ΔΔ by SAS and by SAS and SSSSSS

Review of Review of ΔΔss

Triangles that are the same shape and Triangles that are the same shape and size are congruent.size are congruent.

Each triangle has three sides and three Each triangle has three sides and three angles.angles.

If all six of the corresponding parts are If all six of the corresponding parts are congruent then the triangles are congruent then the triangles are congruent.congruent.

Congruence TransformationsCongruence Transformations

Congruency amongst triangles does Congruency amongst triangles does not change when you…not change when you…

slide, slide,

turn, turn,

or flip or flip

… … the triangles.the triangles.

So, to prove So, to prove ΔΔs s must we prove must we prove ALL sides & ALL ALL sides & ALL s are s are ? ?

Fortunately, NO!Fortunately, NO!

There are some shortcuts…There are some shortcuts…

ObjectivesObjectives

Use the SSS Postulate Use the SSS Postulate

Use the SAS PostulateUse the SAS Postulate

Postulate 4.1 (Postulate 4.1 (SSSSSS))Side-Side-Side Side-Side-Side Postulate Postulate

If 3 sides of one If 3 sides of one ΔΔ are are to 3 to 3 sides of another sides of another ΔΔ, then the , then the ΔΔs are s are ..

More on the SSS PostulateMore on the SSS Postulate

If seg AB If seg AB seg ED, seg AC seg ED, seg AC seg EF, & seg EF, & seg BC seg BC seg DF, then seg DF, then ΔΔABC ABC ΔΔEDF.EDF.

E

D

F

A

B

C

Given: QR Given: QR UT, RS UT, RS TS, QS = 10, US = 10 TS, QS = 10, US = 10Prove: Prove: ΔΔQRS QRS ΔΔUTSUTS

Q

R S T

U

10 10

Example 1:Example 1:

QU

R S T

10 10

Statements Reasons________Statements Reasons________

1. 1. QR QR UT, RS UT, RS TS, TS, 1. Given1. Given

QS=10, US=10QS=10, US=10

2. QS = US 2. Substitution2. QS = US 2. Substitution

3. QS 3. QS US US 3. Def of 3. Def of segs. segs.

4. 4. ΔΔQRS QRS ΔΔUTS 4. SSS PostulateUTS 4. SSS Postulate

Example 1:Example 1:

Postulate 4.2 (Postulate 4.2 (SASSAS))Side-Angle-Side Side-Angle-Side Postulate Postulate

If 2 sides and the included If 2 sides and the included of of one one ΔΔ are are to 2 sides and the to 2 sides and the included included of another of another ΔΔ, then , then the 2 the 2 ΔΔs are s are ..

If seg BC If seg BC seg YX, seg AC seg YX, seg AC seg ZX, & seg ZX, & C C X, then X, then ΔΔABC ABC ΔΔZXY.ZXY.B

A C X

Y

Z)(

More on the SAS PostulateMore on the SAS Postulate

Given: WX Given: WX XY, VX XY, VX ZX ZX Prove: Prove: ΔΔVXW VXW ΔΔZXYZXY

1 2

W

V

XZ

Y

Example 2:Example 2:

Statements Reasons_______Statements Reasons_______

1. WX 1. WX XY; VX XY; VX ZX ZX 1. Given 1. Given

2. 2. 1 1 2 2. Vert. 2 2. Vert. s are s are

3. 3. ΔΔ VXW VXW ΔΔ ZXY 3. SAS Postulate ZXY 3. SAS PostulateW

X

Z

V

Y

12

Example 2:Example 2:

Given: RS Given: RS RQ and ST RQ and ST QT QT Prove: Prove: ΔΔ QRT QRT ΔΔ SRT. SRT.

Q

R

S

T

Example 3:Example 3:

Statements Reasons________Statements Reasons________

1. RS 1. RS RQ; ST RQ; ST QT QT 1. Given 1. Given

2. RT 2. RT RT RT 2. Reflexive 2. Reflexive

3. 3. ΔΔ QRT QRT ΔΔ SRT SRT 3. SSS 3. SSS PostulatePostulate

Q

R

S

T

Example 3:Example 3:

Given: DR Given: DR AG and AR AG and AR GR GR

Prove: Prove: ΔΔ DRA DRA ΔΔ DRG. DRG.

D

AR

G

Example 4:Example 4:

Statements_______Statements_______1. DR 1. DR AG; AR AG; AR GR GR2. DR 2. DR DR DR3.3.DRG & DRG & DRA are DRA are

rt. rt. ss4.4.DRG DRG DRA DRA5. 5. ΔΔ DRG DRG ΔΔ DRA DRA

Reasons____________Reasons____________1. Given 1. Given 2. Reflexive Property2. Reflexive Property3. 3. lines form 4 rt. lines form 4 rt. s s

4. Right 4. Right s Theorem s Theorem

5. SAS Postulate5. SAS Postulate

D

A GR

Example 4:Example 4: