Basic Proportionality Theorem

The Basic Proportionality Theorem, also known as the Thales' Theorem, states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

Verification of the Basic Proportionality Theorem

To verify the Basic Proportionality Theorem, we will need a triangle and a parallel line drawn to one of its sides. We will label the triangle as ABC, with side BC as the base. Let the parallel line intersect AB at point D and AC at point E.

Construction:
1. Draw a triangle ABC with side BC as the base.
2. Draw a line parallel to BC passing through point A. Let this line intersect AB at point D and AC at point E.

Verification:
To verify the theorem, we need to show that the ratio of the lengths of the segments AD and DB is equal to the ratio of the lengths of the segments AE and EC.

Proof:
1. Since line DE is parallel to side BC, we can apply the Alternate Interior Angles Theorem to show that angle ADE is equal to angle ACB.
2. Similarly, we can show that angle AED is equal to angle ABC.
3. From the above two statements, we can conclude that triangle ADE is similar to triangle ACB (by Angle-Angle similarity).
4. By the property of similar triangles, the ratio of the lengths of corresponding sides of two similar triangles is equal.
5. Therefore, we have AD/AC = AE/AB.
6. Rearranging the above equation, we get AD/DB = AE/EC, which is the required proportionality for the theorem.

Conclusion:
Thus, we have verified the Basic Proportionality Theorem using a parallel line drawn to one side of a triangle. This theorem is a fundamental concept in geometry, and it helps in solving various problems related to similar triangles.