Question

What are the measures of the angles in the right triangles formed by the two regular pentagons shown? The measures of the angles in the night triangles formed by the two regular pentagons are

Answer 1

We use the following formula to calculate the measure of an interior angle of a regular figure:

$\frac{180(n-2)}{n}$

Where n is the number of sides of the regular figure, in this case, since it is a pentagon, n=5.

Thus, each internal angle in a pentagon measures:

$\frac{180(5-2)}{5}=\frac{180(3)}{5}=\frac{540}{5}=108$

The following image represents which angles measure 108°:

This can help us to find the measure of one of the angles on the triangles.

We take just one of the triangles as an example:

The yellow angle called x for reference is one of the angles of the triangle.

We can see that the sum of the angle x and the right angle of 90° has to be equal to 108:

$x+90=108$

We solve this for x:

$\begin{gathered} x=108-90 \\ x=18 \end{gathered}$

The yellow angle (the smallest angle in the triangle) measures 18°.

Now we only need to find the third angle:

We have two angles, and we need the third one, which we can call "y" for reference.

We use the following rule:

• The sum of all of the internal angles of a triangle is equal to 180°.

Thus, we add the angles and equal them to 180°:

$90+18+y=180$

We solve for y:

$\begin{gathered} 108+y=180 \\ y=180-108 \\ y=72 \end{gathered}$

Thus, we have the measures are:

Answer: 18, 72, 90