Jim Smith JCHS Sections 4-2, 4-3, 4-5

When we talk about congruent triangles, we mean everything about them Is congruent. All 3 pairs of corresponding angles are equal…. And all 3 pairs of corresponding sides are equal

For us to prove that 2 people are identical twins, we don’t need to show that all “2000” body parts are equal. We can take a short cut and show 3 or 4 things are equal such as their face, age and height. If these are the same I think we can agree they are twins. The same is true for triangles. We don’t need to prove all 6 corresponding parts are congruent. We have 5 short cuts or methods.

SSS If we can show all 3 pairs of corr. sides are congruent, the triangles have to be congruent.

SAS Show 2 pairs of sides and the included angles are congruent and the triangles have to be congruent. Includedangle Non-includedangles

This is called a common side. It is a side for both triangles. We’ll use the reflexive property.

Which method can be used to prove the triangles are congruent

Common side SSS Parallel lines alt int angles Common side SAS Vertical angles SAS

ASA, AAS and HL ASA – 2 angles and the included side A S A AAS – 2 angles and The non-included side AA S

HL ( hypotenuse leg ) is used only with right triangles, BUT, not all right triangles. HL ASA

When Starting A Proof, Make The Marks On The Diagram Indicating The Congruent Parts. Use The Given Info, Properties, Definitions, Etc. We’ll Call Any Given Info That Does Not Specifically State Congruency Or Equality A PREREQUISITE

SOME 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

A B C D E 12 Given: AB = BD EB = BC EB = BC Prove: ∆ABE ˜ ∆DBC = SAS Our Outline P rerequisites S ides A ngles S ides Triangles ˜ =

AC D Given: AB = BD EB = BC EB = BC Prove: ∆ABE ˜ ∆DBC = B E 12 SAS none AB = BD Given 1 = 2 Vertical angles EB = BC Given ∆ABE ˜ ∆DBC SAS = STATEMENTS REASONS PSAS∆’s

A B C 1 2 Given: CX bisects ACB A ˜ B A ˜ B Prove: ∆ACX ˜ ∆BCX X = = AAS PAAS∆’s CX bisects ACB Given 1 = 2 Def of angle bisc 1 = 2 Def of angle bisc A = B Given A = B Given CX = CX Reflexive Prop CX = CX Reflexive Prop ∆ACX ˜ ∆BCX AAS =

Can you prove these triangles are congruent?