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