Lesson 6.1 Congruent Segments pp. 208-214

Objectives: 1. To develop skill in deductive proofs. 2. To prove theorems involving points, segments, and lines.

Proving theorems can be an exciting, creative skill to learn Proving theorems can be an exciting, creative skill to learn. To become successful at this new skill, you must attempt to prove new theorems and accept the challenge that each one presents. There is no set way of proving theorems. Each one presents some challenge or variation. However, as you prove more and more theorems, you will improve your ability to think.

There are some general hints that you should keep in mind when you are attempting to prove a theorem. 1. Identify the premise (or given information). 2. Identify the conclusion you are trying to obtain. 3. Draw a picture to make sure you understand the theorem.

There are some general hints that you should keep in mind when you are attempting to prove a theorem. 4. Write down any definitions, postulates, or previously proved theorems that relate to the theorems you are trying to prove. 5. Work backwards, if necessary, starting with the conclusion.

Remember, to prove a theorem, you can use definitions, postulates, and previously proved theorems. Then use deductive reasoning to derive the desired conclusion.

There are two types of proofs There are two types of proofs. The deductive type of reasoning called the Law of Deduction and the two-column proof. We will be doing mainly two-column proofs.

EXAMPLE 1 Prove: A line and a point not on that line are contained in one and only one plane. (This is Theorem 1.2)

EXAMPLE 2 A line and a point not on that line are contained in one and only one plane. Given: a line l, a point K not on line l Prove: a plane p containing l and K

STATEMENTS REASONS 1. There is a line l, and 1. Premise (Given) a point K, not on l 2. Line l contains two 2. Expansion Post. points, X and Y 3. X, Y, and K are 3. Def. of noncollinear noncollinear points 4. Points X, Y, and K 4. Plane Postulate determine exactly one plane p 5. Line l and point K 5. Flat Plane Postulate are in plane p

Theorem 6.1 Congruent Segment Bisector Theorem. If two congruent segments are bisected, then the four resulting segments are congruent.

Given: Two congruent segments: XY  KL Prove: If A and B are the midpoints of the congruent segments XY and KL, then AY  XA  KB  BL X Y A K L B

Theorem 6.2 Segment congruence is an equivalence relation.

Homework pp. 212-214 The reasons for 1-17 are either “Given” or a specific definition, property, or postulate.

►B. Exercises 19. Given: LM = PQ Prove: LP = MQ L M X P Q

►B. Exercises 19. Statements Reasons 1. LM = PQ 1. Given 2. MP = MP 2. Reflexive prop. 3. LM + MP = 3. Addition prop. MP + PQ 4. LM + MP = LP; 4. Definition of MP + PQ = MQ betweenness 5. LP = MQ 5. Substitution

►B. Exercises 21. Given: AB  CD Prove: CD  AB

►B. Exercises 21. Statements Reasons 1. AB  CD 1. Given 2. AB = CD 2. Def. of cong. 3. CD = AB 3. Symmetric prop. 4. CD  AB 4. Def. of cong.

■ Cumulative Review State the first five postulates. 26. Expansion Postulate

■ Cumulative Review State the first five postulates. 27. Line Postulate

■ Cumulative Review State the first five postulates. 28. Plane Postulate

■ Cumulative Review State the first five postulates. 29. Flat Plane Postulate

■ Cumulative Review State the first five postulates. 30. Plane Intersection Postulate