Congruent Triangles & Proofs: Co ngruent figures have congruent corresponding parts--> sides and angles.

SSS Congruence Side-Side-Side Postulate: If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. Examples:

SAS Congruence Side-Angle-Side Congruence: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Examples:

ASA Congruence Angle-Side-Angle Congruence: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. Examples:

AAS Congruence Angle-Angle-Side Congruence: If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent. Examples:

HL Theorem Hypotenuse Leg Theorem: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another triangle, then the triangles are congruent. Example:

Proving Triangles Congruent: Given: JK ≅ LM, JM ≅ LK Prove: ΔJKM ≅ ΔLMK J K L M Statements Reasons 1) JK ≅ LM, JM ≅ LK 1) Given 2) MK ≅ MK 2) Reflexive Property of Congruence 3) ΔJKM ≅ ΔLMK 3) SSS Postulate Move blue box for answer

Given: X is the midpoint of AG and of NR Prove: ΔANX ≅ ΔGRX A N X R G 1 2 Statements Reasons 1) X is the midpoint of AG and of NR 1) Given 2) AX ≅ GX, NX ≅ RX 2) Definition of midpoint 3) <1 ≅ <2 3) Vertical angles are congruent 4) ΔANX ≅ ΔGRX 4) SAS Theorem

Given: AB CM, AB DB, M is the midpoint of AB, CM ≅ DB Prove: ΔAMC ≅ ΔMBD C A M B D StatementsReasons 1) AB CM, AB DB, M is the midpoint of AB, CM ≅ DB 1) Given 2) <CMA & <DBM are right angles=90 2) Definition of perpendicular 3) <CMA ≅ <DBM 3) All right angles are congruent 4) AM ≅ BM 4) Definition of midpoint ΔAMC ≅ ΔMBD 5) SAS Theorem 5)5)

Given: SQ bisects <PSR, <P ≅ <R Prove: ΔPSQ ≅ ΔRSQ Given ΔPSQ ≅ ΔRSQ Given P Q S R <P ≅ <R SQ bisects <PSR <PSQ ≅ <RSQ Definition of angle bisector FLOW PROOF SQ ≅ SQ Reflexive Property AAS

Given: AE || BD, AE ≅ BD, <E ≅ <D Prove: ΔAEB ≅ ΔBDC A E B D C Statements Reasons 1)1) AE || BD, AE ≅ BD, <E ≅ <D 1) Given 2) <A ≅ <C 2) Corresponding angles postulate 3)3) ΔAEB ≅ ΔBDC 3) ASA Theorem