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Understanding Congruent Triangles and Proof Strategies in Geometry

This guide explores the fundamentals of congruent triangles, including naming and labeling corresponding parts. Learn how to identify congruence transformations and understand that triangles are congruent when all six corresponding parts are congruent (CPCTC). We also delve into shortcuts for proving triangle congruence using the SSS (Side-Side-Side) and SAS (Side-Angle-Side) postulates, offering examples and assignments for practice. Strengthen your geometric skills through these essential concepts.

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Understanding Congruent Triangles and Proof Strategies in Geometry

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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 - 27 • Pre-AP Geometry: Pg. 195 #9 – 16, 22 – 27, 29 - 30

  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 #7, 8, 10, 14 – 16, 22 - 25 • Pre-AP Geometry: Pg. 204 #12, 14 – 18, 22 - 25

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