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

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.

Example 1 :

In the diagram given above, PQ ≅ XY. Prove XY ≅ PQ.

Solution :

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

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

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