– 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).
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.
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
4.1
How Can I Measure An Object?
Pg. 3Units of Measure
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.
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.
4.2 – STATEMENTS AND REASONSWhen 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.
4.3 – REASONSThe reasons for certain statements come from definitions, properties, postulates, and theorems. Below are some commonly used reasons.
Name Property of Equality
Addition Property
If a = b, then
Subtraction Property
If a = b, then
Multiplication Property
If a = b, then
Division Property
If a = b, then
Distribution Property
If a(x + b), then
a + c = b + c
a - c = b - c
ac = bc
a/c = b/c
ax + ab
Substitution Property
If a = b, then
Reflexive Property
Transitive Property
If a = b and b = c, then
b can replace a
a = a
a = c
a. Write the reason for each statement
Statement Reason
If 4(x + 7), then 4x + 28
If 2x + 5 = 9, then 2x = 4
If x – 7 = 2, then x = 9
If 4x = 12, then x = 3
If a = 20, then 5a = 5(20)
If and , then X Y Y Z X Z
BD BD
Distributive Prop
Reflexive Prop
Subtraction Prop
Transitive Prop
Addition Prop
Division prop.
substitution
Statements Reasons
1. 5(x – 3) = 4(x + 2) 1. __________________
2. 5x – 15 = 4x + 8 2. _________________
3. x – 15 = 8 3. __________________
4. x = 23 4. __________________
c. write a reason for each statement.
given
Distributive Prop
Subtraction Prop
Addition Prop
GIVEN: M is the midpoint of , AM = 6inPROVE: AB = 12in
AB
Statements Reasons
1. M is the midpoint of 1. ___________________
2. AM = MB 2. ___________________
3. AM = 6in 3. ___________________
4. AM + MB = AB 4. ___________________
5. AM + AM = AB 5. ___________________
6. 6 + 6 = AB 6. ___________________
7. AB = 12in 7. ___________________
AB
given
Def. of midptgivenSegment Add. Post.SubstitutionSubstitutionSimplify
GIVEN: bisects ABC, mABD = 20°PROVE: ABC = 40°
BD
Statements Reasons
1. 1. bisects ABC Given
2. 2.mABD = mDBC Def. of angle bisector
3. 3.mABD = 20° given
4. 4.ABC = ABD + DBC Angle Addition Prop.
BD
5. 5.ABC = ABD + ABD Substitution
6. 6.ABC = 20° + 20° Substitution
7. 7.ABC = 40° Simplify
Statements Reasons
1. intersects 1. _______________________
2. 2. _______________________
3. 3. _______________________
4. 4. _______________________
5. 5. _______________________
6. 6. _______________________
CDAB given
Supplementary angles
Supplementary angles
Transitive prop.
subtraction
Def. of congruence
given
alt. interior =supplementary
1 + 3 = 180given
1 + 4 = 180 Def. of supplementaryTransitive prop.
3 = 4p // r Alt. interior =
MATH is a parallelogram givenOpp. sides of parallelogram =
given
MH MN Transitive prop.
Base in isos. ∆ =
Def. of congruence
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