PROVING STATEMENTS ABOUT SEGMENTS
About "Proving statements about segments"
Proving statements about segments :
A true statement that follows as a result of other statements is called a theorem. All theorems must be proved. We can prove a theorem using a two-column proof. A two-column proof has numbered statements and reasons that show the logical order of an argument.
Theorem : Properties of Segment Congruence
REFLEXIVE
SYMMETRIC
TRANSITIVE
For any segment AB, AB ≅ AB
If AB ≅ CD, then CD ≅ AB
If AB ≅ CD, and CD ≅ EF, then AB ≅ EF
A proof which is written in paragraph form is called as paragraph proof.
Here is a paragraph proof for the Symmetric Property of Segment Congruence.
Paragraph Proof :
We 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.
Proving statements about segments - Examples
Example 1 :
In the diagram given above, PQ ≅ XY. Prove XY ≅ PQ
Solution :
PQ ≅ XY
PQ = XY
XY = PQ
XY ≅ PQ
Given
Definition of congruence statements
Symmetric property of equality
Definition of congruence segments
Example 2 :
Use the diagram and the given information to complete the missing steps and reasons in the proof.
Given : LK = 5, JK = 5, JK ≅ JL
Prove : LK ≅ JL
A
B
LK = JK
LK ≅ JK
JK ≅ JL
D
Given
Given
Transitive property of equality
C
Given
Transitive property of congruence
Solution :
A. LK = 5
B. JK = 5
C. Definition of congruence segments
D. LK ≅ JL
Example 3 :
In the diagram given below, Q is the midpoint of PR.
Show that PQ and QR are each equal to 1/2 ⋅ PR.
Solution :
Given : Q is the midpoint of PR
Prove : PQ = 1/2 ⋅ PR and QR = 1/2 ⋅ PR
Q is the midpoint of PR
Given
PQ = QR
PQ + QR = PR
PQ + PQ = PR
2 ⋅ PQ = PR
PQ = 1/2 ⋅ PR
QR = 1/2 ⋅ PR
Definition of midpoint
Segment Addition Postulate
Substitution property of equality
Distributive property
Division property of equality
Substitution property of equality
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