in triangle abc angle c is 90.write a indirect proof that angle a is less than 90.
To prove that angle A in triangle ABC is less than 90 degrees using an indirect proof, we need to assume the opposite and show that it leads to a contradiction.
Assumption: Assume that angle A is greater than or equal to 90 degrees.
Proof:
1. In triangle ABC, angle C is given as 90 degrees.
2. Assume that angle A is greater than or equal to 90 degrees.
3. Since the sum of the angles in a triangle is always 180 degrees, angle B must be less than 90 degrees.
4. As angle A is assumed to be greater than or equal to 90 degrees, the sum of angles A and B would be greater than 180 degrees.
5. This leads to a contradiction as the sum of the angles in a triangle cannot be greater than 180 degrees.
6. Therefore, our assumption (that angle A is greater than or equal to 90 degrees) must be false.
7. Consequently, angle A in triangle ABC is less than 90 degrees.
This indirect proof demonstrates that angle A in triangle ABC must be less than 90 degrees.