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Chapter 4

Chapter 4. 4-4 congruent triangles. SAT Problem of the day. Objectives. Use properties of congruent triangles. Prove triangles congruent by using the definition of congruence. Congruent triangles.

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Chapter 4

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  1. Chapter 4 4-4 congruent triangles

  2. SAT Problem of the day

  3. Objectives Use properties of congruent triangles. Prove triangles congruent by using the definition of congruence.

  4. Congruent triangles • Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides. • Two polygons are congruent polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape are congruent.

  5. Properties

  6. Helpful Hint Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices. properties • To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS. • In a congruence statement, the order of the vertices indicates the corresponding parts.

  7. Example#1 • Given: ∆PQR ∆STW • Identify all pairs of corresponding congruent parts. • Solution: • Angles: P  S, Q  T, R  W • Sides: PQ ST, QR  TW, PR  SW

  8. Example#2 • If polygon LMNP polygon EFGH, identify all pairs of corresponding congruent parts. • Solution: • Angles: L  E, M  F, N  G, P  H • Sides: LM EF, MN  FG, NP  GH, LP  EH

  9. Student guided practice • Do problems 3 to 8 in your book page 242

  10. Using corresponding parts • Given: ∆ABC ∆DBC. • Find the value of x.

  11. Example • Given: ∆ABC ∆DBC. • Find mDBC.

  12. Example • Given: ∆ABC  ∆DEF • Find the value of x.

  13. Student guided PRACTICE • Do problems 9 and 10 in your book page 242

  14. Proofs • Given:YWXandYWZ are right angles. • YW bisects XYZ. W is the midpoint of XZ. XY  YZ. • Prove: ∆XYW  ∆ZYW

  15. 5.W is mdpt. of XZ 6.XW ZW 7.YW YW 9.XY YZ solution 1.YWX and YWZ are rt. s. 1. Given 2.YWX  YWZ 2. Rt.   Thm. 3.YW bisects XYZ 3. Given 4.XYW  ZYW 4. Def. of bisector 5. Given 6. Def. of mdpt. 7. Reflex. Prop. of  8.X  Z 8. Third s Thm. 9. Given 10.∆XYW  ∆ZYW 10. Def. of  ∆

  16. Example of proof • Given:ADbisectsBE. • BEbisectsAD. • ABDE, A D • Prove:∆ABC  ∆DEC

  17. 4.ABDE 5.ADbisectsBE, 6.BC EC, AC DC BE bisects AD 1. A D 1. Given 2.BCA  DCE 2. Vertical s are . 3.ABC DEC 3. Third s Thm. 4. Given 5. Given 6. Def. of bisector 7.∆ABC  ∆DEC 7. Def. of  ∆s

  18. Application • The diagonal bars across a gate give it support. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent. • Given: PR and QT bisect each other. • PQS  RTS, QP  RT • Prove: ∆QPS ∆TRS

  19. 1.QP RT 3.PR and QT bisect each other. 4.QS TS, PS  RS solution 1. Given 2.PQS  RTS 2. Given 3. Given 4. Def. of bisector 5.QSP  TSR 5. Vert. s Thm. 6.QSP  TRS 6. Third s Thm. 7. ∆QPS  ∆TRS 7. Def. of  ∆s

  20. Student guided practice • Do problem 11 in your book page 242

  21. Homework • Do problems 13-20 in your book page 243

  22. Have a great day!!!!

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