Chapter 4 Chapter 4 Triangle CongruenceTriangle Congruence

By: Emily Gorges, Janie Eyerman, By: Emily Gorges, Janie Eyerman, Andie Jamison, and Maria OngAndie Jamison, and Maria Ong

4-1 Congruence and 4-1 Congruence and TransformationsTransformations

Vocab:Vocab: dilationdilation-changes size, not shape of -changes size, not shape of

a coordinate figurea coordinate figure reflectionreflection- a figure reflected over a - a figure reflected over a

lineline translationtranslation-- the same figure moved the same figure moved

to another place on coordinate gridto another place on coordinate grid rotationrotation- a figure rotated around a - a figure rotated around a

vertex to a certain degreevertex to a certain degree

4-2 Classifying Triangles4-2 Classifying Triangles

Terms:Terms: RightRight ObtuseObtuse AcuteAcute ScaleneScalene Equilateral Equilateral

IsoscelesIsosceles Equiangular Equiangular

Example Example

Classify each triangle-Classify each triangle-

rightright obtuse acuteobtuse acute

scalene equilateral isoscelesscalene equilateral isosceles

4-3 Angle Relationships in 4-3 Angle Relationships in TrianglesTriangles

• auxiliary lineauxiliary line-- a line that is added to a a line that is added to a figure to aid in a prooffigure to aid in a proof

Exterior Angle TheoremExterior Angle Theorem

The measure of the exterior angle of The measure of the exterior angle of a triangle is equal to the sum of its a triangle is equal to the sum of its remote interior angles.remote interior angles.

Third Angle TheoremThird Angle Theorem

If two angles of a triangle are If two angles of a triangle are congruent to angles of another congruent to angles of another triangle then the third angles of both triangle then the third angles of both triangles are congruent.triangles are congruent.

4-4 Congruent Triangles4-4 Congruent Triangles

Terms:Terms: corresponding anglescorresponding angles corresponding sidescorresponding sides congruent polygonscongruent polygons overlapping trianglesoverlapping triangles

Proof exampleProof example

Given: <ACD=<BDC,AC=BDProve: ACD= BDC

<ACD=<BDC GAC=BD GCD=CD Reflexive ACD= BDC SAS*

*See slide 11 for SAS

4-5 Triangle Congruence: 4-5 Triangle Congruence: SSS and SASSSS and SAS

TermsTerms included angle-included angle- the angle in the angle in

between the 2 given sidesbetween the 2 given sides sside ide sside ide sside-ide- if all 3 sides of if all 3 sides of

a triangle are congruent to the a triangle are congruent to the other triangle, then both other triangle, then both triangles are congruenttriangles are congruent

sside ide aangle ngle sside- ide- the twothe two sides and the included angle sides and the included angle are congruent to the other are congruent to the other triangle, then both triangles triangle, then both triangles are congruent are congruent

4-6 Triangle Congruence: 4-6 Triangle Congruence: ASA, AAS, and HLASA, AAS, and HL

included side- included side- side between the 2 given side between the 2 given anglesangles

aangle ngle sside ide aangle- when the two ngle- when the two angles and included side are angles and included side are congruent to the other triangle, then congruent to the other triangle, then both triangles are congruentboth triangles are congruent

aangle ngle aangle ngle sside- ide- when two angles and when two angles and a not included side are congruent to the a not included side are congruent to the other triangle, then both triangles are other triangle, then both triangles are congruentcongruent

hhypotenuse ypotenuse lleg- eg- in right triangles when in right triangles when the hypotenuse and one leg are the hypotenuse and one leg are congruent to the other triangle, then congruent to the other triangle, then both triangles are congruentboth triangles are congruent

4-7 Triangle Congruence: CPCTC4-7 Triangle Congruence: CPCTC

Given: CED is isosceles, AE=BEProve: AC=BD

CED is isos. GAE=BE G AEC=BED verticleCE=ED Def. of isos AEC= BED SASAC=BD CPCTC

E

4-9 Isosceles and 4-9 Isosceles and Equilateral TrianglesEquilateral Triangles

Isosceles Triangles- Isosceles Triangles- a triangle with two sides a triangle with two sides congruent and the two corresponding angles are congruent and the two corresponding angles are congruentcongruent

Given: AD bisects ABC,Prove: ABC is isosceles

A

CB

D

Try It Yourself!

Equilateral Triangle- Equilateral Triangle- a triangle with all sides a triangle with all sides and angles are congruentand angles are congruent

See, all sides and angles ARE congruent!