When we talk about congruent triangles, we mean everything about them is congruent (or exactly the same) (or exactly the same) 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 4 short cuts or rules to help us prove that two triangles are congruent.

SSS If 3 corresponding sides are exactly the same length, then the triangles have to be congruent. Side-Side-Side (SSS)

SAS If 2 pairs of sides and the included angles are exactly the same then the triangles are congruent. Side-Angle-Side (SAS) Includedangle Non-includedangles

AAS If 2 angles and a side are exactly the same then the triangles have to be congruent. Angle-Angle-Side (AAS)  

RHS If 2 triangles have a right-angle, a hypotenuse, and a side, which are exactly the same then the triangles are congruent. Right angle-Hypotenuse-Side (RHS)

This is called a common side. It is a side for both triangles. Sometimes it is shown by a ‘squiggly’ line placed on the common line

Why is AAA (Angle, Angle, Angle) NOT a proof for CONGRUENT TRIANGLES??? Discuss!!

Which method can be used to prove the triangles are congruent a) b) c) d)

3Common Sides (SSS) Parallel lines alternate angles Common side (AAS) Included angle, 2 sides equal 2 sides equal(SAS) RHS Right angle Hypotenuse & a side equal

RHS is used only with right-angled triangles, BUT, not all right triangles. RHS ASA

When Starting A Proof, Make The Marks On The Diagram Indicating The Congruent Parts. Use The Given Information to do this

ABCD E Given: AB = BD EB = BC EB = BC Aim: Prove ∆ABE  ∆DBC

A B C D E Proof: AB = BD (Given) SIDE (S)  ABC =  CBD (Vertically Opposite) ANGLE (A) EB = BC (Given) SIDE (S)  ∆ABE  ∆DBC (SAS) Given: AB = BD EB = BC EB = BC Mark these on the diagram Aim: Prove ∆ABE  ∆DBC

A Given:  CX bisects  ACB  CAB =  CBA  CAB =  CBA Prove: ∆ACX  ∆BCX B C X

B C X A  Proof:  ACX =  BCXCX bisects  ACB (ANGLE) (A)  CAB =  CBA (Given) ANGLE (A) CX is common SIDE (S)  ∆ACX  ∆BCX (AAS)

Given: MN ║ QR MN = QR Prove: ∆MNP  ∆QRP MNP Q R

Proof:  MNP =  QRPAlternate angles MN║QR (ANGLE) (A)  MPN =  QPR Vertically Opposite angles (ANGLE) (A) MN = QRGiven (SIDE) (S)  ∆MNP  ∆QRP(AAS) MNP Q R  

Given: XZ = AC XY = AB XY = AB  XYZ=  ABC = 90   XYZ=  ABC = 90  Prove: ∆ABC  ∆XYZ X Y Z A B C

X Y Z A BCProof:  ABC =  XYZ = 90  Given (RIGHT-ANGLE) (R) XZ = AC Given (Hypotenuse) (H) XY = ABGiven (SIDE) (S)  ∆XYZ  ∆ABC (RHS)

PQ R S Given: PQ = RS QR = SP QR = SP Prove: ∆PQR  ∆RSP

P Q R S Proof: PQ = RSGiven (SIDE) (S) QR = SP Given (SIDE) (S) XY = ABCommon to both (SIDE) (S)  ∆PQR  ∆RSP (SSS)