To find the missing angles in the triangle, we can use the Triangle Angle Sum Theorem. This theorem says that all the angles in a triangle add up to 180 degrees.
We have two angles given: one is \(2x + 6\) degrees and the other is \(3x - 26\) degrees. The third angle, which is the right angle, measures 90 degrees because it's a right triangle.
So, our equation looks like this:
\[ (2x + 6) + (3x - 26) + 90 = 180 \]
Now, we can simplify this step-by-step:
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Combine the angles: \[ 2x + 6 + 3x - 26 + 90 = 180 \]
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Start by combining like terms:
- For the \(x\) terms: \(2x + 3x = 5x\)
- For the constant terms: \(6 - 26 + 90 = 70\)
So it becomes: \[ 5x + 70 = 180 \]
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Now, we want to get \(5x\) by itself. We can do that by subtracting 70 from both sides: \[ 5x = 180 - 70 \] This simplifies to: \[ 5x = 110 \]
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Next, divide both sides by 5 to find \(x\): \[ x = \frac{110}{5} = 22 \]
Now that we know \(x\), we can find the actual angles:
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For the first angle, \(2x + 6\): \[ 2(22) + 6 = 44 + 6 = 50 \] So, the first angle is 50 degrees.
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For the second angle, \(3x - 26\): \[ 3(22) - 26 = 66 - 26 = 40 \] So, the second angle is 40 degrees.
Finally, we can summarize the angles of the triangle:
- One angle is 90 degrees (the right angle),
- Another angle is 50 degrees,
- And the last angle is 40 degrees.
In total, the three angles are 90°, 50°, and 40°. And they all add up to 180 degrees, just like the Triangle Angle Sum Theorem tells us!
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