Question

quick check

the sum of the measures of the 4 angles of a pentagon is 465^{\circ}. find the measure of the remaining angle.

Explanation:

🆕 New Concept Discovered: Sum of interior angles in a polygon
Total degrees inside any closed straight-sided shape

Step 1: Find the total sum of interior angles in a pentagon

A pentagon has \( n = 5 \) sides. We can find the sum of its interior angles using the formula:

$$ S = (n - 2) \times 180^\circ $$

Substituting \( n = 5 \):

$$ S = (5 - 2) \times 180^\circ $$

$$ S = 3 \times 180^\circ = 540^\circ $$

This means the 5 interior angles of any pentagon must always add up to exactly \( 540^\circ \).

Step 2: Calculate the measure of the remaining angle

We are given that the sum of 4 of these angles is \( 465^\circ \). To find the fifth (remaining) angle, we subtract this sum from the total interior sum:

$$ \text{Remaining angle} = 540^\circ - 465^\circ $$

$$ \text{Remaining angle} = 75^\circ $$

Answer:

\( 75^\circ \)