2.6 prove statements about segments and angles objectives: 1.to understand the role of proof in a...
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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
AmazingAmazing
Your job in constructing the proof is to link pp to qq using definitions, properties, postulates, and previously proven theorems.
pp
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