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Two-column proof and properties of equality in geometry

This article explains how to write a two-column proof for a given situation in geometry and discusses the properties of equality used in the process. It also includes examples and guided practice questions.

kennethwright
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Two-column proof and properties of equality in geometry

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  1. Write a two-column proof for the situation in Example 4 from Lesson 2.5. 4. m∠ 1+m∠ 2=m∠ DBC m∠ 1=m∠ 3 GIVEN: m∠ EBA=m∠ DBC PROVE: 4. Angle Addition Postulate REASONS m∠ EBA= m∠ DBC STATEMENT 5. 5. Transitive Property of Equality 1. 1. m∠ 1=m∠ 3 Given 2. Angle Addition Postulate 2. m∠ EBA=m∠ 3+m∠ 2 3. Substitution Property of Equality 3. m∠ EBA=m∠ 1+m∠ 2 EXAMPLE 1 Write a two-column proof

  2. 1. Four steps of a proof are shown. Give the reasons for the last two steps. for Example 1 GUIDED PRACTICE GIVEN :AC = AB + AB PROVE :AB = BC

  3. ANSWER GIVEN :AC = AB + AB PROVE :AB = BC REASONS STATEMENT 1. 1. AC = AB + AB Given 2. 2. AB + BC = AC Segment Addition Postulate 3. 3. AB + AB = AB + BC Transitive Property of Equality 4. 4. AB = BC Subtraction Property of Equality for Example 1 GUIDED PRACTICE

  4. Theorems • A statement that can be proven. Once you have proven a theorem, you can use the theorem As a reason in other proofs.

  5. Reflexive/Symmetric/Transitive

  6. a. IfRTandTP, then RP. b. IfNKBD, thenBDNK. a. Transitive Property of Angle Congruence b. Symmetric Property of Segment Congruence EXAMPLE 2 Name the property shown Name the property illustrated by the statement. SOLUTION

  7. 2. CD CD ANSWER Reflexive Property of Congruence 3. If Q V, then V Q. ANSWER Symmetric Property of Congruence for Example 2 GUIDED PRACTICE Name the property illustrated by the statement.

  8. Prove this property of midpoints: If you know that Mis the midpoint of AB,prove that ABis two times AMand AMis one half of AB. GIVEN: Mis the midpoint of AB. a. AB = 2 AM PROVE: AM = AB b. 1 2 EXAMPLE 3 Use properties of equality

  9. STATEMENT REASONS 1. 1. Mis the midpoint of AB. Given AMMB 2. 2. Definition of midpoint AM= MB 3. 3. Definition of congruent segments AM + MB = AB 4. 4. Segment Addition Postulate 5. AM + AM = AB 5. Substitution Property of Equality 1 a. 6. 2AM = AB 6. Distributive Property 2 b. 7. AM = AB 7. Division Property of Equality EXAMPLE 3 Use properties of equality

  10. Shopping Mall EXAMPLE 4 Solve a multi-step problem Walking down a hallway at the mall, you notice the music store is halfway between the food court and the shoe store. The shoe store is halfway between the music store and the bookstore. Prove that the distance between the entrances of the food court and music store is the same as the distance between the entrances of the shoe store and bookstore.

  11. Draw and label a diagram. Draw separate diagrams to show mathematical relationships. EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 STEP 2 State what is given and what is to be proved for the situation. STEP 3 Then write a proof.

  12. GIVEN: Bis the midpoint of AC. Cis the midpoint of BD. 4. Transitive Property of Congruence PROVE: AB = CD REASONS STATEMENT 1. 1. Bis the midpoint of AC. Given Cis the midpoint of BD. ABBC 2. 2. Definition of midpoint 3. 3. BCCD Definition of midpoint ABCD 4. 5. AB = CD 5. Definition of congruent segments EXAMPLE 4 Solve a multi-step problem

  13. ANSWER No; Because the critical factor is the midpoint ANSWER Food Court and Bookstore for Example 4 GUIDED PRACTICE 5. In Example 4, does it matter what the actual distances are in order to prove the relationship between ABand CD? Explain. 6. In Example 4, there is a clothing store halfway between the music store and the shoe store. What other two store entrances are the same distance from the entrance of the clothing store?

  14. Homework • P. 116: 1-12

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