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 … the triangles.

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

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

4.4 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 .

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

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

Example 1: 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. ΔQRS  ΔUTS 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 .

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

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

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

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

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

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

Example 4: 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 D R G A

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