In a right triangle, the sum of the angles is always 180 degrees. Since one angle is a right angle (90 degrees), we can set up the equation for the two acute angles.
Let the two acute angles be given as:
- First angle: \( A = 2x + 6 \)
- Second angle: \( B = 3x - 26 \)
According to the Triangle Angle Sum Theorem, we have:
\[ A + B + 90 = 180 \]
This simplifies to:
\[ A + B = 90 \]
Now, substituting the expressions for angles A and B:
\[ (2x + 6) + (3x - 26) = 90 \]
Now combine like terms:
\[ 2x + 3x + 6 - 26 = 90 \]
\[ 5x - 20 = 90 \]
Now, add 20 to both sides:
\[ 5x = 110 \]
Dividing both sides by 5 gives:
\[ x = 22 \]
Now we can find the measures of the acute angles by substituting \( x \) back into the expressions for the angles:
- First angle:
\[ A = 2x + 6 = 2(22) + 6 = 44 + 6 = 50 \text{ degrees} \]
- Second angle:
\[ B = 3x - 26 = 3(22) - 26 = 66 - 26 = 40 \text{ degrees} \]
Thus, the measures of the angles in the triangle are:
- First acute angle: \( 50 \) degrees
- Second acute angle: \( 40 \) degrees
- Right angle: \( 90 \) degrees
In summary, the angles of the triangle are \( 50 \) degrees, \( 40 \) degrees, and \( 90 \) degrees.
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