Glencoe Math: Course 3, Volume 2
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Glencoe Math: Course 3, Volume 2 View details
4. Polygons and Angles
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Exercise 26 Page 403

We are given the three following triangles.

The traingles

The smallest triangle is equilateral. The two larger triangles are made of copies of the smallest triangle. We want to find the measure of each interior angle in each of the three given triangles. Let's start with the smallest triangle. Recall the rule for the sum of the measures of the interior angles of a polygon.

Interior Angle Sum of a Polygon

The sum of the measures of the interior angles of a polygon is where represents the number of sides.

To find the measure of one interior angle of an equilateral triangle, we will start by finding the sum of its interior angles. Let's substitute for in this expression.
Evaluate
The sum of the interior angles of a triangle is Recall that an equilateral triangle is also known as equiangular triangle, because all angles are congruent in this triangle. Therefore, it is a regular polygon and it has angles with the same measure. To find the measure of one angle, we will divide the sum of the angles by
We got that the measure of each angle of the smallest triangle is Now let's consider the two larger triangles.
The traingles

Since the two larger triangles are made of copies of the smallest triangle, we can write all the missing angle measures in these triangles.

The traingles

Finally, we got that the measure of each interior angle in each of the three given triangles is Moreover, we can see that the two larger triangles are also equilateral. If the side length of the smallest triangle is the side length of the two larger triangles are and

The traingles

Therefore, we can make a conjecture that no matter how large an equilateral triangle is, each interior angle measure of the triangle is