If the exterior angle of a regular polygon is equal to the acute angle of a right-angled isosceles triangle, then the polygon is a regular

Options

A. pentagon
B. hexagon
C. octagon
D. heptagon

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the acute angle of a right-angled isosceles triangle
A right-angled isosceles triangle has one right angle (90 degrees) and the other two angles are equal. Since the sum of angles in a triangle is 180 degrees, each of the other two angles is:
\[ \text{Acute angle} = \frac{180^\circ - 90^\circ}{2} = 45^\circ \]

### Step 2: Set the exterior angle of the polygon
According to the problem, the exterior angle of the regular polygon is equal to the acute angle of the right-angled isosceles triangle, which is 45 degrees.

### Step 3: Relate the exterior angle to the number of sides
The formula for the exterior angle of a regular polygon with \( n \) sides is given by:
\[ \text{Exterior angle} = \frac{360^\circ}{n} \]
Setting this equal to 45 degrees, we have:
\[ \frac{360^\circ}{n} = 45^\circ \]

### Step 4: Solve for \( n \)
To find \( n \), we can rearrange the equation:
\[ n = \frac{360^\circ}{45^\circ} \]
Calculating this gives:
\[ n = 8 \]

### Step 5: Identify the polygon
A regular polygon with 8 sides is called an octagon.

### Final Answer
Thus, the polygon is a regular octagon.

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