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Sec. 4-2 Δ by SSS and SAS Objective: 1) To prove 2 Δs using the SSS and the SAS Postulate
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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
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ΔABC ΔPQR AB PQ BC QR CA RP B C A Q R P A P B Q C R
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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.
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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
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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
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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
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SAS S A S S S A A B C D YES, ABC CDA
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SAS S A S S S A A B C D YES, ABC CDA A C
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S S S S S S SSS YES, PQR RSP P Q R S
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S A S S S A SAS YES, PQR SQT P Q R S T
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S A SS S A NO, SAS YES,
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S A S S S A NO, SAS YES,
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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
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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
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A S A A A S ASA YES, PQR PST P Q R S T
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AA S A S A AAS YES, ABC DCB C D B A
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A A S A S A AAS YES, ABC CDA A B C D
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NO, AAS YES, A A S A S A SS SAS YES, AA ASA YES,
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SSS SAS ASA AAS HL
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