Chapter 2 Review Reasoning and Proof

Theorems are accepted as true. Vocab:State whether each sentence is true or false. If false, replace the incorrect word to make it true. Theorems are accepted as true. In a two-column proof, the properties that justify each step are called reasons. The Reflexive Property of Equality states that for any number a, a=a. The contrapositive of a conditional is formed by switching the hypothesis and conclusion of the original statement.

X, Y and Z are collinear and XY = YZ Inductive Reasoning and Conjecture: Make a conjecture based on the given information. Draw a figure if necessary. and are supplementary X, Y and Z are collinear and XY = YZ

The intersection of two planes is a line. Conditional Statements: a)Write each statement in if-then form, then identify the hypothesis and conclusion. b)Write the converse of the statement. If it is false give a counterexample. March has 31 days. The intersection of two planes is a line.

The intersection of two different lines is a line. Postulates:Determine if each statement is always, sometimes, or never true. Explain.(use a postulate or theorem to explain when possible) The intersection of two different lines is a line. If P is the midpoint of XY, then XP=PY. Four points determine exactly one plane. If MX = MY, then M is the midpoint of XY.

If 3 = CD, and CD=XY, then 3 = XY Algebraic Properties and Proof: State the property that justifies each statement. If 3(x +2)=6, then 3x+6= 6 If 10x = 20, then x = 2 If AB + 20 = 45, then AB = 25 If 3 = CD, and CD=XY, then 3 = XY

Algebraic Proof: Write a two-column proof. If AC = AB, AC = 4x+1 and AB = 6x-13, then x = 7.

Proving Segment Relationship: Write a two-column proof Given: AB=CD Prove: AC = BD A B C D

Angle Relationship: Find the measure of the numbered angle 2 3 73 1 C B E

Proving Angle Relationships: Write a two-column proof Given: Prove: E C A B

ANSWERS Slide 1: 1. Postulates are accepted as true. 2. True 3. True 4. The converse of a conditional is formed by switching the hypothesis and conclusion of the original statement.

ANSWERS Slide 2: 1. Angles A and B are a linear pair. 2. Y is the midpoint of XZ.

ANSWERS Slide 3: 1. a) If it is March, then the month has 31 days. - hyp- It is March concl.- the month has 31 days b) If the month has 31 days, then it is March. -false= There are 31 days in the month of July a) If two planes intersect, then the intersection is a line. -hyp- two planes intersect concl.- it is a line b) If the intersection is a line, then two planes intersected. -true

ANSWERS Slide 4: 1. Never, because the intersection of two lines is exactly one point. 2. Always, that is the definition of a midpoint. 3. Sometimes, because 3 points determine exactly one plane, the fourth point could be on a second plane. 4. Sometimes, X and Y could be collinear so the lines with X and Y would be intersecting at M and therefore M would not be a midpoint.

ANSWERS Slide 5: 1. Distributive Property 2. Division Property 3. Subtraction Property 4. Transitive Property

ANSWERS Given: AC = AB, AC = 4x + 1, AB = 6x – 13 Prove: x = 7 AC = AB, AC = 4x + 1, 1. Given AB = 6x – 13 4x + 1 = 6x – 13 2. Substitution Prop 4x + 1 – 4x= 6x – 13 – 4x 3. Subtraction Prop 1 = 2x – 13 4. Substitution Prop 1 + 13 = 2x – 13 + 13 5. Addition Prop 14 = 2x 6. Substitution Prop 14/2 = 2x/2 7. Division Prop 7 = x 8. Substitution Prop x = 7 9. Symmetric Prop

ANSWERS Slide 7: AB = CD 1. Given AB + BC = AC 2. Segment Addition Postulate BC + CD = BD CD + BC = AC 3. Substitution Prop AC = BD 4. Substitution Prop Given: AB = CD Prove: AC = BD A B C D

ANSWERS Slide 8: 1. angle 1 + 73 = 90 angle 1 = 17 2. perpendicular lines =rt angles angle2 = 90 3. angles 2 and 3 are vertical angles. Vertical angles are congruent. angle 3= 90

ANSWERS Given: Prove: 3. ABE + EBF = ABF 3. Angle Addition Postulate C E Given: Prove: A B ABF is congruent to EBC 1. Given ABF = EBC 2. Definition of congruent angles 3. ABE + EBF = ABF 3. Angle Addition Postulate EBF + FBC = EBC 4. ABE + EBF = EBF + FBC 4. Substitution Prop 5. ABE + EBF – EBF = 5. Subtraction Prop EBF + FBC - EBF 6. ABE = FBC 6. Substitution Prop *Angle symbols should be used in front of each set of 3 letters