Geometry Unit 4

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.

Non-included angles Included angle

ASA, AAS and HL

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

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

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 Vertical angles SAS Parallel lines alt int angles Common side SAS

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 MIDPOINT DEF OF A BISECTOR VERT ANGLES ARE CONGRUENT DEF OF PERPENDICULAR BISECTOR REFLEXIVE PROPERTY (COMMON SIDE) PARALLEL LINES ….. ALT INT ANGLES

A E 1 B 2 SAS C Given: AB = BD EB = BC Prove: ∆ABE = ∆DBC D

Our Outline P rerequisites S ides A ngles S ides Triangles ˜

A 1 B 2 C Given: AB = BD EB = BC Prove: ∆ABE = ∆DBC E SAS D STATEMENTS REASONS P S A S ∆’s none AB = BD Given 1 = 2 Vertical angles EB = BC Given ∆ABE ˜ ∆DBC SAS

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

Can you prove these triangles are congruent?