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. Please

Answer 1

In a triangle, the sum of the angles is always 180 degrees. For a right triangle, one of the angles is 90 degrees, meaning the sum of the other two angles must be 90 degrees.

Let the acute angles be:
1. \( A = 2x + 6 \) degrees
2. \( B = 3x - 26 \) degrees

Since the sum of angles \( A \) and \( B \) must equal 90 degrees, we can write the equation:

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

Combining like terms:

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

This simplifies to:

\[
5x - 20 = 90
\]

Next, add 20 to both sides:

\[
5x = 110
\]

Now, divide by 5:

\[
x = 22
\]

Now, we can find the measures of angles \( A \) and \( B \):

1. For angle \( A \):

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

2. For angle \( B \):

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

Thus, we have found the angles of the triangle:

- Angle \( A = 50 \) degrees
- Angle \( B = 40 \) degrees
- The right angle \( C = 90 \) degrees

To summarize, the measures of the angles in the right triangle are \( 50 \) degrees, \( 40 \) degrees, and \( 90 \) degrees.