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Vocabulary theorem two-column proof

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1 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

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

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4 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

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 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

8 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 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 s supp &   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

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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

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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 s supp &   Rt s HONORS – PROVE ALL THEOREMS


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