We are given the three following triangles.

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 $(n - 2) 180,$ where $n$ 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

3

for

n

in this expression.

(n - 2) 180

Substitute

$n = 3$

(3 - 2) 180

▼

Evaluate

SubTerm

Subtract term

(1) 180

IdPropMult

Identity Property of Multiplication

180

The sum of the interior angles of a triangle is

18 0^{\circ} .

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

3

angles with the same measure. To find the measure of one angle, we will divide the sum of the angles by

3 .

Sum of Angles : One Angle : 18 0^{\circ} \frac{1 8 0}{3} = 6 0^{\circ}

We got that the measure of each angle of the smallest triangle is

6 0^{\circ} .

Now let's consider the two larger triangles.

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

Finally, we got that the measure of each interior angle in each of the three given triangles is $6 0^{\circ} .$ Moreover, we can see that the two larger triangles are also equilateral. If the side length of the smallest triangle is $a,$ the side length of the two larger triangles are $2 a$ and $3 a .$

Therefore, we can make a conjecture that no matter how large an equilateral triangle is, each interior angle measure of the triangle is $6 0^{\circ} .$

Exercise