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.

$Converse of Midpoint Theorem Proof$

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

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

$Converse of Midpoint Theorem$

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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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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