4.3  Δs

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

 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.

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)

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

So, we can only prove Δs  if ALL sides AND ALL s are . There are some shortcuts…

4.3 Proving Δs are  : SSS and SAS

Objectives Use the SSS Postulate Use the SAS Postulate

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 .

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

Given: QR  UT, RS  TS, QS=10, US=10 Prove: ΔQRS  ΔUTS

Proof Statements Reasons 2. QS=US 2. Substitution 1. QR  UT, RS  TS, 1. Given QS=10, US=10 2. QS=US 2. Substitution 3. QS  US 3. Def of  segs. 4. Δ QRS  Δ UTS 4. SSS Postulate

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 .

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

Given: WX  XY, VX  ZX Prove: Δ VXW  Δ ZXY 1 2 Y V

Proof Statements Reasons 1. WX  XY; VX  ZX 1. Given 2. 1  2 2. Vert s Thm. 3. Δ VXW  Δ ZXY 3. SAS Postulate

Given: RS  RQ and ST  QT Prove: Δ QRT  Δ SRT.

Proof Statements Reasons 1. RS  RQ; ST  QT 1. Given 2. RT  RT 2. Reflexive 3. Δ QRT  Δ SRT 3. SSS Postulate

Given: DR  AG and AR  GR Prove: Δ DRA  Δ DRG.

Proof Statements Reasons 1. DR  AG; AR  GR 1. Given 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 Postuate

Assignment Pre-AP: Pg. 195 #9 – 16, 22 – 25 Pg. 204 #14 – 19, 22 – 25