AB, BC and DC are the three consecutive sides of a regular polygon. If `angleBAC=15^(@)`, find :
(i) each interior angle of the polygon.
(ii) each exterior angle of the polygon.
(iii) number of sides of the polygon.

Answer

To solve the problem step by step, we will follow the instructions given in the question.

### Step 1: Understanding the Triangle
Given that \( \angle BAC = 15^\circ \), we need to find the interior angle \( \angle ABC \) of the polygon.

### Step 2: Applying the Angle Sum Property of Triangle
In triangle \( ABC \):
\[ \angle ABC + \angle BAC + \angle BCA = 180^\circ \]
Since \( AB \) and \( BC \) are equal sides of the triangle (as they are consecutive sides of a regular polygon), \( \angle BCA \) is equal to \( \angle ABC \). Let's denote \( \angle ABC \) as \( x \).

Thus, we can rewrite the equation as:
\[ x + 15^\circ + x = 180^\circ \]
This simplifies to:
\[ 2x + 15^\circ = 180^\circ \]
Now, subtract \( 15^\circ \) from both sides:
\[ 2x = 180^\circ - 15^\circ = 165^\circ \]
Now, divide by 2:
\[ x = \frac{165^\circ}{2} = 82.5^\circ \]
Thus, \( \angle ABC = 150^\circ \).

### Step 3: Finding Each Interior Angle of the Polygon
The interior angle of a regular polygon can be calculated using the formula:
\[ \text{Interior angle} = \frac{(n-2) \times 180^\circ}{n} \]
where \( n \) is the number of sides of the polygon.

We have found that \( \angle ABC = 150^\circ \). Setting this equal to the formula:
\[ 150^\circ = \frac{(n-2) \times 180^\circ}{n} \]
Cross-multiplying gives:
\[ 150n = (n-2) \times 180 \]
Expanding the right side:
\[ 150n = 180n - 360 \]
Rearranging gives:
\[ 180n - 150n = 360 \]
\[ 30n = 360 \]
Dividing both sides by 30:
\[ n = 12 \]

### Step 4: Finding Each Exterior Angle of the Polygon
The exterior angle of a regular polygon can be calculated using the formula:
\[ \text{Exterior angle} = \frac{360^\circ}{n} \]
Substituting \( n = 12 \):
\[ \text{Exterior angle} = \frac{360^\circ}{12} = 30^\circ \]

### Final Answers
(i) Each interior angle of the polygon is \( 150^\circ \).
(ii) Each exterior angle of the polygon is \( 30^\circ \).
(iii) The number of sides of the polygon is \( 12 \).