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Zorluk: OrtaAngles, Parallel Lines, and Polygons

A convex polygon with nn sides has a total interior angle sum of 1440∘1440^\circ. If (n−4)(n - 4) of its interior angles each measure 150∘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?

Cevap: 135 degrees

Cevap

The measure of one of the remaining interior angles is 135∘135^\circ.
Using the interior angle sum formula (n−2)×180∘=1440∘(n - 2) \times 180^\circ = 1440^\circ, we find n−2=8n - 2 = 8, which means the polygon has n=10n = 10 sides. The number of angles measuring 150∘150^\circ is 10−4=610 - 4 = 6. Their total measure is 6×150∘=900∘6 \times 150^\circ = 900^\circ. The sum of the remaining four equal angles is 1440∘−900∘=540∘1440^\circ - 900^\circ = 540^\circ. Dividing 540∘540^\circ by 4 gives 135∘135^\circ for each remaining interior angle.

Adım Adım Çözüm

1
Calculate the total number of sides nn of the convex polygon.
n=10n = 10
The sum of interior angles of an nn-sided polygon is given by (n−2)×180∘(n - 2) \times 180^\circ. Setting (n−2)×180∘=1440∘(n - 2) \times 180^\circ = 1440^\circ gives n−2=8n - 2 = 8, so n=10n = 10.
2
Determine the number of interior angles that measure 150∘150^\circ each and find their combined sum.
6 angles totaling 900∘900^\circ
There are (n−4)=10−4=6(n - 4) = 10 - 4 = 6 interior angles of 150∘150^\circ each. Their sum is 6×150∘=900∘6 \times 150^\circ = 900^\circ.
3
Calculate the sum of the remaining four equal interior angles.
540∘540^\circ
Subtracting the sum of the known angles from the total interior angle sum yields 1440∘−900∘=540∘1440^\circ - 900^\circ = 540^\circ.
4
Find the measure of one of the remaining four equal angles.
135∘135^\circ
Dividing the remaining sum equally among the 4 angles gives 540∘/4=135∘540^\circ / 4 = 135^\circ.

Anahtar Kavram

Polygon interior angle sum theorem
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Angles, Parallel Lines, and Polygons

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