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 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: ____________
Insert illustration here
Statements
Reasons
1.
1.
2.
2.
3.
3.
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.
Assignment
• Textbook PP. 116-119:
3,4, 10-13, 16, 21, 22