in triangle ABC angle B =90 and BD PERPENDICULAR AC proof that angle ABD =angle ACB

To prove that angle ABD is equal to angle ACB, we can use the fact that in a right-angled triangle, the two acute angles are complementary. Therefore, the sum of the two acute angles in a right-angled triangle is always 90 degrees.

Here's a step-by-step explanation of how you can prove that angle ABD is equal to angle ACB:

Step 1: Given triangle ABC with angle B = 90 degrees and BD perpendicular to AC.

Step 2: Since angle B is 90 degrees, triangle ABC is a right-angled triangle.

Step 3: In a right-angled triangle, angle A + angle B + angle C = 180 degrees (sum of angles in a triangle).

Step 4: Since angle B is 90 degrees, angle A + 90 degrees + angle C = 180 degrees.

Step 5: Subtract 90 degrees from both sides of the equation to isolate angle A: angle A + angle C = 90 degrees.

Step 6: In triangle ABC, angle A and angle C are both acute angles (less than 90 degrees).

Step 7: Due to the complementary property, the sum of two acute angles in a triangle is 90 degrees.

Step 8: Therefore, angle ABD (which is equal to angle A) is equal to angle ACB (which is equal to angle C).

Hence, we have proven that angle ABD is equal to angle ACB in triangle ABC with angle B = 90 degrees and BD perpendicular to AC.

Note that you have two right triangles, with a common angle. So, they are similar, with all angles equal.