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Reasoning with Properties
of Algebra &
Proving Statements About
Segments
CCSS: G-CO.12
A proof is an argument that uses logic, definitions,
properties, and previously proven statements to show
that a conclusion is true.
An important part of writing a proof is giving
justifications to show that every step is valid.
You learned in Chapter 1 that segments with
equal lengths are congruent and that angles with
equal measures are congruent. So the Reflexive,
Symmetric, and Transitive Properties of Equality
have corresponding properties of congruence.
Theorem
• A true statement that follows as a result of
other true statements.
• All theorems MUST be proved!
2-Column Proof
• Numbered statements and corresponding
reasons in a logical order organized into 2
columns.
statements
reasons
1.
1.
2.
2.
3.
3.
etc.
Properties of Segment and
Angle Congruence
Properties of Segment and
Angle Equality
• *We use these to prove properties of
congruence
Theorem 2.1- Properties of Segment
Congruence
• Segment congruence is reflexive,
symmetric, & transitive.
For any AB, AB  AB.
If AB  BC and BC  CD, then AB  CD.
If AB  BC, then BC  AB.
Proof of symmetric part of thm.
2.1
Statements
1. AB  BC
2. AB = BC
3. BC = AB
4. BC  AB
Reasons
1.
2.
3.
4.
Given
Defn. of congruent segs.
Symmetric prop of =
Defn. of congruent segs.
Paragraph Proof
• Same argument as a 2-column proof, but
each step is written as a sentence; therefore
forming a paragraph.
P
X
Y
Q
• You are given that line segment PQ is congruent
with line segment XY. By the definition of congruent
segments, PQ=XY. By the symmetric property of
equality XY = PQ. Therefore, by the definition of
congruent segments, it follows that line segment XY
congruent to line segment PQ.
Ex: Given: LK = 5
JK = 5
JK ≅ JL
Prove: LK≅ JL
Statements
1.
2.
3. LK = JK
4. LK ≅ JK
5. JK ≅ JL
6.
K
Reasons
1. Given
2. Given
3. Trans. Prop of =
4.
5. Given
6. Trans. Prop of
congruence
Ex: Given: Q is the midpoint of PR.
PR
Prove: PQ and QR =
2
1.
2.
3.
4.
5.
6.
Statements
Q is midpt of PR
PQ=QR
PQ+QR=PR
QR+QR=PR
2QR=PR
QR= PR
2
PR
7. PQ=
2
1.
2.
3.
4.
5.
6.
Reasons
Given
Defn. of midpt
Seg + post
Subst. prop of =
Simplify
Division prop of =
7. Subst. prop
Classwork
• Page 105: problems 6-11 all
• We’ll go over the answers after