2.5 Vocabulary Two-Column Proof A proof is deductive reasoning that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them. Definitions Postulates Properties Theorems Hypothesis Conclusion

Remember… The Distributive Property: a(b + c) = ab + ac The Commutative Property: a + b = b + c

Remember… Definitions (Biconditional ↔ ) ≅ ↔ = Mid pt ↔ 2≅ Bis ↔ 2≅ Rt ↔ 90 St ↔ 180 (you can “see” a st ) Acute ↔ 0<A<90 Obtuse ↔ 90<Obtuse<180 Supplementary ↔ 2∡ = 180 Complementary ↔ 2∡ = 90 Postulates: Seg Addition Postulate Angle Addition Postulate

Example 2. Two Column Proof Given: A and B are supplementary and mA = 45° Prove: mB = 135° STATEMENTS REASONS 1. A and B are supplementary, mA = 45° Given 2. mA + mB = 180° Supplementary ↔ 2∡s = 180 3. 45° + mB = 180° Subst. Prop of = Steps 1, 2 4. mB = 135° Subtr. Prop of =

Given: B is the midpoint of AC, AB  EF Prove: BC  EF Check It Out! Example 3 Given: B is the midpoint of AC, AB  EF Prove: BC  EF STATEMENTS REASONS 1. B is the midpoint of AC. 2. AB  BC 3. AB  EF 4. BC  EF

Example 4. Given: 1  3 Prove: mDBA = mEBC STATEMENTS REASONS D E 2 3 C B A STATEMENTS REASONS

Example 5. Given: M is midpoint of AB Prove: AM = ½ AB STATEMENTS REASONS