In a triangle ABC,right angled at B, BD is drawn perpendicular to AC. prove that:

(i) angle ABD = angle c
(ii) angle CBD = angle A

(i) ABD is the complement of A.

C is also the complement of A
Therefore ABD = C

(ii)
CBD is the complement of C
A is the complement of C.
Therefore CBD = A

It will be easier to see if you draw the figure. B, ADB and BDC are right angles.

To prove that angle ABD is equal to angle C, and angle CBD is equal to angle A, in triangle ABC, we can use the concept of similar triangles and the properties of right-angled triangles.

Let's start with proving that angle ABD is equal to angle C:

(i) Proving angle ABD = angle C:
To show that angle ABD is equal to angle C, we need to establish that triangle ABD is similar to triangle ABC.

In triangle ABD and triangle ABC, we have:
- Angle ABD is equal to angle ABC (right angle at B).
- Angle B is shared by both triangles.
- Since angle ABD is equal to angle ABC and angle BAD is equal to angle BAC (both right angles), we can conclude that angle A is equal to angle C.

Therefore, we have proved that angle ABD is equal to angle C.

Now, let's move on to proving that angle CBD is equal to angle A:

(ii) Proving angle CBD = angle A:
To show that angle CBD is equal to angle A, we again need to establish that triangle CBD is similar to triangle ABC.

In triangle CBD and triangle ABC, we have:
- Angle CBD is equal to angle BAC (right angle at B).
- Angle B is shared by both triangles.
- Since angle CBD is equal to angle BAC and angle CBD is equal to angle ABC (proved in part i), we can conclude that angle A is equal to angle CBD.

Hence, we have proved that angle CBD is equal to angle A.

By using the concept of similar triangles and the properties of right-angled triangles, we have shown that angle ABD is equal to angle C, and angle CBD is equal to angle A in the given triangle ABC, where angle B is a right angle.

To prove the given statements:

(i) angle ABD = angle C
(ii) angle CBD = angle A

We can use the concept of angles in a right-angled triangle and the properties of alternate angles.

Given: Triangle ABC is right-angled at B, and BD is drawn perpendicular to AC.

Proof:
(i) To prove angle ABD = angle C

In triangle ABD and triangle CBD:
1. AD = CD (common side, as both triangles share the side BD)
2. BD = BD (common side, as both triangles share the side BD)
3. Angle ABD = angle CBD (Right angles, as BD is perpendicular to AC)

By the Side-Angle-Side (SAS) congruence criterion, triangles ABD and CBD are congruent.

Thus, angle ABD = angle C.

(ii) To prove angle CBD = angle A

In triangle ABD and triangle CBD:
1. AD = CD (common side, as both triangles share the side BD)
2. BD = BD (common side, as both triangles share the side BD)
3. Angle ABD = angle CBD (Right angles, as BD is perpendicular to AC)

By the Side-Angle-Side (SAS) congruence criterion, triangles ABD and CBD are congruent.

Thus, angle CBD = angle A.

Therefore, we have proved that:
(i) angle ABD = angle C
(ii) angle CBD = angle A.