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Before Section 3.3 One more way to prove triangles congruent: Angle-Angle-Side (AAS) Postulate

B. Y. A. X. C. Z. Before Section 3.3 One more way to prove triangles congruent: Angle-Angle-Side (AAS) Postulate.

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Before Section 3.3 One more way to prove triangles congruent: Angle-Angle-Side (AAS) Postulate

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  1. B Y A X C Z Before Section 3.3 One more way to prove triangles congruent: Angle-Angle-Side (AAS) Postulate If two angles and the nonincluded side of one triangle are congruent to two angles and the nonincluded side of a second triangle, then the two triangles are congruent.

  2. C B D M F Given B  C Given D F Given Definition of midpoint DM = MF Definition of congruent segments AAS ∆BDM ∆CFM

  3. 3.3 CPCTC and Circles • Objectives: • Use CPCTC in proofs • Know and use basic properties of circles

  4. Congruent triangles: triangles whose corresponding parts are congruent. CPCTC Corresponding Part of Congruent Triangles are Congruent. This can be used in a proof only AFTER triangles have been proven congruent.

  5. Circle:the set of points in a plane equidistant from the one point, the center. Notation: circle C or C. Radius:the distance from the center of the circle to any point on the circle. Theorem 19: All radii of a circle are congruent. C

  6. Circumference of a Circle: C = 2r Area of a Circle: A = r2 Example 1: Find the circumference and area of a circle with a radius of 12 units.

  7. N Example 2: Given: O Prove: ∆NOL ∆NOM M O L All radii of a circle are congruent Given Definition of perpendicular NOL and NOMare right angles All right angles are congruent NOL  NOM ∆NOL ∆NOM SAS

  8. W Example 3: Given: Z is the midpoint of Yand W are complementary to V Prove: Z V X Y • Z is the midpoint of Definition of midpoint WZ = ZY Definition of congruent Given Y and W are comp. to V Y  W Congruent Complements Theorem Vertical Angles Theorem VZY  WZX ∆VZY ∆XZW ASA CPCTC

  9. N K P M L Example 4: Given: P Prove: All radii of a circle are congruent • Vertical Angles Theorem KPL  NPM SAS ∆VZY ∆XZW CPCTC

  10. T Example 5: Given: Q RT = TS Prove: TRQ  TSQ Q R S All radii of a circle are congruent • Given • RT = RS Definition of congruent • SSS ∆TRQ ∆TSQ • TRQ  TSQ CPCTC

  11. F Example 6: Given: C is the midpoint of AC = CE Prove: ∆ABF  ∆EDF A E B C D FCA and FCE are right angles Definition of perpendicular FCA FCE All right angles are congruent Reflexive Property AC = CE Given Definition of congruent SAS ∆FCA ∆FCE CPCTC A E Continued on next slide

  12. F Example 6: Given: C is the midpoint of AC = CE Prove: ∆ABF  ∆EDF A E B C D Given BC = CD Definition of midpoint Definition of congruent Subtraction Property Given C is the midpoint of ∆ABF ∆EDF SAS

  13. A B C Example 7: G H D E F Definition of bisect Definition of bisect Vertical Angles Theorem AGD BGE SAS • ∆AGD ∆BGE CPCTC Continued on next slide

  14. A B C Example 7: G H D E F Definition of bisect Definition of bisect BHE CHF Vertical Angles Theorem SAS ∆BHE  ∆CHF CPCTC Transitive Property

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