– Justifications and Proofs

1. – Justifications and Proofs Measuring, describing, and transforming: these are three major skills in geometry that you have been developing. In this chapter, you will focus on comparing; you will explore ways to determine if two figures have the same shape (called similar) and if they have the same size (congruent).

2. Making logical and convincing arguments that support specific ideas about the shape you are studying is another important skill. In this chapter you will learn how you can document facts to support a conclusion in a flowchart and two-column proof.

3. In this chapter, you will learn: • how to support a mathematical statement using flowcharts and two-column proofs • what a converse of a conditional statement is and how to recognize whether or not the converse is true • how to disprove a statement with a counterexample • about the special relationships between shapes that are similar or congruent • how to determine if triangles are similar and/or congruent

5. 4.1 How Can I Measure An Object? Pg. 3 Units of Measure

6. 4.1 – What Is A Proof? Types of Proof Whenever you buy a new product that needs to be put together, you are given a set of directions. The directions are written in a specific order that must be followed closely in order to get the desired finished product. Sometimes they clarify their directions by explaining why you are completing each step. This is the same idea we use in geometry in proofs.

7. 4.1 – ORDERING STATEMENTS When you write a proof, the statements must be in a specific order, building off of each other. You can't just jump to the end without breaking down each part. To illustrate this, with your group explain how to make a peanut butter and jelly sandwich. Work with your team to include all steps to make sure the sandwich will be made correctly.

8. 4.2 – STATEMENTS AND REASONS When you write a proof in geometry, each statement you make must have a reason to support it. This helps people understand why each statement was listed. This can be done in a flowchart proof or a two-column proof. Examine the two types below. Notice where the statements and reasons are. Also, notice how the statements are in a specific order.

11. 4.3 – REASONS The reasons for certain statements come from definitions, properties, postulates, and theorems. Below are some commonly used reasons.

12. a + c = b + c a - c = b - c ac = bc a/c = b/c ax + ab

13. b can replace a a = a a = c

14. a. Write the reason for each statement

15. Distributive Prop Reflexive Prop Subtraction Prop Transitive Prop Addition Prop Division prop. substitution

16. c. write a reason for each statement. given Distributive Prop Subtraction Prop Addition Prop

17. GIVEN: M is the midpoint of , AM = 6in PROVE: AB = 12in given Def. of midpt given Segment Add. Post. Substitution Substitution Simplify

18. GIVEN: bisects ABC, mABD = 20° PROVE: ABC = 40° 1. 1. Given bisects ABC mABD = mDBC 2. 2. Def. of angle bisector mABD = 20° 3. 3. given 4. Angle Addition Prop. 4. ABC = ABD + DBC 5. Substitution 5. ABC = ABD + ABD 6. Substitution 6. ABC = 20° + 20° 7. Simplify 7. ABC = 40°

19. given Supplementary angles Supplementary angles Transitive prop. subtraction Def. of congruence

20. given alt. interior = supplementary 1 + 3 = 180 given 1 + 4 = 180 Def. of supplementary Transitive prop. 3 = 4 p // r Alt. interior =

21. MATH is a parallelogram given Opp. sides of parallelogram = given Transitive prop. Base  in isos. ∆ = Def. of congruence