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Theorem – 1. A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

Given : - PQ is drawn parallel to BC meeting AB at P and AC at Q.

To Prove : AP : PB = AQ : QC

Construction : - BQ and CD are joined.

Statement	Reason
1.	1.
2.	2.
3.	3.are on the same base PQ and between the same parallels PQ and BC.
4.	4. Statements (1), (2) and (3)

Theorem – 2. If a straight line meets two sides of a triangle proportionally it is parallel to the third side.

Given: - A line PQ meets AB at P and AC at Q such that AP : PB = AQ : QC

To Prove: - PQ is parallel to BC

Proof: - If PQ is not parallel to BC, let PR is parallel to BC meeting AC at R.

Statement	Reason
1.	1. Given
2. PR || BC	2. Assumption
3.	3. Statement (2) by B.P.T.
4.	4. Statement (1) and (3)
5.	5. Statement (4)
6.	6. Statement (5)
7.	7. Statement (6)
8. QC = RC	8. Statement (7)
9. Q and R are same line	9. Statement (8)
10. PQ || BC QED.	10. Statement (9)