Given : AB || CD AE  BE

1. WARM UP Given: AB || CD AE  BE Prove: 3  4 E 3 4 C D 1 2 A B

2. Statements Reasons 1. AB || CD 1. Given 2. If lines are parallel, then corresponding angles are congruent. 2. 1 3, 2 4 3. AE BE 3. Given 4. If two sides of a triangle are congruent, then the angles opposite those sides are congruent. 4. 1 2 5. 2 3 5. Substitution 6. 3 4 6. Substitution

3. Section 4-5 AAS, HL Theorems & Proofs

4. AAS Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. A ABC   ___ DEF D C B F E

5. C Example 1) ABC   ___ DBC AAS Theorem D A B Example 2) A D ABC   ___ EDC C AAS Theorem B E

6. HL Theorem: If the hypotenuse and a leg of one right triangle are congruent to corresponding parts of another right triangle, then the triangles are congruent. E C A F B D ABC   ___ DFE

7. B Example 1) ABC   ___ DBC HL Theorem A D C Example 2) C D ABC   ___ DCB HL Theorem A B

8. PROOF EXAMPLE 1: V Given: 1  2 CD bisects VCW Prove: DV  DW 1 3 D C 4 2 W Statements Reasons 1. CD bisects VCW 1. Given 2. 3  4 2. Def. of Angle Bisector 3. 1  2 6. DV  DW 3. Given 4. CD  CD 4. Reflexive Property 5. ΔCVD ΔCWD 5. AAS Theorem 6. CPCTC

9. PROOF EXAMPLE 2: X Given: W and Y are right angles WX  YX Prove: WZ  YZ Y W Z Statements Reasons 1. W and Y are right angles 1. Given 2. ΔXWZ andΔXYZ are right triangles. 2. Def. of Right Triangle 3. WX  YX 6. WZ  YZ 3. Given 4. XZ  XZ 4. Reflexive Property 5. ΔXWZ ΔXYZ 5. HL Theorem 6. CPCTC

10. PROOF EXAMPLE 3: Given: KL  LA; KJ  JA; AK bisects LAJ Prove: LK  JK L 1 K A 2 J

11. Statements Reasons 1. Given 1. KL  LA; KJ  JA 2. Land J are right angles 2. Def. of Perpendicular Lines 3. mL = 90; mJ = 90 3. Def. of Right Angles 4. mL= mJ; L J 4. Substitution • AK bisects LAJ 5. Given 6. 1 2 6. Def. of Angle Bisector 7. KA  KA 7. Reflexive Property 8. ΔLKA ΔJKA 8. AAS Theorem 9. LK  JK 9. CPCTC