Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To...
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Transcript of Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To...
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