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Chapter 3: Problem 30

Prove that the exterior angles of a regular polygon are equal.

Short Answer

Expert verified
The exterior angles of a regular polygon are equal because each is \[\frac{360^\text{掳}}{n}.\]

Step by step solution

01

Define Exterior Angles

Exterior angles are the angles formed between one side of the polygon and the extension of the adjacent side. For a regular polygon, every exterior angle is formed in the same way.
02

Sum of Exterior Angles

The sum of the exterior angles of any polygon is always 360 degrees, regardless of the number of sides. This is because an exterior angle at each vertex makes a full circle (360 degrees) when summed up.
03

Calculate Each Exterior Angle

For a regular polygon, all exterior angles are equal. If the polygon has n sides, the measure of each exterior angle can be found by dividing the sum of exterior angles (360 degrees) by the number of sides (n). Thus, each exterior angle is \[\text{Each exterior angle} = \frac{360^\text{掳}}{n}.\]
04

Conclusion

Since each exterior angle is calculated by dividing 360 degrees by the number of sides (n) and this division results in an equal angle for each side, the exterior angles of a regular polygon are equal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

regular polygon
A regular polygon is a polygon that has all sides of equal length and all angles of equal measure. Examples include equilateral triangles and squares. Because of their symmetry, regular polygons have some interesting properties that make them easier to study in geometry.
In a regular polygon, not only are the sides and internal angles equal, but the exterior angles are also identical. This symmetry simplifies many calculations, including the calculation of angles.
sum of exterior angles
One of the key concepts of polygons is the sum of their exterior angles. The exterior angle is formed by extending one side of the polygon and measuring the angle between this extension and the adjacent side.
Importantly, the sum of the exterior angles of any polygon鈥攔egular or irregular鈥攊s always 360 degrees. This may seem surprising at first, but it works because as you walk around the polygon and make one full loop, you turn full circle around the shape, which corresponds to 360 degrees.
For example:
  • If a polygon has 4 sides, like a square, the sum of its exterior angles is still 360 degrees.
  • If it has 6 sides, like a regular hexagon, the sum is still 360 degrees.
  • So, regardless of the number of sides, this rule holds true.
angle calculation
In a regular polygon, since all exterior angles are equal, the measure of one exterior angle can be calculated by dividing the total sum of the exterior angles (360 degrees) by the number of sides (n).
The formula is: \[\text{Each exterior angle} = \frac{360^\text{掳}}{n}.\]
This simple division gives you the measure of each exterior angle in any regular polygon. For example:
  • For an equilateral triangle (3 sides), each exterior angle is \[\frac{360^\text{掳}}{3} = 120^\text{掳}.\]
  • For a square (4 sides), each exterior angle is \[\frac{360^\text{掳}}{4} = 90^\text{掳}.\]
  • For a regular hexagon (6 sides), each exterior angle is \[\frac{360^\text{掳}}{6} = 60^\text{掳}.\]
These calculations make it easier to understand and work with regular polygons.

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Most popular questions from this chapter

There will not be a SSA pattern of congruence of triangles. Investigate this with geometric software or by drawing figures. Show an example of two non congruent triangles with two pairs of congruent sides and one pair of congruent non included angles.

In Exercises \(17-22\), assume that \(\ell, m,\) and \(n\) are three distinct lines in a plane and \(P\) is a point in the plane. If \(\ell \perp m\) and \(\ell \perp n,\) is \(m \perp n ?\)

Prove that every point on the bisector of an angle is equidistant from the sides of the angle.

Find the number of sides of a polygon if the sum of its angles is twice the sum of its exterior angles.

Solve the equation \(a=\frac{(n-2) 180^{\circ}}{n}\) for \(n\) when \(a\) is a given value. Find the number of sides of each polygon (if possible) if the given value is the measure of one interior angle of a regular polygon. $$145^{\circ}$$
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