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:
- \( A = 2x + 6 \) degrees
- \( B = 3x - 26 \) 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 \( B \):
- For angle \( A \):
\[ A = 2(22) + 6 = 44 + 6 = 50 \text{ degrees} \]
- 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.