Triangle ABC is right-angled at B and A is perpendicular

To prove that triangle ABC is right-angled at B and that A is perpendicular, we need to show that angle B is a right angle. This can be done by demonstrating that side AB is perpendicular to side BC.

Proof:

1. Given: Triangle ABC, with angle B as a right angle.

2. To prove: A is perpendicular to BC.

3. Let's assume that A is not perpendicular to BC.

4. In a triangle, the sum of the angles is always 180 degrees. Since angle B is a right angle (90 degrees), the sum of angles A and C must be 90 degrees as well.

5. If A is not perpendicular to BC, then angle A must be acute (less than 90 degrees). Let's assume angle A is x degrees.

6. If angle A is x degrees, then angle C must be (90 - x) degrees to maintain the sum of angles A and C as 90 degrees.

7. Now, we have angle A as x degrees and angle C as (90 - x) degrees.

8. According to the given information, angle B is a right angle (90 degrees).

9. To satisfy the sum of angles in a triangle, the sum of angles A, B, and C must be 180 degrees.

10. Therefore, x + 90 + (90 - x) = 180.

11. Simplifying the equation, we get 180 = 180, which is always true.

12. This means that our assumption in step 3 is incorrect, and A must be perpendicular to BC.

13. Hence, triangle ABC is right-angled at B, and A is perpendicular to BC.

Conclusion:

Therefore, we have proved that triangle ABC is right-angled at B, and A is perpendicular to BC.