Sec. 4-2 Δ  by SSS and SAS Objective: 1) To prove 2 Δs  using the SSS and the SAS Postulate

R P Q A B C A  P B  Q C  R AB  PQ BC  QR CA  RP If ABC  PQR then find the corresponding parts CPCTC Theorem C C PT C orresponding arts ongruent riangles ongruent in are

ΔABC  ΔPQR AB  PQ BC  QR CA  RP B C A Q R P A  P B  Q C  R

In Sec. 4-1 we learn that if all the sides and all the  s are  of 2Δs then the Δs are . But we don’t need to know all 6 corresponding parts are . There are short cuts.

POSTULATE 4-1 (SSS) POSTULATE Side - Side - Side (SSS) Congruence Postulate Side MNQR Side PMSQ Side NPRS If If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. then  MNP  QRS SSS SSS C ONGRUENCE P OSTULATE

Included – A word used frequently when referring to the  s and the sides of a Δ. Means – “in the middle of” What  is included between the sides BX and MX?  X What side is included between  B and  M? BM BM X

POSTULATE 4-2 (SAS) POSTULATE Side-Angle-Side (SAS) Congruence Postulate Side PQWX Side QSXY then  PQS  WXY Angle QX If If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. ASS SAS C ONGRUENCE P OSTULATE

SAS S A S S S A A B C D YES,  ABC   CDA

SAS S A S S S A A B C D YES,  ABC   CDA A C

S S S S S S SSS YES,  PQR   RSP P Q R S

S A S S S A SAS YES,  PQR   SQT P Q R S T

S A SS S A NO, SAS YES,

S A S S S A NO, SAS YES,

ASA C ONGRUENCE P OSTULATE S POSTULATE 4-3 (ASA) POSTULATE Angle - Side - Angle (ASA) Congruence Postulate Side PNSR If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. then  MNP  QRS S Angle NR Angle P S If AS A

AAS C ONGRUENCE P OSTULATE SS POSTULATE 4-4 (AAS) POSTULATE Angle - Angle - Side (AAS) Congruence Postulate Side PMSQ If two angles and the NON included side of one triangle are congruent to two angles and the NON included side of a second triangle, then the two triangles are congruent. then  MNP  QRS Angle NR Angle P S If AAS

A S A A A S ASA YES,  PQR   PST P Q R S T

AA S A S A AAS YES,  ABC   DCB C D B A

A A S A S A AAS YES,  ABC   CDA A B C D

NO, AAS YES, A A S A S A SS SAS YES, AA ASA YES,

SSS SAS ASA AAS HL