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Warm Up

Prove Statements about Segments and Angles. Warm Up. Lesson Presentation. Lesson Quiz. 1. If m 1 = m 3, then m 3 = ?. m 1. ANSWER. 2. If AB = CD and CD = TU, then ?. AB = TU. ANSWER. Warm-Up. Use a property of equality to complete the statement.

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Warm Up

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  1. Prove Statements about Segments and Angles Warm Up Lesson Presentation Lesson Quiz

  2. 1.Ifm 1 = m 3, thenm 3 = ? . m 1 ANSWER 2.IfAB = CD andCD = TU, then ? . AB = TU ANSWER Warm-Up Use a property of equality to complete the statement.

  3. 3.IfRS = WX, then ? + AB = ? + AB. RS; WX ANSWER 4. Ifm EFG = 28º andm GFH = 62º,then ? + 62º = m EFG + m GFH. 28º ANSWER Warm-Up Use a property of equality to complete the statement.

  4. 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 STATEMENTS 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 for the situation in Example 4 from Lesson 2.5.

  5. 1. Four steps of a proof are shown. Give the reasons for the last two steps. REASONS STATEMENT 1. 1. AC = AB + AB Given 2. 2. AB + BC = AC Segment Addition Postulate 3. AB + AB = AB + BC 4. AB = BC Guided Practice GIVEN :AC = AB + AB PROVE :AB = BC ANSWER 3. Transitive Property of Equality 4. Subtraction Property of Equality

  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 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 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= 2AM PROVE: AM=AB b. 1 2 Example 3

  9. STATEMENTS 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 SOLUTION

  10. 4. WHAT IF? Look back at Example 3. What would be different if you were proving that AB = 2 MB and that MB = ABinstead? ANSWER In steps 5,6, and 7,AM would be replaced by MB. 1 2 Guided Practice

  11. Shopping Mall 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. Example 4

  12. STEP 1 Draw and label a diagram. STEP 2 Draw separate diagrams to show mathematical relationships. STEP 3 State what is given and what is to be proved for the situation. Then write a proof. Example 4 SOLUTION

  13. GIVEN: Bis the midpoint of AC. Cis the midpoint of BD. 4. Transitive Property of Congruence PROVE: AB = CD REASONS STATEMENTS 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

  14. ANSWER No; the critical factor is the midpoint. ANSWER Food Court and Bookstore 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?

  15. ? ? ? ? Lesson Quiz Copy and complete the proof. 1. GIVEN: MA = TH GIVEN:MA = TH PROVE:MT = AH PROVE: MT = AH REASONS STATEMENTS 1. 1. MA = TH Given Reflexive Prop. Of Eq. 2. 2. AT = AT Addition Prop. Of Eq. 3. 3. MA + AT = AT + TH 4. Segment Add. Post. MA + AT = MT; AT + TH =AH 4. ? Substitution Prop. Of Eq. 5. 5. MT = AH

  16. o GIVEN: m 1 + m 2 = 90 ; m 1 = 59 o o PROVE: m 2 = 31 ANSWER Statements (Reasons) o o 1. (Given) m 1 + m 2 = 90 ; m 1 = 59 o 2. m 2 = 90 – m 1 (Subtraction Prop. Of Eq.) o o 3. m 2 = 90 –59 (Substitution Prop. Of Eq.) o 4. m 2 = 31 (Simplify) Lesson Quiz Use the given information to prove the statement. 2.

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