Angle-Angle-Side (AAS) Postulate 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. A B C X Y Z

Definition of midpoint C B D M F Statements Reasons 1. 2. 3. 4. 5. 6. B  C Given D  F Given Given DM = MF Definition of midpoint Definition of congruent segments ∆BDM  ∆CFM AAS

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

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.

Notation: circle C or ⨀C. 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

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.

N M O L Example 2: Given: ⨀O 𝑁𝑂 ⊥ 𝐿𝑀 Prove: ∆NOL  ∆NOM 𝑁𝑂 ⊥ 𝐿𝑀 Prove: ∆NOL  ∆NOM L M O N Statements Reasons 1. ⨀O 1. Given 2. 3. 4. 5. All radii of a circle are congruent Given NOL and NOM are right angles Definition of perpendicular NOL  NOM All right angles are congruent ∆NOL  ∆NOM SAS

Given: Z is the midpoint of Y and W are complementary to V Prove: W Example 3: Given: Z is the midpoint of Y and W are complementary to V Prove: W Z V X Y Statements Reasons 1. 1. Given 2. 3. 4. 5. 6. 7. 8. Z is the midpoint of WZ = ZY Definition of midpoint Definition of congruent Y and W are comp. to V Given Y  W Congruent Complements Theorem VZY  WZX Vertical Angles Theorem ∆VZY  ∆XZW ASA CPCTC

N K P M L Example 4: Given: ⨀P Prove: Statements Reasons 1. ⨀𝑃 1. Given 2. 3. 4. 5. All radii of a circle are congruent KPL  NPM Vertical Angles Theorem ∆VZY  ∆XZW SAS CPCTC

T Q R S Example 5: Given: ⨀Q RT = TS Prove: TRQ  TSQ Statements Reasons 1. ⨀Q 1. Given 2. 3. 4. 5. 6. All radii of a circle are congruent RT = RS Given Definition of congruent ∆TRQ  ∆TSQ SSS TRQ  TSQ CPCTC

F A E B C D Example 6: Given: C is the midpoint of AC = CE Prove: ∆ABF  ∆EDF A E B C D Statements Reasons 1. 1. Given 2. 3. 4. 5. 6. 7. 8. FCA and FCE are right angles Definition of perpendicular FCA  FCE All right angles are congruent Reflexive Property AC = CE Given Definition of congruent ∆FCA  ∆FCE SAS A  E CPCTC Continued on next slide

F A E B C D Example 6: Given: C is the midpoint of AC = CE Prove: ∆ABF  ∆EDF A E B C D Statements Reasons 9. 10. 11. 12. 13. 14. C is the midpoint of Given BC = CD Definition of midpoint Definition of congruent Subtraction Property Given ∆ABF  ∆EDF SAS

A B C G H D E F Example 7: Definition of bisect Definition of bisect Statements Reasons 1. 1. Given 2. 3. 4. 5. 6. Definition of bisect Definition of bisect AGD  BGE Vertical Angles Theorem ∆AGD  ∆BGE SAS CPCTC Continued on next slide

A B C G H D E F Example 7: Definition of bisect Definition of bisect Statements Reasons 7. 7. Given 8. 9. 10. 11. 12. 13. Definition of bisect Definition of bisect BHE  CHF Vertical Angles Theorem ∆BHE  ∆CHF SAS CPCTC Transitive Property