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Let's analyze the diagram and the problem step by step.

### Problem Breakdown:

- We are given a triangle \( \triangle ABC \) with point \( D \) on \( AB \). The angle \( DCB = a^\circ \), and the triangle is isosceles with \( AB = AD \) and \( DB = DC \).
  
### Part (a)
We need to find the following angles in terms of \( a \), providing reasons:
1. \( \angle DBC \)
2. \( \angle ADB \)
3. \( \angle DBA \)

#### Step 1: \( \angle DBC \)
- In \( \triangle DBC \), \( DCB = a^\circ \), and since the triangle is isosceles (given that \( DB = DC \)), \( \angle DBC = \angle DCB = a^\circ \).
  
Thus, \( \angle DBC = a^\circ \).

#### Step 2: \( \angle ADB \)
- \( \triangle ABD \) is isosceles, and since \( AB = AD \), we know that \( \angle ADB = \angle ABD \).
- The angle sum in a triangle is \( 180^\circ \), so in \( \triangle ADB \):

\[
\angle ADB + \angle ABD + \angle DAB = 180^\circ
\]

- Since \( \triangle ABD \) is isosceles, \( \angle ABD = \angle ADB \). Let these equal \( x \).
Thus,

\[
x + x + \angle DAB = 180^\circ \quad \Rightarrow \quad 2x + \angle DAB = 180^\circ
\]

Since \( \angle DAB = \angle DBC + \angle DCB = 2a^\circ \) (by exterior angle theorem),

\[
2x + 2a = 180^\circ \quad \Rightarrow \quad x + a = 90^\circ
\]

So, \( \angle ADB = 90^\circ - a^\circ \).

#### Step 3: \( \angle DBA \)
- By the property of isosceles triangles, \( \angle DBA = \angle ADB = 90^\circ - a^\circ \).

### Part (b)
We are asked to prove that \( \angle ABC \) is a right angle.

#### Step 1: Use of the exterior angle theorem
- We know that \( \angle ADB = 90^\circ - a^\circ \) and \( \angle DBC = a^\circ \).
- Since the total angle at point \( B \) in triangle \( ABC \) is:

\[
\angle ABC = \angle ABD + \angle DBC
\]

Substituting \( \angle ABD = 90^\circ - a^\circ \) and \( \angle DBC = a^\circ \):

\[
\angle ABC = (90^\circ - a^\circ) + a^\circ = 90^\circ
\]

Thus, \( \angle ABC \) is a right angle.

---

Do you need any further clarification or additional details?

Here are 5 related questions:
1. How can we generalize the exterior angle theorem for other types of triangles?
2. How do we calculate unknown angles in a triangle if only one angle is known?
3. What are the properties of isosceles triangles that are useful in angle calculations?
4. Can we prove the sum of angles in any triangle always equals \(180^\circ\)?
5. How does the Pythagorean theorem relate to right-angled triangles like \( \triangle ABC \)?

**Tip**: In any isosceles triangle, the angles opposite the equal sides are always equal.

In triangle ABC, with point D on AB, where DCB = a°, and triangle ABD is isosceles with AB = AD and DB = DC. Find the angles DBC, ADB, and DBA in terms of a. Show that ABC is a right angle.

Learn how to solve an isosceles triangle problem where the angles DBC, ADB, and DBA are found in terms of angle a, and prove that ABC is a right angle using the exterior angle theorem and angle sum property.

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