CONGRUENT TRIANGLES Sections 4-3, 4-4, 4-5

This is called a common side. It is a side for both triangles. You need to mark it!! It is called the Reflexive property when a side or angle is  to itself.

Things you can assume : Anything marked in the picture is True. Anything written in words is True. Vertical angles are congruent. Reflexive Property

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

When you prove triangles congruent, you only need a specific combination of sides and angles.

Let’s look at all the possible combinations SSS ASA SAS AAS SSA AAA Which ones do you think work and why???

If we can show all 3 pairs of corr. sides are congruent, the triangles SSS If we can show all 3 pairs of corr. sides are congruent, the triangles have to be congruent. J A C B K L  

Included angle- An angle between two sides Ex. Name the included angle C B A AC&BC_____ AB&BC_____ AC&AB_____ C  B  A

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

A B J L K M N D C B Z A X C Y Which method can be used to prove the triangles are congruent? Write an congruence statement Vertical angles A B J L Reflexive SSS K M ∆ABC≅ ∆CDA N SAS D C  JKN   MKL B Z A SSS X SAS C Y  ABC   XYZ  IJH   KML

What about SSA?

Will AAA work??? AAA doesn’t preserve the size of the triangles

Included side- A side between 2 angles. Name the included side. A&B_____ C&B_____ A&C_____      

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

C D B E J A L M N K ∆JKL≅ ∆NLM:ASA ∆ABE≅ ∆CDB: AAS ∆ABC≅ ∆ADC : ASA

How do we find the missing side of a right triangle? So, if we had a leg and hypotenuse, then we could find the 3rd side.. Proving triangles congruent 5 ? 4 3

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

Decide if the following triangles are congruent by SSS,SAS,ASA,AAS, or HL. If they are not congruent, then write not congruent.

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

SAS Our Outline P rerequisites S ides A ngles Triangles ˜ A C = B 1 2 Given: AB = BD EB = BC Prove: ∆ABE ˜ ∆DBC A C = B 1 2 Our Outline P rerequisites S ides A ngles Triangles ˜ SAS E D =

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

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

Proofs with perpendicular lines ⟘   A B 1 Statement Reason 2 C D