Proving Triangles Congruent

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.

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

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

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

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. ASA HL

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

Proof 1) O is the midpoint of MQ and NP 2) 3) 4) 1) Given Given: O is the midpoint of MQ and NP Prove: 1) O is the midpoint of MQ and NP 2) 3) 4) 1) Given 2) Def of midpoint 3) Vertical Angles 4) 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 =

Proof Given: Prove: 1) 2) 3) 1) Given 2) Reflexive Property 3) SSS

Proof 1) 2) 3) 4) 1) Given 2) Alt. Int. <‘s Thm Prove: 1) 2) 3) 4) 1) Given 2) Alt. Int. <‘s Thm 3) Reflexive Property 4) SAS