advertisement

Proof – A logical argument that shows a statement is true Theorem – A statement that has been formally proven Theorem 2.1: Segments congruence is reflexive, symmetric, and transitive. Reflexive property of : For any segment AB , ________________. Examples: (a) JK _______ (b) XY _______ Symmetric property of : If AB CD , then ___________________. Example: (a) If GH WO , then _________________ (b) If JK LM , then _________________ Transitive property of : If AB CD and CD EF , then _________________. Example: (a) If JK LM and LM HA , then _________________ (b) If XY SU and SU TK , then _________________ Theorem 2.2: Angle congruence is reflexive, symmetric, and transitive. Reflexive property of : For any angle A, ______________. Examples: (a) K _______ (b) Y _______ Symmetric property of : If A B, then _________________. Example: (a) If G H , then _________________ (b) If LMN RST , then _________________ Transitive property of : If A B and B C, then __________________. Example: (a) If J Z and Z H , then _________________ (b) If YAB LOG and LOG UVT , then ____________________ A two-column geometric proof consists of a list of statements, and the reasons to show that those statements are true. 1. Read the given(s) and what is to be proved. This is the beginning and end to the proof. 2. Draw a figure that illustrates what is to be proved, if one is not already given. 3. Mark the figure according to what you can deduce about it from the information given. 4. Write the steps down carefully, without skipping even the simplest one, and be sure to number each step. 1. E Given: m2 m3 , mAXD mAXC Prove: m1 m4 D 2 3 A 1 X 4 B C A 2. Given: BD bisects ABC Prove: mABD 2 mABD B D C 1. BD bisects ABC 1. ___________________ 2. __________________________ 2. Definition of angle bisector 3. __________________________ 3. Definition of congruent angles 4. mABD mDBC mABC 4. ___________________________ 5. mABD _________ mABC 5. Substitution Property of Equality 6. __________________________ 6. Distributive Property (Simplify) 3. Given: AB CD M is the midpoint of AB N is the midpoint of CD Prove: AM CN Sketch: Statements 1. ______________________ Reasons 1. ________________________________ ______________________ ______________________ 2. AB = CD 2. ________________________________ 3. AM = MB, CN = ND 3. ________________________________ 4. ______________________ 4. Segment Addition Post. ______________________ 5. AM + MB = CN + ND 5. ________________________________ 6. AM + AM = CN + CN 6. ________________________________ 7. 2AM = 2CN 7. ________________________________ 8. ______________________ 8. Division Property of = 9. AM CN 9. ________________________________