2.6 prove statements about segments and angles objectives: 1.to understand the role of proof in a...

2.6 Prove Statements about Segments 2.6 Prove Statements about Segments and Anglesand Angles

Objectives:

1.To understand the role of proof in a deductive system

2.To write proofs using geometric theorems

Premises in Geometric Premises in Geometric ArgumentsArguments

The following is a list of premises that can be used in geometric proofs:

1.Definitions and undefined terms

2.Properties of algebra, equality, and congruence

3.Postulates of geometry

4.Previously accepted or proven geometric conjectures (theorems)

AmazingAmazing

Usually we have to prove a conditional statement. Think of this proof as a maze, where the hypothesishypothesis is the starting point and the conclusionconclusion is the ending.

pp

qq

AmazingAmazing

Your job in constructing the proof is to link pp to qq using definitions, properties, postulates, and previously proven theorems.

pp

qq

Example 1Example 1

Construct a two-column proof of:If m1 = m3, then mDBC = mEBA.

Example 1Example 1

Given: m1 = m3

Prove: mDBC = mEBA

Statements Reasons

Example 1Example 1

Given: m1 = m3

Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3

Example 1Example 1

Given: m1 = m3

Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

Example 1Example 1

Given: m1 = m3

Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

2. m1 + m2 = m3 + m2

Example 1Example 1

Given: m1 = m3

Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

2. m1 + m2 = m3 + m2 2.Addition Property

Example 1Example 1

Given: m1 = m3

Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

2. m1 + m2 = m3 + m2 2.Addition Property

3. m1 + m2 = mDBC

Example 1Example 1

Given: m1 = m3

Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

2. m1 + m2 = m3 + m2 2.Addition Property

3. m1 + m2 = mDBC 3.Angle Addition Postulate

Example 1Example 1

Given: m1 = m3

Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

2. m1 + m2 = m3 + m2 2.Addition Property

3. m1 + m2 = mDBC 3.Angle Addition Postulate

4. m3 + m2 = mEBA

Example 1Example 1

Given: m1 = m3

Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

2. m1 + m2 = m3 + m2 2.Addition Property

3. m1 + m2 = mDBC 3.Angle Addition Postulate

4. m3 + m2 = mEBA 4.Angle Addition Postulate

Example 1Example 1

Given: m1 = m3

Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

2. m1 + m2 = m3 + m2 2.Addition Property

3. m1 + m2 = mDBC 3.Angle Addition Postulate

4. m3 + m2 = mEBA 4.Angle Addition Postulate

5. mDBC = mEBA

Example 1Example 1

Given: m1 = m3

Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

2. m1 + m2 = m3 + m2 2.Addition Property

3. m1 + m2 = mDBC 3.Angle Addition Postulate

4. m3 + m2 = mEBA 4.Angle Addition Postulate

5. mDBC = mEBA 5.Substitution Property

Two-Column ProofTwo-Column Proof

Notice in a two-columntwo-column proof, you first list what you are givengiven (hypothesis) and what you are to proveprove (conclusion).

The proof itself resembles a T-chart with numbered statementsstatements on the left and numbered reasonsreasons for those statements on the right.

Before you begin your proof, it is wise to try to map out the maze from pp to qq.

Generic Two-Column ProofGeneric Two-Column Proof

Given: ____________

Prove: ____________

Statements Reasons

1. 1.

2. 2.

3. 3.

Insert illustration here

Properties of EqualityProperties of Equality

Maybe you remember these from Algebra.

Reflexive Property of Reflexive Property of EqualityEquality

For any real number a, a = a.

Symmetric Property of Symmetric Property of EqualityEquality

For any real numbers a and b, if a = b, then b = a.

Transitive Property of Transitive Property of EqualityEquality

For any real numbers a, b, and c, if a = b and b = c, then a = c.

Theorems of CongruenceTheorems of Congruence

Congruence of SegmentsCongruence of SegmentsSegment congruence is reflexive, symmetric,

and transitive.

Congruence of AnglesCongruence of AnglesAngle congruence is reflexive, symmetric, and

transitive.

Theorems of CongruenceTheorems of Congruence

Given:Given:M is the midpoint of AB Prove: Prove: AB is twice AM and AM is one half of AB.

– M is the midpoint of AB

– AM MB≅– AM=MB– AM+MB=AB– AM+AM=AB– 2AM=AB– AM= AB/2

• Given• Definition of midpoint• Def of congruence• Segment Add Pos• Substitution• Simplify• Division prop of equal

AssignmentAssignment

• P. 116-119: 3,4, 10-13, 16, 21, 22

• Finish for homework