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.(4 points)

Answer 1

In a triangle, the sum of all angles is always \(180\) degrees. Since we are given a right triangle, one angle measures \(90\) degrees, and the remaining two angles must therefore sum to \(90\) degrees.

Let the two angles be:

1. \( \theta_1 = 2x + 6 \) degrees
2. \( \theta_2 = 3x - 26 \) degrees

According to the Triangle Angle Sum Theorem, we have:

\[
\theta_1 + \theta_2 + 90 = 180
\]

This simplifies to:

\[
\theta_1 + \theta_2 = 90
\]

Substituting the expressions for \( \theta_1 \) and \( \theta_2 \):

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

Combining like terms:

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

\[
5x - 20 = 90
\]

Now, add \(20\) to both sides:

\[
5x = 110
\]

Next, divide by \(5\):

\[
x = 22
\]

Now we can find the measures of the angles:

1. For \( \theta_1 = 2x + 6 \):

\[
\theta_1 = 2(22) + 6 = 44 + 6 = 50 \text{ degrees}
\]

2. For \( \theta_2 = 3x - 26 \):

\[
\theta_2 = 3(22) - 26 = 66 - 26 = 40 \text{ degrees}
\]

Thus, the measures of the angles in the right triangle are:

- One angle is \(90\) degrees (the right angle).
- The other two angles are \(50\) degrees and \(40\) degrees.

Final answer:
- The angles of the triangle are \(90\) degrees, \(50\) degrees, and \(40\) degrees.