In a right triangle, the sum of all angles is 180 degrees. Since one angle is a right angle (90 degrees), the sum of the remaining two acute angles must be 90 degrees.
Given the acute angles measure \( 2x + 6 \) degrees and \( 3x - 26 \) degrees, we can set up the equation:
\[ (2x + 6) + (3x - 26) = 90 \]
Now, simplify the equation:
- Combine like terms: \[ 2x + 3x + 6 - 26 = 90 \]
This simplifies to: \[ 5x - 20 = 90 \]
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Next, add 20 to both sides: \[ 5x = 110 \]
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Now, divide by 5: \[ x = 22 \]
Now that we have the value of \( x \), we can find the measures of the two acute angles:
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For the first angle: \[ 2x + 6 = 2(22) + 6 = 44 + 6 = 50 \text{ degrees} \]
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For the second angle: \[ 3x - 26 = 3(22) - 26 = 66 - 26 = 40 \text{ degrees} \]
Thus, the measures of the angles in the triangle are:
- \( 50 \) degrees
- \( 40 \) degrees
- \( 90 \) degrees (the right angle)
In summary:
- The two acute angles are 50 degrees and 40 degrees. The right angle is 90 degrees.
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