Question

Geometry The measure of one acute angle in a right triangle is four times the measure of the other acute angle.

Write and solve a system of equations to find the measures of the acute angles.

Answer 1

Step 1. Write the expressions:

It's given that “one acute angle is four times the measure of the other acute angle”.

So, consider that one acute angle is $x$ then the other acute angle is $4x$.

Step 2. Find the acute angle:

Since the given triangle is a right triangle.

So the angles of the triangle are $x$, $4x$, and $90°$.

The sum of interior angles of the right triangle is $180°$.

So,

$\begin{array}{rcl}x+4x+90°& =& 180°\\ & \Rightarrow & 5x=180°-90°\\ & \Rightarrow & x=\frac{90°}{5}\\ & \Rightarrow & =18°\end{array}$

Find the angles:

$x=18°$

And,

$\begin{array}{rcl}4x& =& 4\left(18°\right)\\ & =& 72°\end{array}$

Therefore, the one acute angle is $\mathbf{18}\mathbf{°}$ and the other acute angle is $\mathbf{72}\mathbf{°}$.