ARE THEY CONGRUENT?

HOMEWORK: WS - Congruent Triangles

Proving Δ’s are using: SSS, SAS, HL, ASA, & AAS

SSS If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.

SAS If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

ASA If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

AAS If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

HL If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.

Methods of Proving Triangles Congruent

DIRECT InformationDirect information comes in two forms:

congruent statements in the ‘GIVEN:’ part of a proofmarked in the picture

Example:

GIVEN KL NL, KM NM

PROVE KLM NLMOR

INDIRECT InformationIndirect Information appears in the ‘GIVEN:’ part of the proof but is NOT a congruency statement

Example:Given: JO SH; O is the midpoint of SH Prove: SOJ HOJ

J

HOS

INDIRECT Information

• Perpendicular lines right angles all rt s are ∠ ≅

• Midpoint of a segment 2 segments≅

• Parallel lines AIA

• Parallelogram 2 sets of parallel lines 2 pairs of AIA

• Segment is an angle bisector 2 angles≅

• Segments bisect each other 2 sets of segments≅

• Perpendicular bisector of a segment 2 segments &≅

2 right angles

BUILT-IN InformationBuilt- in information is part of the drawing.

Example:Vertical angles VA

Shared side Reflexive Property

Shared angle Reflexive Property

Any Parallelogram 2 pairs parallel lines 2 pairs of AIA

Steps to Write a Proof1. Take the 1st Given and MARK it on the picture2. WRITE this Given in the PROOF & its reason3. If the Given is NOT a ≅ statement,

write the ≅ stmt to match the marks Continue until there are no more GIVEN4. Do you have 3 ≅ statements?

If not, look for BUILT-IN parts5. Do you have ≅ triangles?

If not, write CNBDIf YES, Write the triangle congruency and reason (SSS, SAS, SAA, ASA, HL)

GIVEN KL NL, KM NM

PROVE KLM NLM

≅ ≅≅

ΔKLM ≅ ΔNLM SSS

given

given

reflexive prop

GIVEN

PROVE

BC DA, BC AD

BC DABC AD

∠BCA ∠DAC

AC AC

given

AIA

reflexive prop

ΔABC ≅ ΔCDA

given

SAS

ΔABC ≅ ΔCDA

≅

≅

≅

≅

Given: A D, C F, Prove: ∆ABC ∆DEF

A B

C

D

E

F A D given

C F given given

∆ABC ∆DEF AAS

Given: bisects IJK, ILJ JLK

Prove: ΔILJ ΔKLJ

bisects IJKGiven IJL IJH Definition of angle bisector ILJ JLK Given Reflexive Prop

ΔILJ ΔKLJ ASA

J

K

I

L

Given: , Prove: ΔTUV ΔWXV

GivenGiven

TVU WVX Vertical angles

ΔTUV ΔWXV SAS

VTW

U

X

Given: , H L Prove: ΔHIJ ΔLKJ

GivenH L GivenIJH KJL Vertical angles

ΔHIJ ΔLKJ ASAL

J

KI

H

Given: , PRT STR Prove: ΔPRT ΔSTR

GivenPRT STR Given

Reflexive Prop

ΔPRT ΔSTR SAS

S

P T

R

Given: is perpendicular bisector of

Prove:

is perpendicular bisector of given∠ABM & PBM are rt s∠ ∠ def lines

≅ def bisector ∠ABM PBM ≅ ∠ all rt s are ∠ ≅

≅ reflexive prop.

ΔABM ΔPBM ≅ SAS

Given: O is the midpoint of and

Prove: ΔMON ≅ ΔPOQ

O is the midpoint of and given ≅ def. midpoint ≅ def. midpoint

∠MON ≅ ∠ VA

ΔMON Δ≅ SAS

Given: ≅ ; || Prove: ΔABD ≅ ΔCDB

≅ given || given

∠ADB CBD≅ ∠ AIA ≅ reflexive prop.

ΔABD ΔCDB≅ SAS

Given: ; O is the midpoint of Prove: SOJ HOJ

J

S H0

Given: HJ GI, GJ JIProve: ΔGHJ ΔIHJ

JG

H

I

Given: 1 2; A E ; C is midpt of AEProve: ΔABC ΔEDC

21

C

D

EA

B

Given: , , and Prove: ΔPQR ΔPSR

Given PQR = 90° Def. lines Given PSR = 90° Def. linesPQR PSR all right s are Given Reflexive Prop

ΔPQR ΔPSR HL

S

RP

Q

CheckpointDecide if enough information is given to prove the

triangles are congruent. If so, state the congruence postulate you would use.

Given: LJ bisects IJK, ILJ JLK Prove: ΔILJ ΔKLJ

J

K

I

L

Given: 1 2, A E and Prove: ΔABC ΔEDC

1 2 GivenA E Given Given

ΔABC ΔEDC ASA

21

C

D

EA

B

Given: , Prove: ΔABD ΔCBD

Given Given Reflexive Prop

ΔABD ΔCBD SSS

B

C

A

D