A regular polygon has an interior angle that is 140^\circ greater than its exterior angle.… — ACT

A regular polygon has an interior angle that is $140^\circ$ greater than its exterior angle. How many sides does this polygon have?

The regular polygon has 18 sides.

The correct answer is 18. By setting up the equations for the interior angle

I

and exterior angle

E

, we have

I = E + 140^\circ

and

I + E = 180^\circ

. Substituting the first equation into the second gives

(E + 140^\circ) + E = 180^\circ

, which simplifies to

2E + 140^\circ = 180^\circ \implies 2E = 40^\circ \implies E = 20^\circ

. The number of sides

n

of a regular polygon is given by

n = 360^\circ / E

. Thus,

n = 360^\circ / 20^\circ = 18

Set up the algebraic relationship between the interior angle (

I

) and the exterior angle (

E

I = E + 140^\circ

The problem states that the interior angle is

140^\circ

greater than the exterior angle.

Use the supplementary relationship between any interior angle and its corresponding exterior angle.

I + E = 180^\circ

An interior angle and its adjacent exterior angle form a linear pair, which sums to

180^\circ

Substitute the expression for

I

from Step 1 into the equation from Step 2 and solve for

E

(E + 140^\circ) + E = 180^\circ \implies 2E + 140^\circ = 180^\circ \implies 2E = 40^\circ \implies E = 20^\circ

Substituting allows us to solve for a single variable representing the exterior angle.

Calculate the number of sides (

n

) using the sum of the exterior angles of a convex polygon.

n = \frac{360^\circ}{E} = \frac{360^\circ}{20^\circ} = 18

The sum of the exterior angles of any convex polygon is

360^\circ

, and in a regular polygon, all

n

exterior angles are equal.

Relationship between interior and exterior angles of a regular polygon

Estimated Time:1m 30s

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