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THEOREM - 3. If a perpendicular drawn from the right angle to the hypotenuse of a right – angled triangle, the two triangle so formed are similar to the original triangle and to one another.

Given : - ABC is a right angle triangle right angled at A. AD is drawn perpendicular to the hypotenuse BC.

To Prove : - are similar to one another.

Proof : -

Statement	Reason
1.	1. (i) Common angles (ii) Each being a right angle.
2.	2. Statement (1.) by AAA.
3.	3. (i) Common angle (ii) Each being a right angle.
4.	4. Statement (3) by AAA
5.	5. Statements (2) and (4)
6. QED	6. Statements (2), (4) and (5)

THEOREM – 4. The areas of similar triangles are in the ratio of the squares on their corresponding sides.

Given : -

To Prove : -

Construction : AM and DN are drawn perpendiculars to BC and EF respectively.

Proof : -

Statement	Reason
1.	1. Given
2.	2. Statement (1)
3.	3.
4.	4. Statement (3)
5.	5. Statement (2) and (4)
6.	6. Statement (5)
7.	7. Statement (1)
8.	8. Statement (6) and (7)