CONGRUENT TRIANGLES

When we talk about congruent triangles,When we talk about congruent triangles,we mean everything about them is we mean everything about them is congruentcongruent (or exactly the same) (or exactly the same) All 3 pairs of corresponding angles are equal….All 3 pairs of corresponding angles are equal….and all 3 pairs of corresponding sides are equaland all 3 pairs of corresponding sides are equal

For us to prove that 2 people are For us to prove that 2 people are identical twins, we don’t need to show identical twins, we don’t need to show that all “2000 +” body parts are equal. that all “2000 +” body parts are equal. We can take a short cut and show 3 or 4 We can take a short cut and show 3 or 4 things are equal such as their face, age things are equal such as their face, age and height. If these are the same I think and height. If these are the same I think we can agree they are twins. The same we can agree they are twins. The same is true for triangles. We don’t need to is true for triangles. We don’t need to prove all 6 corresponding parts are prove all 6 corresponding parts are congruent. We have 4 short cuts or congruent. We have 4 short cuts or rules to help us prove that rules to help us prove that two triangles are congruent.two triangles are congruent.

SSSSSSIf 3 corresponding If 3 corresponding sidessides are exactly are exactly the same length, then the triangles the same length, then the triangles

have to be congruent.have to be congruent.Side-Side-Side (SSS)Side-Side-Side (SSS)

SASSASIf 2 pairs of sides and the If 2 pairs of sides and the

included angles are exactly the same included angles are exactly the same then the triangles are congruent.then the triangles are congruent.

Side-Angle-Side (SAS)Side-Angle-Side (SAS)

IncludedIncludedangleangle

Non-includedNon-includedanglesangles

AASAASIf 2 angles and a side If 2 angles and a side

are exactly the same then the triangles are exactly the same then the triangles have to be congruent.have to be congruent.

Angle-Angle-Side (AAS)Angle-Angle-Side (AAS)

RHSRHSIf 2 triangles have a right-angle, a If 2 triangles have a right-angle, a hypotenuse, and a side, which are hypotenuse, and a side, which are

exactly the same then the triangles are exactly the same then the triangles are congruent.congruent.

Right angle-Hypotenuse-Side (RHS)Right angle-Hypotenuse-Side (RHS)

This is called a common side.This is called a common side.It is a It is a sideside for both triangles. for both triangles.

Sometimes it is shown by a ‘squiggly’ Sometimes it is shown by a ‘squiggly’ line placed on the common lineline placed on the common line

Why is AAA (Angle, Angle, Angle)Why is AAA (Angle, Angle, Angle)NOTNOT a proof for a proof for

CONGRUENT TRIANGLES???CONGRUENT TRIANGLES???Discuss!!Discuss!!

Which method can be used toWhich method can be used toprove the triangles are congruentprove the triangles are congruent

a)a) b)b)

c)c)

d)d)

3Common 3Common

Sides Sides (SSS)(SSS)

Parallel linesParallel linesalternate anglesalternate anglesCommon sideCommon side

(AAS)(AAS)

Included angle,Included angle, 2 sides equal 2 sides equal

(SAS)(SAS)

RHSRHSRight angleRight angleHypotenuseHypotenuse& a side & a side equalequal

RHSRHS is used is usedonly with right-angled triangles, only with right-angled triangles,

BUT, not BUT, not allall right triangles. right triangles.

RHSRHS ASAASA

When Starting A Proof, Make TheWhen Starting A Proof, Make TheMarks On The Diagram IndicatingMarks On The Diagram Indicating

The Congruent Parts. The Congruent Parts.

Use The GivenUse The GivenInformation to do thisInformation to do this

AA

BB

CC

DDEE

Given: AB = BDGiven: AB = BD EB = BCEB = BCAim: Prove Aim: Prove ∆ABE ∆ABE ∆DBC∆DBC

AA

BB

CC

DDEEProof:Proof:AB = BD AB = BD (Given) SIDE ((Given) SIDE (SS))ABC = ABC = CBD (Vertically Opposite) ANGLE (CBD (Vertically Opposite) ANGLE (AA) ) EB = BC EB = BC (Given) SIDE ((Given) SIDE (SS))

∆ ∆ABE ABE ∆DBC (∆DBC (SASSAS))

Given: AB = BDGiven: AB = BD EB = BCEB = BCMark these on the diagramMark these on the diagram

Aim: Aim: Prove Prove ∆ABE ∆ABE ∆DBC∆DBC

AA

Given: Given: CX bisects CX bisects ACB ACB CAB = CAB = CBACBA

Prove: Prove: ∆ACX∆ACX ∆BCX ∆BCX

BB

CC

XX

BB

CC

XXAA

Proof:Proof: ACX = ACX = BCX BCX CX bisects CX bisects ACB (ANGLE) ( ACB (ANGLE) (AA))CAB = CAB = CBA (Given) ANGLE (CBA (Given) ANGLE (AA) ) CX is commonCX is common SIDE ( SIDE (SS))

∆ ∆ACXACX ∆BCX ∆BCX ((AASAAS))

Given: Given: MN ║ QR MN ║ QR MN = QR MN = QR

Prove: Prove: ∆MNP∆MNP ∆QRP ∆QRP

MM NN

PP

QQRR

Proof:Proof: MNP = MNP = QRP QRP Alternate angles MNAlternate angles MN║║QR (ANGLE) (QR (ANGLE) (AA))MPN = MPN = QPR Vertically Opposite angles (ANGLE) (QPR Vertically Opposite angles (ANGLE) (AA) ) MN = QRMN = QR Given (SIDE) (Given (SIDE) (SS))

∆ ∆MNPMNP ∆QRP ∆QRP((AASAAS))

MM NN

PP

QQRR

Given: Given: XZ = ACXZ = AC XY = AB XY = AB

XYZXYZ= = ABCABC = 90 = 90 Prove: Prove: ∆ABC∆ABC ∆XYZ ∆XYZ

XX YY

ZZAABB

CC

XX YY

ZZAABB

CC

Proof:Proof: ABC = ABC = XYZ = 90 XYZ = 90 Given (RIGHT-ANGLE) (Given (RIGHT-ANGLE) (RR))XZ = AC Given (Hypotenuse) (XZ = AC Given (Hypotenuse) (HH) ) XY = ABXY = AB Given (SIDE) (Given (SIDE) (SS))

∆ ∆XYZXYZ ∆ABC ∆ABC ((RHSRHS))

PP QQ

RRSS

Given: Given: PQ = RSPQ = RS QR = SP QR = SP

Prove: Prove: ∆PQR∆PQR ∆RSP ∆RSP

PP QQ

RRSSProof:Proof:PQ = RSPQ = RS Given (SIDE) (Given (SIDE) (SS))QR = SP Given (SIDE) (QR = SP Given (SIDE) (SS) ) XY = ABXY = AB Common to both (SIDE) (Common to both (SIDE) (SS))

∆ ∆PQRPQR ∆RSP ∆RSP ((SSSSSS))