Theorem Of Parallelograms

Converse Mid-Point Theorem

The line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.

Converse of mid-point theorem
The line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.
Given:A triangle ABC,E is mid-point of AB. Line l drawn through E and parallel to BC meeting AC at F.
To prove. AF=FC
Construction: Through C, draw a line m parallel BA to meet line l at D.

Proof:
Statements
Reasons
1. EBCD is a parallelogram
1. EF∥BC (given), BA∥CD (const.).
2. BE=CD
2. Opp.sides of a parallelogram are equal.
3. EA=BE
3. E is mid-point of AB(given).
4. EA=CD
4. From 2 and 3.
In △AEF and △CDF
5. ∠EAF=∠DCF
5. Alt. ∠s, CD ∥BA and AC is a transversal.
6. ∠EFA=∠DFC 6. Vert.opp.∠s.
7. EA=CD
7. From 4
8. △AEF≌△CDF 8. AAS rule of ccongruency
9. AF=FC
9. c.p.c.t.
Hence, F is mid-point of AC.