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Triangle Congruence: ASA and AAS

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 is the included side of < A and <B.

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Triangle Congruence: ASA and AAS

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  1. Triangle Congruence: ASA and AAS Geometry (Holt 4-6) K. Santos

  2. Included Side Included side---is the common side of two consecutive angles in a triangle A B C is the included side of < A and <B

  3. 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 <D A D E <B <F B C F Then: DFE

  4. 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 nonincludedside of another triangle, then the triangles are congruent. Given: <A <D A D E <B <F B C F Then: DFE Another way the triangles could have been congruent by AAS would be to use the same angles with

  5. 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: ABC and DEF are right triangles Then: DEF

  6. 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

  7. Example Write a congruence statement for each pair of triangles. Name the postulate or theorem that justifies your statement. M 1. Given: < P < N < PMO < NMO P O N So, by AAS • V W by ASA Z Y

  8. Example Determine if you can use ASA or AAS to prove M O is a midpoint of N O P Q Since O is a midpoint of then You know: <Q <M And you know vertical angles are congruent <NOQ <POM So, by AAS

  9. Proof 1: Given: < S < Q P bisects <SRQ S Q Prove: R Statements Reasons • < S < Q 1. Given 2. bisects <SRQ 2. Given 3. < PRS < PRQ 3. definition of an angle bisector 4. 4. Reflexive Property of congruence 5.5. AAS Theorem (1, 3, 4)

  10. Proof 2: Given: ⟘ A < B < D Prove: DC B C D Statements Reasons 1. ⟘ 1. Given 2. <ACB and <ACD are right 2. definition of perpendicular angles lines • <ACB 3. all right angles are congruent 4. < B < D 4. Given 5. 5. Reflexive Property of congruence 6. DC 6. AAS Theorem (3, 4, 5)

  11. Proof 3: Given: |A B | D is a midpoint of Prove: C E D Statements Reasons 1. | 1. Given 2. <ACD <BDE 2. Corresponding angles postulate 3. | 3. Given 4. <ADC <BED 4. Corresponding angles postulate 5. D is a midpoint of 5. Given 6. 6. definition of a midpoint 7. 7. ASA Postulate (2, 6, 4)

  12. Proof 4: Given: ⟘ S R T Prove: P Statements Reasons 1. ⟘ 1. Given 2. < PRS and < PRT are right 2. definition of perpendicular angles lines 3. PRT are right 3. definition of a right triangle triangles 4. 4. Given 5. 5. Reflexive Property of congruence 6. 6. HL Theorem (3-4, 5)

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