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Name———————————————————————— Lesson 4.5 Date ————————————— Study Guide For use with the lesson “Prove Triangles Congruent by SAS and HL” Use sides and angles to prove congruence. Lesson 4.5 goal Vocabulary In a right triangle, the sides adjacent to the right angle are called the legs. The side opposite the right angle is called the hypotenuse of the right triangle. Postulate 20 Side-Angle-Side (SAS) Congruence Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. Theorem 5 Hypotenuse-Leg Congruence Theorem: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. example 1 Use the SAS Congruence Postulate Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. Prove that nABC > nDEF. B F D C A E Solution } } } } > EF , and ∠ B > ∠ E. The marks on the diagram show that AB > DE , BC So, by the SAS Congruence Postulate, n ABC > n DEF. Exercises for Example 1 Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Postulate. 1. n PQT, n RQS P 2. n NKJ, nLKM R J 3. nWXY, n ZXY X Z L K Q T S W N Y M Geometry Chapter Resource Book 4-67 Name———————————————————————— Lesson Lesson 4.5 4.5 Date ————————————— Study Guide continued For use with the lesson “Prove Triangles Congruent by SAS and HL” example 2 Use the Hypotenuse-Leg Theorem Write a proof. } } } } } } Given: AB > DC , BA ⊥ AC, CD ⊥ DB A C B D Prove: n ABC > n DCB Solution B Redraw the triangles so they are side by side with the corresponding parts in the same position. Mark the given information in the diagram. C A C B D Exercises for Example 2 Write a proof. } } } } 4. Given: AB > DB , BC ⊥ AD 5. Given: m∠ JKL 5 m∠ MLK 5 908 } } Prove: n ABC > n DBCJL > MK Prove:n JKL > n MLK B A 4-68 Geometry Chapter Resource Book C K J L M D Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. StatementsReasons } } } } 1. Given 1. BA ⊥ AC, CD ⊥ DB 2. Definition of ⊥ lines 2. ∠ A and ∠ D are right angles. 3. n ABC and n DCB are right triangles. 3. Definition of a right triangle } } 4. Reflexive Property of Congruence H 4. CB > BC } } 5. Given L 5. AB > DC 6. HL Congruence Theorem 6. n ABC > n DCB Lesson 4.5 Prove Triangles Congruent by SAS and HL, continued answers } 5. Statements } 12. not enough 13. RM ù FB 14. ∠ J ù ∠ D } } } } } } } ; Vertical Angles Theorem; Given; CB ù BD } 15. JM ù DB or JR ù DF 16. Given; AB ù BE ; SAS Congruence Postulate 17. Given; Alternate Interior Angles Theorem; Given; Reflexive Property of Congruence; SAS Congruence Postulate Practice Level C 1. not enough 2. enough; HL 3. not enough } } 4. enough; SAS 5. ∠ BCA; ∠ EDF 6. BC ; ED } } 7. AC; FD 8. They are congruent by SAS. 9. Definition of perpendicular lines; n PRS and n QSR are right triangles; Reflexive Property of Congruence; HL Congruence Theorem 10. ∠ OML and ∠ OMN are right angles; ∠ OML ù ∠ OMN; Given; Reflexive Property of Congruence; SAS Congruence Postulate Reasons 1. ∠ JKL and ∠ MLK 1. Given are right angles. 2. n JKL and n MLK 2. Def. of a right are right triangles. triangle } } 3. Given 3. JL > MK } } 4. Reflexive Property 4. KL > LK of Congruence 5. n JKL > n MLK 5. HL Congruence Theorem Problem Solving Workshop: Mixed Problem Solving } } 1. a. The legs are MN and NL . The hypotenuse is } ML . b. 45° to 60° y 2. a. C 4 B A x 2 D Study Guide E 1. Yes; You are given that two sides and the 4. Reasons Statements } } H 1. AB > DB 1. Given } } 2. Given 2. BC ⊥ AD 3. ∠ ACB and ∠ DCB 3. Def. of ⊥ lines are right angles. 4. n ABC and n DCB 4. Def. of a right are right triangles. triangle } } 5. Reflexive Property L 5. BC > BC of Congruence 6. n ABC > n DBC 6. HL Congruence Theorem A52 Geometry Chapter Resource Book CS10_CC_G_MECR710761_C4AK.indd 52 F } b. n ABC is a right triangle because AB has a } 1 slope of } 2 and BC has a slope of 22. nDEF is 1 } a right triangle because DE has a slope of } 2 and } } } EF has a slope of 22. c. The legs AB and DE } both have a length of 3Ï5 and the hypotenuses } } } AC and DF both have a length of 3Ï10 . So, by the Hypotenuse-Leg Congruence Theorem, n ABC > nDEF. } } } 3. a. 2Ï3x 1 5Ï3x 1 3Ï3x 5 180 b. 36°, 90°, 54° c. right triangle 4. Yes; because the flag is a rectangle, you know that ∠ PJL, ∠ JLM, ∠ LMP, and } } } } . JL i PM and ∠ MPJ are right angles and JP > LM } } JP i LM by the Lines Perpendicular to a Transversal Theorem. Using the Alternate Interior Angles Theorem, you know that ∠ JKP > ∠ MNL. So, nJKP > nMNL by the AAS Congruence Theorem. 5. 458 6. Answers will vary. 7. a. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. included angle of one triangle are congruent to two sides and the included angle of another triangle. 2. Yes; ∠ JKN and ∠ MKL are congruent because they are vertical angles. So you have two sides and the included angle of one triangle that are congruent to two sides and the included angle of another triangle. 3. No; You have two sides in nWXY that are congruent to two sides in n ZXY, but the angle in n ZXY is not the included angle. Reasons Statements } } 1. Given 1. BD > AC } } , 2. Given 2. AB ⊥ BC } } DC ⊥ BC 4/28/11 6:14:09 PM