Question

The measure of one of the small angles of a right triangle is$26$more than $3$times the measure of the other small angle. Find the measure of both angles.

Short answer

The measure of the angles are $16$and $74$.

Step-by-step solution

Step 1 Given information is,

The measure of one of the small angles of a right triangle is $26$more than $3$times the measure of the other small angle.

Step 2 To find:

The measure of the two angles.

Step 3 Let us consider,

$x$represents the measure of the first angle.

$y$represents the measure of the second angle.

We know that,

$$\begin{gathered} x+y+90=180 \\ x+y=180-90 \\ x+y=90 \end{gathered}$$

Step 4 From the given information,

The measure of one of the small angles of the right triangles is$26$more than$3$times of the measure of the other small angle.

So, $x=26+3y$.

Step 5 From the step3 and4

The obtained linear equations are,

$$\begin{gathered} x+y=90 \\ x=26+3y \end{gathered}$$

Now we are going to solve these equations by the substitution method.

Step 6 Find the required values by using the substitution method.

Now substitute $x=26+3y$in the equation.

$$\begin{gathered} 26+3y+y=90 \\ 26+4y=90 \end{gathered}$$

Now solve for$y$.

$$\begin{gathered} 4y=90-26 \\ 4y=64 \\ y=\frac{64}{4} \\ y=16 \end{gathered}$$

Step 7 Find the value of x

Substitute the value of $y$in the equation $x+y=90$.

$$\begin{gathered} x+16=90 \\ x=90-16 \\ x=74 \end{gathered}$$

Let us check the answer

Substitute $x=74,y=16$into the equation

$$\begin{gathered} x+y=90 \\ 74+16=90 \\ 90=90 \end{gathered}$$

which is true.