. How many different triangles are possible in Step 4? 9. Repeat the activity again using a measure of 90 in Step 2 and a circle with a radius of 8.25 cm in Step 3. Classify the resulting triangle by its angle measures. 10. Based on your results, complete the following conjecture. In a Side-Side-Angle combination, if the corresponding nonincluded angles are?, then only one triangle is possible. 4-5 Technology Lab 251
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42 2 Make a Plan Draw the course using vertical lines to show north-south directions. Then use these parallel lines and the alternate interior angles to help find angle measures of ABC. 3 Solve m CAB = = 35 m CBA = ( ) = 99 You know the measures of CAB and CBA and the length of the included side AB. Therefore by ASA, a unique triangle ABC is determined. 4 Look Back One and only one triangle can be made using the information in the table, so the table does give enough information to determine the location of all the checkpoints. 1. What if...? If 7.6 km is the distance from B to C, is there enough information to determine the location of all the checkpoints? Explain. EXAMPLE 2 Applying ASA Congruence Determine if you can use ASA to prove UVX WVX. Explain. UXV WXV as given. Since WVX is a right angle that forms a linear pair with UVX, WVX UVX. Also VX VX by the Reflexive Property. Therefore UVX WVX by ASA. 2. Determine if you can use ASA to prove NKL LMN. Explain. Construction Congruent Triangles Using ASA Use a straightedge to draw a segment and two angles, or copy the given segment and angles. Construct CD congruent to the given segment. Construct C congruent to one of the angles. Construct D congruent to the other angle. CDE Label the intersection of the rays as E. 4-5 Triangle Congruence: ASA, AAS, and HL 253
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44 Theorem Hypotenuse-Leg (HL) Congruence THEOREM HYPOTHESIS CONCLUSION If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. ABC DEF You will prove the Hypotenuse-Leg Theorem in Lesson 4-8. EXAMPLE 4 Applying HL Congruence Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. A VWX and YXW According to the diagram, VWX and YXW are right triangles that share hypotenuse WX. WX XW by the Reflexive Property. It is given that WV XY, therefore VWX YXW by HL. B VWZ and YXZ This conclusion cannot be proved by HL. According to the diagram, VWZ and YXZ are right triangles, and WV XY. You do not know that hypotenuse WZ is congruent to hypotenuse XZ. 4. Determine if you can use the HL Congruence Theorem to prove ABC DCB. If not, tell what else you need to know. THINK AND DISCUSS 1. Could you use AAS to prove that these two triangles are congruent? Explain. 2. The arrangement of the letters in ASA matches the arrangement of what parts of congruent triangles? Include a sketch to support your answer. 3. GET ORGANIZED Copy and complete the graphic organizer. In each column, write a description of the method and then sketch two triangles, marking the appropriate congruent parts. 4-5 Triangle Congruence: ASA, AAS, and HL 255
45 4-5 Exercises KEYWORD: MG7 4-5 GUIDED PRACTICE KEYWORD: MG7 Parent 1. Vocabulary A triangle contains ABC and ACB with BC closed in between them. How would this help you remember the definition of included side? SEE EXAMPLE 1 p. 252 Surveying Use the table for Exercises 2 and 3. A landscape designer surveyed the boundaries of a triangular park. She made the following table for the dimensions of the land. A A to B B to C C to A 115 ft Bearing E S 25 E N 62 W Distance 115 ft?? 2. Draw the plot of land described by the table. Label the measures of the angles in the triangle. C B 3. Does the table have enough information to determine the locations of points A, B, and C? Explain. SEE EXAMPLE 2 p. 253 Determine if you can use ASA to prove the triangles congruent. Explain. 4. VRS and VTS, given that 5. DEH and FGH VS bisects RST and RVT SEE EXAMPLE 3 p Use AAS to prove the triangles congruent. Given: R and P are right angles. QR SP Prove: QPS SRQ Proof: SEE EXAMPLE 4 p. 255 Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 7. ABC and CDA 8. XYV and ZYV 256 Chapter 4 Triangle Congruence
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49 4-6 Triangle Congruence: CPCTC Objective Use CPCTC to prove parts of triangles are congruent. Vocabulary CPCTC Why learn this? You can use congruent triangles to estimate distances. CPCTC is an abbreviation for the phrase Corresponding Parts of Congruent Triangles are Congruent. It can be used as a justification in a proof after you have proven two triangles congruent. EXAMPLE 1 Engineering Application SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. To design a bridge across a canyon, you need to find the distance from A to B. Locate points C, D, and E as shown in the figure. If DE = 600 ft, what is AB? D B, because they are both right angles. DC CB,because DC = CB = 500 ft. DCE BCA, because vertical angles are congruent. Therefore DCE BCA by ASA or LA. By CPCTC, ED AB, so AB = ED = 600 ft. 1. A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? EXAMPLE 2 Proving Corresponding Parts Congruent Given: AB DC, ABC DCB Prove: A D Proof: 2. Given: PR bisects QPS and QRS. Prove: PQ PS 260 Chapter 4 Triangle Congruence
50 EXAMPLE 3 Using CPCTC in a Proof Given: EG DF, EG DF Prove: ED GF Proof: Work backward when planning a proof. To show that ED GF, look for a pair of angles that are congruent. Then look for triangles that contain these angles. Statements 1. EG DF 2. EG DF 3. EGD FDG 4. GD DG 5. EGD FDG 6. EDG FGD 7. ED GF Reasons 1. Given 2. Given 3. Alt. Int. Thm. 4. Reflex. Prop. of 5. SAS Steps 1, 3, and 4 6. CPCTC 7. Converse of Alt. Int. Thm. 3. Given: J is the midpoint of KM and NL. Prove: KL MN You can also use CPCTC when triangles are on a coordinate plane. You use the Distance Formula to find the lengths of the sides of each triangle. Then, after showing that the triangles are congruent, you can make conclusions about their corresponding parts. EXAMPLE 4 Using CPCTC in the Coordinate Plane Given: A (2, 3), B (5, -1), C (1, 0), D (-4, -1), E (0, 2), F (-1, -2) Prove: ABC DEF Step 1 Plot the points on a coordinate plane. Step 2 Use the Distance Formula to find the lengths of the sides of each triangle. D = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 AB = (5-2) 2 + (-1-3) 2 DE = (0 - (-4)) 2 + (2 - (-1)) 2 = = 25 = 5 = = 25 = 5 BC = (1-5) 2 + (0 - (-1)) 2 EF = (-1-0) 2 + (-2-2) 2 = = 17 = = 17 AC = (1-2) 2 + (0-3) 2 DF = (-1 - (-4)) 2 + (-2 - (-1)) 2 = = 10 = = 10 So AB DE, BC EF, and AC DF. Therefore ABC DEF by SSS, and ABC DEF by CPCTC. 4. Given: J (-1, -2), K (2, -1), L (-2, 0), R (2, 3), S (5, 2), T (1, 1) Prove: JKL RST 4-6 Triangle Congruence: CPCTC 261
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