We have here a right triangle, and that is why we have two acute angles (that is, the measure of each of them is less than 90 degrees).
We also know that the sum of the inner angles of a triangle is 180 degrees.
Having this information at hand, we can proceed as follows:
[tex](3x+8)+(2x+12)+90=180[/tex]
This is the equation. Now, we need to solve this equation to find x, and then we need to use the algebraical equations to find each of the acute angles.
Solving the equation
1. Sum the like terms (like terms have the same variable or they are constants.)
[tex]3x+2x+8+12+90=180[/tex]
Then, we have:
[tex]5x+110=180\Rightarrow5x=180-110\Rightarrow5x=70[/tex]
2. We need to divide each side of the equation by 5 to isolate x:
[tex]\frac{5x}{5}=\frac{70}{5}\Rightarrow x=14[/tex]
Now, we have x = 14. Therefore, the values for each of the acute angles are (we need to substitute the value of x in each equation):
a. 3x + 8 ---> 3 * (14) +8 = 42 + 8 =50. Hence, one acute angle measures 50 degrees.
b. 2x + 12 ---> 2 * (14) + 12 = 28 + 12 = 40 degrees. Therefore, the other acute angle measures 40 degrees.
In summary, the equation to solve is:
[tex](3x+8)+(2x+12)+90=180[/tex]
And the values for each of the acute angles are 50 and 40 degrees.
