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4.3  Δ s

4.3  Δ s. Objectives. Name and label corresponding parts of congruent triangles Identify congruence transformations.  Δ s. Triangles that are the same shape and size are congruent. Each triangle has three sides and three angles.

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4.3  Δ s

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  1. 4.3 Δs

  2. Objectives • Name and label corresponding parts of congruent triangles • Identify congruence transformations

  3. Δs • Triangles that are the same shape and size are congruent. • Each triangle has three sides and three angles. • If all six of the corresponding parts are congruent then the triangles are congruent.

  4. CPCTC • CPCTC – Corresponding Parts of Congruent Triangles are Congruent • Be sure to label Δs with proper mappings (i.e. if D  L, V  P, W  M, DV  LP, VW  PM, and WD  ML then we must write ΔDVW ΔLPM)

  5. Congruence Transformations • Congruency amongst triangles does not change when you… • slide, • turn, • or flip • … the triangles.

  6. Assignment • Geometry: Pg. 195 #9 – 16, 22 - 25 • Pre-AP Geometry: Pg. 195 #9 – 16, 22 - 27

  7. So, to prove Δs  must we prove ALL sides & ALL s are  ? Fortunately, NO! • There are some shortcuts…

  8. 4.4 Proving Δs are  : SSS and SAS

  9. Objectives • Use the SSS Postulate • Use the SAS Postulate

  10. Postulate 4.1 (SSS)Side-Side-Side  Postulate • If 3 sides of one Δ are  to 3 sides of another Δ, then the Δs are .

  11. E A F C D B More on the SSS Postulate If seg AB  seg ED, seg AC  seg EF, & seg BC  seg DF, then ΔABC ΔEDF.

  12. Given: QR  UT, RS  TS, QS = 10, US = 10 Prove: ΔQRS ΔUTS Example 1: U U Q Q 10 10 10 10 R R S S T T

  13. Example 1: Statements Reasons________ 1. QR  UT, RS  TS,1. Given QS=10, US=10 2. QS = US 2. Substitution 3. QS  US 3. Def of  segs. 4. ΔQRS ΔUTS 4. SSS Postulate

  14. Postulate 4.2 (SAS)Side-Angle-Side  Postulate • If 2 sides and the included  of one Δ are  to 2 sides and the included  of another Δ, then the 2 Δs are .

  15. More on the SAS Postulate • If seg BC  seg YX, seg AC  seg ZX, & C X, then ΔABC  ΔZXY. B Y ) ( A C X Z

  16. Given: WX  XY, VX  ZX Prove: ΔVXW ΔZXY Example 2: W Z X 1 2 V Y

  17. Example 2: Statements Reasons_______ 1. WX  XY; VX  ZX 1. Given 2. 1 2 2. Vert. s are  3. Δ VXW Δ ZXY 3. SAS Postulate W Z X 1 2 V Y

  18. Given: RS  RQ and ST  QT Prove: Δ QRT  Δ SRT. Example 3: S Q R T

  19. Example 3: Statements Reasons________ 1. RS  RQ; ST  QT 1. Given 2. RT  RT 2. Reflexive 3. Δ QRT Δ SRT 3. SSS Postulate Q S R T

  20. Given: DR  AG and AR  GR Prove: Δ DRA  Δ DRG. Example 4: D R A G

  21. Statements_______ 1. DR  AG; AR  GR 2. DR  DR 3.DRG & DRA are rt. s 4.DRG   DRA 5. Δ DRG  Δ DRA Reasons____________ 1. Given 2. Reflexive Property 3.  lines form 4 rt. s 4. Right s Theorem 5. SAS Postulate Example 4: D R G A

  22. Assignment • Geometry: Pg. 204 #14, 16, 18, 22 - 25 • Pre-AP Geometry: Pg. 204 #14 – 19, 22 - 25

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