Vocabulary theorem two-column proof 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… 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

A theorem is any statement that you can prove A theorem is any statement that you can prove. Once you have proven a theorem, you can use it as a reason in later proofs. Theorems 2.1 and 2.2 (Segments and Angles)

a=b b=c thus a=c by transitive OR substitution Numbers or variables representing numbers may be equal (=). Figures that are the “same” are congruent (), not =. Congruence is a “relationship”. Remember! For equations the transitive property is a special case of substitution. Hence, you can always use substitution as a justification when you have an equation (= sign) but not always transitive. a=b b=c thus a=c by transitive OR substitution x=y+2 y=7 thus x=7+2 by substitution but not transitive For congruent things (angles, segments, figures) always use transitive. You can not substitute things

Example 1: Identifying Property of Equality and Congruence Identify the property for each A. QRS  QRS B. m1 = m2 so m2 = m1 C. AB  CD and CD  EF, so AB  EF. D. 32° = 32° E. x = y and y = z, so x = z. F. DEF  DEF G. AB  CD, so CD  AB.

Remember! A geometric proof begins with Given and Prove statements, which restate the hypothesis and conclusion of the conjecture. In a two-column proof, you list the steps of the proof in the left column. You write the matching reason for each step in the right column.

Example 2 Write a justification for each step, Given that B is the midpoint of AC and AB  EF. Prove BC  EF 1. B is the midpoint of AC. 2. AB  BC 3. AB  EF 4. BC  EF

Theorems 1. Linear Pair  Supp (Theorem, not Postulate) 2. Supps of Same   (or Supps of    ) 3. All Rt Angles are  4. Comps of Same   (or Comps of    ) 5. Transitive for Supp & Comp with s 6. Congruent Addition Thrms (segments & angles) (  ’s +  ‘s   ) 7. Vertical s   8. Two s supp &   Rt s HONORS – PROVE ALL THEOREMS

Proof?

Example 3 Complete a two-column proof of one case of the Congruent Supplements Theorem (Supps of Same →  ). Given: 1 and 2 are supplementary, and 2 and 3 are supplementary. Prove: 1  3

Example 4: Writing a Two-Column Proof from a Plan Use the given plan to write a two-column proof for one case of Congruent Complements Theorem. Given: 1 comp 2, 3 comp 4, 2  3 Prove: 1  4 Plan: The measures of complementary angles add to 90° by definition. Use substitution to show that the sums of both pairs are equal. Use the Subtraction Property and the definition of congruent angles to conclude that 1  4.

Example 5: 1. Use the given plan to write a two-column proof. Given: 1 and 2 are supplementary, and 1  3 Prove: 3 and 2 are supplementary. (TRANSITIVE PROPERTY) Plan: Use the definitions of supplementary and congruent angles and substitution to show that m3 + m2 = 180°. By the definition of supplementary angles, 3 and 2 are supplementary.

2. Write a two-column proof. Given: 1, 2 , 3, 4 Prove: m1 + m2 = m1 + m4 Plan: ?.

Congruent Addition Theorem

Theorems Through Ch 2: 1. Linear Pair  Supp (Theorem, not Postulate) 2. Supps of Same   (or Supps of    ) 3. All Rt Angles are  4. Comps of Same   (or Comps of    ) 5. Transitive for Supp & Comp with  s 6. Congruent Addition Thrms (segments & angles) (  ’s +  ‘s   ) 7. Vertical s   8. Two s supp &   Rt s HONORS – PROVE ALL THEOREMS