# Converse of Midpoint Theorem

The straight line drawn through the midpoint of one side of a triangle parallel to another bisects the third side.

Given: In ∆PQR, S is the midpoint of PQ, and ST is drawn parallel to QR.

To prove: ST bisects PR, i.e., PT = TR.

Construction: Join SU where U is the midpoint of PR.

Proof:

 Statement Reason 1. SU ∥ QR and SU = $$\frac{1}{2}$$QR. 1. By Midpoint Theorem. 2. ST ∥QR and SU ∥ QR. 2. Given and statement 1. 3. ST ∥ SU. 3. Two lines parallel to the same line are parallel themselves. 4. ST and SU are not the same line. 4. From statement 3. 5. T and U are coincident points. 5. From statement 4. 6. T is the midpoint of PR (Proved). 6. From statement 5.

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