Examkin

Question

Difficulty: MediumPolygon Angles and Properties

A convex polygon has nn sides. The sum of the measures of n−1n-1 of its interior angles is 2030∘2030^\circ. What is the measure, in degrees, of the remaining interior angle?

Answer: 130 degrees

Answer

The measure of the remaining interior angle is 130 degrees.
The sum of the interior angles of a convex polygon must be a multiple of 180∘180^\circ. The multiple of 180∘180^\circ immediately greater than 2030∘2030^\circ is 2160∘2160^\circ (which is 12×180∘12 \times 180^\circ). The difference between the total sum and the sum of the n−1n-1 angles is 2160∘−2030∘=130∘2160^\circ - 2030^\circ = 130^\circ. Since 130∘130^\circ is less than 180∘180^\circ, this is a valid interior angle for a convex polygon.

Step-by-Step Solution

1
Write the formula for the total sum of the interior angles of a convex polygon.
The sum of the interior angles of a polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ.
This formula relates the number of sides to the total sum of the interior angles.
2
Apply the properties of convex polygons to set up an inequality for the total sum.
Since the remaining interior angle must be greater than 0∘0^\circ and less than 180∘180^\circ, the total sum of all nn angles must satisfy 2030∘<(n−2)×180∘<2030∘+180∘2030^\circ < (n-2) \times 180^\circ < 2030^\circ + 180^\circ, which simplifies to 2030∘<(n−2)×180∘<2210∘2030^\circ < (n-2) \times 180^\circ < 2210^\circ.
By definition, every interior angle of a convex polygon is strictly less than 180∘180^\circ.
3
Solve the inequality to find the integer value of n−2n-2.
Dividing the entire inequality by 180∘180^\circ gives 11.28<n−2<12.2811.28 < n-2 < 12.28. Since the number of sides nn must be an integer, n−2n-2 must be the integer 1212.
A polygon must have a whole number of sides.
4
Calculate the total sum of all interior angles and find the remaining angle.
The total sum is 12×180∘=2160∘12 \times 180^\circ = 2160^\circ. Subtracting the sum of the other n−1n-1 angles gives the remaining angle: 2160∘−2030∘=130∘2160^\circ - 2030^\circ = 130^\circ.
The difference between the total sum of all interior angles and the sum of the n−1n-1 angles is the measure of the final angle.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, where each interior angle is strictly less than 180∘180^\circ.
Rate this question

Topics

Polygon Angles and Properties

More questions from this topic