Geometry 9/5/14 - Bellwork

2.6 Prove Statements about Segments and Angles

Objectives:

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

2. To write proofs using geometric theorems

Premises in Geometric Arguments

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)

Amazing

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

p

q

Amazing

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

p

q

Example 1

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

Example 1

Given: m1 = m3Prove: mDBC = mEBA

Statements Reasons

Example 1

Given: m1 = m3Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3

Example 1

Given: m1 = m3Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

Example 1

Given: m1 = m3Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

2. m1 + m2 = m3 + m2

Example 1

Given: m1 = m3Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

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

Example 1

Given: m1 = m3Prove: mDBC = mEBA

Statements Reasons

1. m1 = m3 1.Given

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

3. m1 + m2 = mDBC

Example 1

Given: m1 = m3Prove: 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 1

Given: m1 = m3Prove: 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 1

Given: m1 = m3Prove: 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 1

Given: m1 = m3Prove: 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 1

Given: m1 = m3Prove: 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 Proof

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

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

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

Generic Two-Column Proof

Given: ____________

Prove: ____________

Statements Reasons

1. 1.

2. 2.

3. 3.

Insert illustration here

Theorems of Congruence

Congruence of SegmentsSegment congruence is reflexive, symmetric,

and transitive.

Congruence of AnglesAngle congruence is reflexive, symmetric, and

transitive.

Theorems of Congruence

Assignment

• Textbook PP. 116-119: 3,4, 10-13, 16, 21, 22