Certainly! Let's thoroughly break down the problem and solution step-by-step:
### Step 1: Determine the Number of Sides
Since we are dealing with a triangle, the number of sides [tex]\( n \)[/tex] is:
[tex]\[ n = 3 \][/tex]
### Step 2: Calculate Each Interior Angle
For a regular polygon with [tex]\( n \)[/tex] sides, the formula for each interior angle is:
[tex]\[ \text{Each interior angle} = \frac{(n - 2) \times 180^\circ}{n} \][/tex]
Plugging in [tex]\( n = 3 \)[/tex]:
[tex]\[ \text{Each interior angle} = \frac{(3 - 2) \times 180^\circ}{3} \][/tex]
[tex]\[ \text{Each interior angle} = \frac{1 \times 180^\circ}{3} \][/tex]
[tex]\[ \text{Each interior angle} = 60^\circ \][/tex]
Thus, each interior angle of a triangle is [tex]\( 60^\circ \)[/tex].
### Step 3: Simplify the Calculation for Verification
To verify, we can use a simpler check by dividing 180 degrees by the number of sides [tex]\( n \)[/tex]. For regular polygons, the exterior angle is given by:
[tex]\[ \text{Exterior angle} = \frac{360^\circ}{n} \][/tex]
Since interior and exterior angles are supplementary:
[tex]\[ \text{Interior angle} = 180^\circ - \text{Exterior angle} \][/tex]
[tex]\[ \text{Interior angle} = 180^\circ - \frac{360^\circ}{3} \][/tex]
[tex]\[ \text{Interior angle} = 180^\circ - 120^\circ \][/tex]
[tex]\[ \text{Interior angle} = 60^\circ \][/tex]
### Step 4: Side Length and Example Angles
Given:
- Each side of the triangle is 5 cm.
- An example angle provided is [tex]\( 10^\circ \)[/tex].
### Summary of Results
The resulting calculations for interior angles of a regular triangle confirm that:
[tex]\[ \text{Each interior angle} = 60^\circ \][/tex]
and simplified check assures us:
[tex]\[ \text{Interior angle per side} = 60^\circ \][/tex]
Therefore, our final detailed solution shows that for a regular triangle (equilateral triangle):
[tex]\[ \text{Each interior angle} = 60^\circ \][/tex]
