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:
- \( \theta_1 = 2x + 6 \) degrees
- \( \theta_2 = 3x - 26 \) 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:
- For \( \theta_1 = 2x + 6 \):
\[ \theta_1 = 2(22) + 6 = 44 + 6 = 50 \text{ degrees} \]
- 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.
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