theorem two-column proof

1. Definitions • Postulates • Properties • Theorems Conclusion Hypothesis 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.

2. 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

4. Remember! Numbers or variables representing numbers may be equal (=). Figures that are the “same” are congruent (), not =. Congruence is a “relationship”. 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

5. 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.

6. 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.

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

8. 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 IN 2.6 • 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 ssupp &   Rts • HONORS – PROVE ALL THEOREMS

9. Proof?

10. 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

12. 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.

13. 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.

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

15. Congruent Addition Theorem

17. 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 ssupp &   Rts • HONORS – PROVE ALL THEOREMS