2.5 Proving Statements about Segments
Objectives:
• To justify statements about congruent
segments.
• Write reasons for steps in a proof.
Definitions
Theorem A true statement that is proven as a result of
other true statements.
Two-column proof Type of proof that has numbered statements
and reasons that show the logical order of
an argument. Also called a direct proof.
Components of 2-column proofs
Some sort of
• Given: “hypothesis”
diagram will
be here.
• Prove: “conclusion”
Statements
Reasons
1. Given information (usually) 1. Given (usually)
2. next logical step
2. geometry “reason” to justify
3. cont. deductive reasoning 3. cont. geometry “reasons”
4. until you prove “conclusion” 4. final geometry “reason”
Properties of Segment Congruence
• Thm 2.1
• Segment congruence is reflexive, symmetric,
and transitive.
• Examples:
• Reflexive: For any segment AB, AB ≅ AB
• Symmetric: If AB ≅ CD, then CD ≅ AB
• Transitive:
If AB ≅ CD, and CD ≅ EF, then AB ≅ EF
Ex1: Write a 2-column proof for the
symmetric property of congruence.
Given: PQ ≅ XY
Prove: XY ≅ PQ
Statements
1.
2.
3.
4.
PQ ≅ XY
PQ = XY
XY = PQ
XY ≅ PQ
Diagram not
needed this time.
Reasons:
1. Given
2. Def. of segments
3. Symmetric Property
4. Def. of
segments
Another type of proof…
• Paragraph proof: A proof written in
paragraph form.
• Here is an example of the symmetric
property of congruence in paragraph form.
• You are given that PQ ≅ 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 XY ≅ PQ.
Ex 2: Using Congruence
• Use the diagram and the given information
to complete the missing steps and reasons in
the proof.
K
• GIVEN: LK = 5, JK = 5, JK ≅ JL
• PROVE: LK ≅ JL
J
L
Statements:
1.
2.
3.
4.
5.
________________
LK = JK
LK ≅ JK
JK ≅ JL
________________
Reasons
1.
2.
3.
4.
5.
Given
_________________
_________________
Given
Transitive Property
Ex3: Using Segment Relationships
• In the diagram, Q is the midpoint of PR.
Show that PQ and QR are equal to ½ PR.
• Given: Q is the midpoint of PR.
• Prove: PQ = ½ PR and QR = ½ PR.
R
Q
P
I don’t really have room for this proof here so I am going
to finish it on the next slide. However, proofs are always
“package deals” with the given, prove, and the diagram.
You must write it all in your assignments.
Statements
1.
2.
3.
4.
5.
6.
7.
Q is the midpoint of PR.
PQ = QR
PQ + QR = PR
PQ + PQ = PR
2 ∙ PQ = PR
PQ = ½ PR
QR = ½ PR
Reasons
1.
2.
3.
4.
5.
6.
7.
Given
Def. of midpoint
Segment Addition
Substitution
Simplify
Division
Substitution
Assignment
• HW 2.5
p. 104 #1-11, 16-19,
28-40 even