LESSON 2.6
Prove Statements about
Segments and Angles
Objectives:
To understand the role of proof in a deductive
system
2. To write proofs using geometric theorems
1.
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 m1 = m3, then mDBC = mEBA.
Example 1
Given: m1 = m3
Prove: mDBC = mEBA
Statements
Reasons
Example 1
Given: m1 = m3
Prove: mDBC = mEBA
Statements
1. m1 = m3
Reasons
Example 1
Given: m1 = m3
Prove: mDBC = mEBA
Statements
Reasons
1. m1 = m3
1.Given
Example 1
Given: m1 = m3
Prove: mDBC = mEBA
Statements
Reasons
1. m1 = m3
1.Given
2. m1 + m2 = m3 + m2
Example 1
Given: m1 = m3
Prove: mDBC = mEBA
Statements
Reasons
1. m1 = m3
1.Given
2. m1 + m2 = m3 + m2
2.Addition Property
Example 1
Given: m1 = m3
Prove: mDBC = mEBA
Statements
Reasons
1. m1 = m3
1.Given
2. m1 + m2 = m3 + m2
2.Addition Property
3. m1 + m2 = mDBC
Example 1
Given: m1 = m3
Prove: mDBC = mEBA
Statements
Reasons
1. m1 = m3
1.Given
2. m1 + m2 = m3 + m2
2.Addition Property
3. m1 + m2 = mDBC
3.Angle Addition Postulate
Example 1
Given: m1 = m3
Prove: mDBC = mEBA
Statements
Reasons
1. m1 = m3
1.Given
2. m1 + m2 = m3 + m2
2.Addition Property
3. m1 + m2 = mDBC
3.Angle Addition Postulate
4. m3 + m2 = mEBA
Example 1
Given: m1 = m3
Prove: mDBC = mEBA
Statements
Reasons
1. m1 = m3
1.Given
2. m1 + m2 = m3 + m2
2.Addition Property
3. m1 + m2 = mDBC
3.Angle Addition Postulate
4. m3 + m2 = mEBA
4.Angle Addition Postulate
Example 1
Given: m1 = m3
Prove: mDBC = mEBA
Statements
Reasons
1. m1 = m3
1.Given
2. m1 + m2 = m3 + m2
2.Addition Property
3. m1 + m2 = mDBC
3.Angle Addition Postulate
4. m3 + m2 = mEBA
4.Angle Addition Postulate
5. mDBC = mEBA
Example 1
Given: m1 = m3
Prove: mDBC = mEBA
Statements
Reasons
1. m1 = m3
1.Given
2. m1 + m2 = m3 + m2
2.Addition Property
3. m1 + m2 = mDBC
3.Angle Addition Postulate
4. m3 + m2 = mEBA
4.Angle Addition Postulate
5. mDBC = mEBA
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: ____________
IMAGE
Statements
Reasons
1.
1.
2.
2.
3.
3.
Properties of Equality
Maybe you remember these from Algebra.
Reflexive Property of
Equality
For any real number a,
= a.
Symmetric Property of
Equality
For any real numbers a and
b, if a = b, then b = a.
Transitive Property of
Equality
For any real numbers a, b,
and c, if a = b and b = c, then
a = c.
a
Theorems of Congruence
Congruence of Segments
Segment congruence is reflexive, symmetric, and transitive.
Theorems of Congruence
Congruence of Angles
Angle congruence is reflexive, symmetric, and transitive.
Given: M is the midpoint of AB
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
Concept Summary
 Writing a Two-Column Proof
 Turn to Page 114 in textbook
Assignment
 P. 116-119: 3,4, 10-13,
16, 21, 22
 Finish for homework