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Triangle Congruence: ASA and AAS Geometry (Holt 4-6) K. Santos Included Side Included side---is the common side of two consecutive angles in a triangle A B C π΄π΅ is the included side of < A and <B Angle-Side-Angle (ASA) Congruence Postulate (4-6-1) 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. Given: <A β <D <B β <F π΄π΅ β π·πΉ A B Then: βπ΄π΅πΆ β βDFE D C E F Angle-Angle-Side (AAS) Congruence Theorem (4-6-2) If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent. Given: <A β <D <B β <F π΄πΆ β π·πΈ A B D C E F Then: βπ΄π΅πΆ β βDFE Another way the triangles could have been congruent by AAS would be to use the same angles with π΅πΆ β πΉπΈ Hypotenuse-Leg (HL) Congruence Theorem (4-6-3) If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another triangle, then the triangles are congruent. B C D A F E Given: βABC and βDEF are right triangles π΅πΆ β πΈπΉ πΆπ΄ β πΉπ· Then: βπ΄π΅πΆ β βDEF Methods to prove two triangles are congruent Five ways to prove two triangles are congruent: SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem Example Write a congruence statement for each pair of triangles. Name the postulate or theorem that justifies your statement. M 1. Given: < P β < N < PMO β < NMO P O N So, βπππ β βNMO by AAS 2. V W βπππ β βYZV by ASA Z Y Example Determine if you can use ASA or AAS to prove βπππ β βPOM M O is a midpoint of ππ N O Q Since O is a midpoint of ππ then ππ β ππ You know: <Q β <M And you know vertical angles are congruent <NOQ β <POM So, βπππ β βPOM by AAS P Proof 1: Given: < Sβ < Q π π bisects <SRQ Prove: βππ π β βQRP P S Q R Statements 1. < Sβ < Q 2. π π bisects <SRQ 3. < PRS β < PRQ 4. ππ β ππ 1. 2. 3. 4. 5. βππ π β βQRP 5. Reasons Given Given definition of an angle bisector Reflexive Property of congruence AAS Theorem (1, 3, 4) Proof 2: A Given: π΄πΆ β π΅π· <Bβ <D Prove: βπ΄π΅πΆ β βADC B Statements 1. π΄πΆ β π΅π· 2. <ACB and <ACD are right angles 3. <ACB β < π΄πΆπ· 4. < B β < D 5. π΄πΆ β π΄πΆ 6. βπ΄π΅πΆ β βADC C D Reasons 1. Given 2. definition of perpendicular lines 3. all right angles are congruent 4. Given 5. Reflexive Property of congruence 6. AAS Theorem (3, 4, 5) Proof 3: Given: π΄πΆ||π΅π· π΄π·|| π΅πΈ D is a midpoint of πΆπΈ Prove: βπ΄π·πΆ β βBED A B C E D Reasons Statements 1. π΄πΆ||π΅π· 2. <ACD β <BDE 1. Given 2. Corresponding angles postulate 3. π΄π·|| π΅πΈ 4. <ADC β <BED 3. Given 4. Corresponding angles postulate 5. D is a midpoint of πΆπΈ 5. Given 6. πΆπ· β πΈπ· 6. definition of a midpoint 7. βπ΄π·πΆ β βBED 7. ASA Postulate (2, 6, 4) Proof 4: Given: ππ β ππ ππ β ππ Prove: β ππ π β βPRT Statements 1. ππ β ππ 2. < PRS and < PRT are right angles 3. β ππ π πππ βPRT are right triangles 4. ππ β ππ 5. ππ β ππ 6. β ππ π β βPRT S R T P Reasons 1. Given 2. definition of perpendicular lines 3. definition of a right triangle 4. Given 5. Reflexive Property of congruence 6. HL Theorem (3-4, 5)