Sections 4-3 - 4-5Sections 4-3 - 4-5

When we talk about congruent triangles,When we talk about congruent triangles,we mean everything about them is congruent. we mean everything about them is congruent. 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. We that all “2000” body parts are equal. We can take a short cut and show 3 or 4 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 5 short congruent. We have 5 short cuts or methods.cuts or methods.

SSSSSSIf we can show all 3 pairs of corr.If we can show all 3 pairs of corr.

sides are congruent, the triangles sides are congruent, the triangles have to be congruent.have to be congruent.

SASSASShow 2 pairs of sides and the Show 2 pairs of sides and the

included angles are congruent and included angles are congruent and the triangles have to be congruent.the triangles have to be congruent.

IncludedIncludedangleangle

Non-includedNon-includedanglesangles

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

We’ll use the reflexive property.We’ll use the reflexive property.

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

Common sideCommon side

SSSSSS

Parallel linesParallel linesalt int anglesalt int angles

Common sideCommon side

Vertical anglesVertical angles

SASSAS

SASSAS

ASA, AAS and HL ASA, AAS and HL

ASA – 2 anglesASA – 2 anglesand the included sideand the included side

AA

SS

AAAAS – 2 angles andAAS – 2 angles andThe non-included sideThe non-included side AAAA

SS

HLHL ( hypotenuse leg ) is used ( hypotenuse leg ) is usedonly with right triangles, BUT, only with right triangles, BUT,

not all right triangles. not all right triangles.

HLHL ASAASA

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

The Congruent Parts. Use The GivenThe Congruent Parts. Use The GivenInfo, Properties, Definitions, Etc. Info, Properties, Definitions, Etc.

We’ll Call Any Given Info That Does We’ll Call Any Given Info That Does Not Specifically State Congruency Not Specifically State Congruency

Or Equality A Or Equality A PREREQUISITEPREREQUISITE

SOME REASONS WE’LL BE USINGSOME REASONS WE’LL BE USING

• DEF OF MIDPOINTDEF OF MIDPOINT

• DEF OF A BISECTORDEF OF A BISECTOR

• VERT ANGLES ARE CONGRUENTVERT ANGLES ARE CONGRUENT

• DEF OF PERPENDICULAR BISECTORDEF OF PERPENDICULAR BISECTOR

• REFLEXIVE PROPERTY (COMMON SIDE)REFLEXIVE PROPERTY (COMMON SIDE)

• PARALLEL LINES ….. ALT INT ANGLESPARALLEL LINES ….. ALT INT ANGLES

AA

BB

CC

DDEE

11 22

Given: AB = BDGiven: AB = BD EB = BCEB = BC

Prove: Prove: ∆ABE ∆ABE ˜ ˜ ∆DBC∆DBC==

SASSASOur OutlineOur OutlineP P rerequisitesrerequisitesS S ideideA A nglengleS S ideideTriangles ˜Triangles ˜ ==

AA CC

DD

Given: AB = BDGiven: AB = BD EB = BCEB = BC

Prove: Prove: ∆ABE ∆ABE ˜ ˜ ∆DBC∆DBC==BB

EE

11 22

SASSAS

<none><none>AB = BD Given AB = BD Given 1 = 2 Vertical angles1 = 2 Vertical anglesEB = BC GivenEB = BC Given∆∆ABE ˜ ∆DBC SASABE ˜ ∆DBC SAS==

STATEMENTS REASONSSTATEMENTS REASONS

PPSSAASS∆’∆’ss

AA BB

CC

11 22

Given: CX bisects ACBGiven: CX bisects ACB A A ˜ B˜ BProve: Prove: ∆ACX∆ACX ˜ ∆BCX˜ ∆BCX

XX

====

AASAAS

PPAAAASS∆’∆’ss

CX bisects ACB GivenCX bisects ACB Given 1 = 2 1 = 2 Def Def of angle biscof angle bisc A = B GivenA = B Given CX = CX Reflexive PropCX = CX Reflexive Prop∆∆ACX ˜ ∆BCX AASACX ˜ ∆BCX AAS==

Can you prove these trianglesCan you prove these trianglesare congruent?are congruent?

AA BB

DD CC

XX

Given: AB ll DCGiven: AB ll DCX is the midpoint of ACX is the midpoint of ACProve: AXB Prove: AXB ˜ CXD˜ CXD==

AA BB

DD CC

XX

Given: AB ll DCGiven: AB ll DCX is the midpoint of ACX is the midpoint of ACProve: AXB Prove: AXB ˜ CXD˜ CXD==

ASAASA

Triangle Congruence

SSS – If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent

SAS – If two sides and the included angle of one triangle are congruent to those of another triangle, then the two triangles are congruent

ASA – If two angles and the included side of one triangle are congruent to those of another triangle, then the two triangles are congruent

AAS – If two angles and a non-included side of one triangle are congruent to those of another triangle, then the two triangles are congruent

HL – If the hypotenuse and one leg of a right triangle are congruent to those of another right triangle, then the two triangles are congruent

.

Triangle Congruence

SSS – If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent

SAS – If two sides and the included angle of one triangle are congruent to those of another triangle, then the two triangles are congruent

ASA – If two angles and the included side of one triangle are congruent to those of another triangle, then the two triangles are congruent

AAS – If two angles and a non-included side of one triangle are congruent to those of another triangle, then the two triangles are congruent

HL – If the hypotenuse and one leg of a right triangle are congruent to those of another right triangle, then the two triangles are congruent

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