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5. The measure of each interior angle of a regular [tex]\(n\)[/tex]-sided polygon is [tex]\(\frac{(n-2) \cdot 180^{\circ}}{n}\)[/tex]. A regular pentagon is shown below. What is the measure of the designated angle?

A. [tex]\(108^{\circ}\)[/tex]
B. [tex]\(144^{\circ}\)[/tex]
C. [tex]\(198^{\circ}\)[/tex]
D. [tex]\(252^{\circ}\)[/tex]
E. [tex]\(288^{\circ}\)[/tex]


Sagot :

To determine the measure of each interior angle of a regular pentagon, follow these steps:

1. A regular [tex]\(n\)[/tex]-sided polygon has its interior angles equal. The formula for calculating the measure of each interior angle of a regular [tex]\(n\)[/tex]-sided polygon is:

[tex]\[ \text{Interior angle} = \frac{(n-2) \times 180^\circ}{n} \][/tex]

2. In this case, the polygon is a pentagon, which means it has [tex]\(n = 5\)[/tex] sides.

3. Substitute [tex]\(n = 5\)[/tex] into the formula:

[tex]\[ \text{Interior angle} = \frac{(5-2) \times 180^\circ}{5} \][/tex]

4. Simplify the expression inside the parentheses:

[tex]\[ \text{Interior angle} = \frac{3 \times 180^\circ}{5} \][/tex]

5. Multiply the values:

[tex]\[ \text{Interior angle} = \frac{540^\circ}{5} \][/tex]

6. Divide the result:

[tex]\[ \text{Interior angle} = 108^\circ \][/tex]

Thus, the measure of each interior angle of a regular pentagon is [tex]\(108^\circ\)[/tex]. Therefore, the answer is:

[tex]\[ \boxed{108^\circ} \][/tex]