Question

A right triangle has acute angles measuring 2x+6 degrees and 3x−26 degrees. Use the Triangle Angle Sum Theorem to find the measures of the missing angles of the triangle. Then write a paragraph explaining how you did it.

Answer 1

In a right triangle, the sum of the angles is always 180 degrees. Since one of the angles is a right angle (90 degrees), the sum of the two acute angles must be 90 degrees. Given the acute angles measuring \(2x + 6\) degrees and \(3x - 26\) degrees, we can set up the following equation:

\[
(2x + 6) + (3x - 26) = 90
\]

Now, combine like terms:

\[
2x + 3x + 6 - 26 = 90
\]
\[
5x - 20 = 90
\]

Next, solve for \(x\):

\[
5x = 90 + 20
\]
\[
5x = 110
\]
\[
x = 22
\]

Now that we have the value of \(x\), we can find the measure of each angle:

1. For \(2x + 6\):
\[
2(22) + 6 = 44 + 6 = 50 \text{ degrees}
\]

2. For \(3x - 26\):
\[
3(22) - 26 = 66 - 26 = 40 \text{ degrees}
\]

Thus, the angles in the triangle are:
- \(50\) degrees
- \(40\) degrees
- \(90\) degrees (the right angle)

In summary, I used the Triangle Angle Sum Theorem, which states that the angles in a triangle add up to 180 degrees, to find the acute angles of the right triangle. First, I set up an equation for the sum of the two acute angles equal to 90 degrees since the third angle is 90 degrees. Then I combined the expressions for the acute angles, simplified the equation, and solved for \(x\). Finally, I substituted the value of \(x\) back into the angle expressions to find the measures of the angles in the triangle, resulting in angles of 50 degrees, 40 degrees, and a right angle of 90 degrees.