A convex polygon with n sides has a total interior angle sum of 1440^\circ. If (n - 4) of … — JAMB UTME

A convex polygon with $n$ sides has a total interior angle sum of $1440^\circ$ . If $(n - 4)$ of its interior angles each measure $150^\circ$ , and the remaining four interior angles are equal in measure, what is the measure of one of the remaining interior angles in degrees?

Answer: 135 degrees

Answer

The measure of one of the remaining interior angles is

135^\circ

Using the interior angle sum formula

(n - 2) \times 180^\circ = 1440^\circ

, we find

n - 2 = 8

, which means the polygon has

n = 10

sides. The number of angles measuring

150^\circ

10 - 4 = 6

. Their total measure is

6 \times 150^\circ = 900^\circ

. The sum of the remaining four equal angles is

1440^\circ - 900^\circ = 540^\circ

. Dividing

540^\circ

by 4 gives

135^\circ

for each remaining interior angle.

Step-by-Step Solution

Calculate the total number of sides

n

of the convex polygon.

n = 10

The sum of interior angles of an

n

-sided polygon is given by

(n - 2) \times 180^\circ

. Setting

(n - 2) \times 180^\circ = 1440^\circ

gives

n - 2 = 8

, so

n = 10

Determine the number of interior angles that measure

150^\circ

each and find their combined sum.

6 angles totaling

900^\circ

There are

(n - 4) = 10 - 4 = 6

interior angles of

150^\circ

each. Their sum is

6 \times 150^\circ = 900^\circ

Calculate the sum of the remaining four equal interior angles.

540^\circ

Subtracting the sum of the known angles from the total interior angle sum yields

1440^\circ - 900^\circ = 540^\circ

Find the measure of one of the remaining four equal angles.

135^\circ

Dividing the remaining sum equally among the 4 angles gives

540^\circ / 4 = 135^\circ

Key Concept

Polygon interior angle sum theorem

Estimated Time:1m 30s

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