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Chapter 13: Problem 257

Prove that any two regular polygons with the same number of sides are similar.

Short Answer

Expert verified
We can prove that any two regular polygons with the same number of sides are similar by following these steps: 1. Observe that the interior angles of both polygons have the same measure (\(\dfrac{(n-2) \cdot 180掳}{n}\)), since they have the same number of sides, n. 2. Show that their corresponding angles are congruent, as their measures are equal. 3. Prove that their corresponding sides are proportional by comparing their perimeters, where the ratio is \(\dfrac{a}{b}\), with a and b being the side lengths of the polygons. 4. Conclude that, since corresponding angles are congruent and corresponding sides are proportional, the polygons are similar according to the definition of similarity.

Step by step solution

01

Understanding the geometrical definition of similar polygons

To prove that two polygons are similar, we need to show that their corresponding angles are congruent, and their corresponding sides are proportional. In other words, polygons are similar if their corresponding angles are equal and sides have the same ratio.
02

Find the measure of a single angle in a regular polygon

Let's consider a regular polygon with n sides. To find the measure of one interior angle, we can use the formula for the sum of the interior angles of a polygon: Sum of interior angles = \((n-2) \cdot 180掳\) Since all angles in a regular polygon are equal, we can find the measure of a single angle by dividing the sum of interior angles by the number of angles: Single interior angle = \(\dfrac{(n-2) \cdot 180掳}{n}\)
03

Check the congruency of corresponding angles in both polygons

Since both regular polygons have the same number of sides, they will also have the same number of angles. Moreover, as shown in step 2, the measure of a single interior angle in each polygon is the same: Single interior angle of polygon 1 = \(\dfrac{(n-2) \cdot 180掳}{n}\) Single interior angle of polygon 2 = \(\dfrac{(n-2) \cdot 180掳}{n}\) As the measures of corresponding angles are equal, we can say that the corresponding angles in both polygons are congruent.
04

Prove that the corresponding sides are proportional

Let's call the side length of the first polygon a and the side length of the second polygon b. We can calculate the ratio of the corresponding sides as: Ratio of corresponding sides = \(\dfrac{a}{b}\) Since both polygons have the same number of sides (n), their perimeters can be expressed as: Perimeter of polygon 1 = n * a Perimeter of polygon 2 = n * b Dividing the perimeters of both polygons, we have: \(\dfrac{n * a}{n * b} = \dfrac{a}{b}\) Notice that the ratio of the perimeters of the polygons is equal to the ratio of their corresponding sides, which means that their corresponding sides are proportional.
05

Conclude that the polygons are similar

We have shown in step 3 that the corresponding angles in both regular polygons are congruent by having the same interior angle measure. We also showed in step 4 that their corresponding sides are proportional. Therefore, according to the definition of similarity, we can conclude that any two regular polygons with the same number of sides are similar.

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Key Concepts

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

Regular Polygons
A regular polygon is a special kind of polygon where all sides are of the same length and all interior angles are equal. It is like having a shape where everything is just right!
Think about a square; it is a regular polygon because all four sides are equal, and all the angles are 90 degrees. Regular polygons can have any number of sides, like a triangle, pentagon, or hexagon.
  • A regular triangle is called an equilateral triangle.
  • A regular quadrilateral is a square.
  • Each regular shape has its own special formula for angles and properties.
These shapes are handy because they make it easy to understand more complex ideas in geometry.
Proportional Sides
When we say the sides of a polygon are proportional, we mean that the lengths are in the same ratio to one another. In the case of similar polygons, their side lengths are consistent in terms of proportion.
For example, if one square has side lengths of 2 cm and another similar square has side lengths of 4 cm, the ratio between the lengths of the two squares' sides is 1:2.
  • This means that each side of the larger square is twice as long as the corresponding side of the smaller square.
  • Proportional sides help us determine similarity between two polygons easily.
The neat part? Even if the polygons are different in size, if the ratio remains constant, they are similar.
Congruent Angles
Congruent angles are angles that have exactly the same measure. When we speak about similar polygons, one key criterion is that they have congruent corresponding angles.
If two polygons each have angles of 45, 60, and 75 degrees in the same order, these angles are congruent. This means that:
  • Every angle in one polygon matches exactly with an angle in the other polygon.
  • Congruent angles imply that the shape remains the same, even though the size may differ.
In our exercise on regular polygons, knowing the angles are congruent is critical to proving similarity.
Polygon Interior Angles
The interior angles of a polygon are crucial because they provide a way to understand the shape of the polygon better. In a regular polygon, each angle is equal since all sides and angles are the same.
To find the sum of a polygon's interior angles, we use the formula: \[(n-2) \cdot 180^{\circ}\]
Where \(n\) is the number of sides. This gives us the sum for all the angles inside the polygon.
Knowing how to find the measure of one angle in a regular polygon is important:
  • Single interior angle = \(\dfrac{(n-2) \cdot 180^{\circ}}{n}\)
  • This formula allows us to compare polygons by checking each angle.
Being able to calculate these angles helps immensely in proving concepts like polygon similarity.

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Most popular questions from this chapter

Let \(\mathrm{ABC}\) be a triangle where \(\mathrm{D}\) is a point on \(\underline{\mathrm{AB}}\), and \(\mathrm{E}\) is a point on \(\underline{\mathrm{AC}}\). Prove that if \(\underline{\mathrm{DE}} \| \underline{\mathrm{BC}}\), then \(\mathrm{AB} / \mathrm{AD}=\mathrm{BC} / \mathrm{DE}\). (See figure.)

Is \(12 / 20=36 / 60\) a proportion?

A boy knows that his height is \(6 \mathrm{ft}\). and his shadow is \(4 \mathrm{ft}\). long. At the same time of day, a tree's shadow is \(24 \mathrm{ft}\). long. How high is the tree?

(a) If 2 triangles are congruent, does it follow that they are similar ? Why? (b) If 2 triangles are similar, does it follow that they are congruent? Why?

Given the A.A.A. (Angle, Angle, Angle) Similarity Theorem, prove the A.A. (Angle, Angle) Similarity Theorem.
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