In a regular pentagon ABCDE, draw a diagonal BE and then find the measure of :
(i) `angleBAE` (ii) `angleABE` (iii) `angleBED`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the measure of each interior angle of the regular pentagon ABCDE.

The formula for finding the sum of interior angles of a polygon is:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]
where \( n \) is the number of sides. For a pentagon, \( n = 5 \).

\[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]

Since it is a regular pentagon, each interior angle is:
\[ \text{Each angle} = \frac{540^\circ}{5} = 108^\circ \]

### Step 2: Find the measure of angle BAE.

Since angle BAE is one of the interior angles of the regular pentagon ABCDE, we have:
\[ \angle BAE = 108^\circ \]

### Step 3: Analyze triangle ABE.

In triangle ABE, since AB = AE (as sides of the regular pentagon), angles opposite to equal sides are equal. Thus:
\[ \angle ABE = \angle AEB \]

Let \( \angle ABE = \angle AEB = x \).

### Step 4: Use the triangle angle sum property.

The sum of angles in triangle ABE is:
\[ \angle BAE + \angle ABE + \angle AEB = 180^\circ \]
Substituting the known values:
\[ 108^\circ + x + x = 180^\circ \]
This simplifies to:
\[ 108^\circ + 2x = 180^\circ \]
\[ 2x = 180^\circ - 108^\circ = 72^\circ \]
\[ x = \frac{72^\circ}{2} = 36^\circ \]

Thus, we find:
\[ \angle ABE = 36^\circ \]

### Step 5: Find the measure of angle BED.

Since angle AED is also an interior angle of the regular pentagon, we have:
\[ \angle AED = 108^\circ \]

Now, we can find angle BED:
\[ \angle BED = \angle AED - \angle ABE = 108^\circ - 36^\circ = 72^\circ \]

### Final Measures:
- \( \angle BAE = 108^\circ \)
- \( \angle ABE = 36^\circ \)
- \( \angle BED = 72^\circ \)

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