A right triangle has acute angles measuring 2x + 6 degrees and 3x - 26. Use the Triangle Angle Sum Theorem to find the measures of the missing angles of the triangle.
The Triangle Angle Sum Theorem states that in any triangle, the sum of the measures of the angles is 180 degrees.
So, in this right triangle, the right angle measures 90 degrees. Therefore, we have:
2x + 6 + 3x - 26 + 90 = 180
5x - 20 + 90 = 180
5x + 70 = 180
5x = 110
x = 22
Now we can find the measures of the acute angles:
First angle:
2x + 6 = 2(22) + 6 = 44 + 6 = 50 degrees
Second angle:
3x - 26 = 3(22) - 26 = 66 - 26 = 40 degrees
Therefore, the missing angles in the right triangle are 50 degrees and 40 degrees.
explain
Sure, I'd be happy to explain the steps further.
We have a right triangle with two acute angles and a right angle. The sum of the measures of the angles in any triangle is always 180 degrees. In this case, the right angle measures 90 degrees.
Let's denote the acute angles as A and B. We know that the right angle is 90 degrees. So, we can write the equation:
A + B + 90 = 180
Given that A = 2x + 6 and B = 3x - 26, we substitute these values into the equation:
(2x + 6) + (3x - 26) + 90 = 180
Next, we solve for x by combining like terms:
5x - 20 + 90 = 180
5x + 70 = 180
5x = 110
x = 22
Now that we have found the value of x, we can substitute it back into the expressions for A and B to find the measures of the acute angles.
For A:
A = 2x + 6
A = 2(22) + 6
A = 44 + 6
A = 50 degrees
For B:
B = 3x - 26
B = 3(22) - 26
B = 66 - 26
B = 40 degrees
Therefore, the missing angles in the right triangle measure 50 degrees and 40 degrees.