A right triangle has acute angles measuring 2x+6 degrees and 3x-26degrees . Use the triangle angle sum theorem to find the measures of the missing angles of the triangle.
The sum of the angles in any triangle is always 180 degrees. In this case, we are given that one angle measures 2x+6 degrees and another angle measures 3x-26 degrees.
Let's denote the third angle as y degrees. Therefore, the sum of the three angles can be written as:
(2x+6) + (3x-26) + y = 180
Simplifying the equation, we get:
5x - 20 + y = 180
5x + y = 200
y = 200 - 5x
Since the triangle is a right triangle, one of the angles must be 90 degrees. This means that either 2x+6 = 90 or 3x-26 = 90. Solving for both cases:
Case 1: 2x+6 = 90
2x = 84
x = 42
Substitute x back into y = 200 - 5x:
y = 200 - 5(42)
y = 200 - 210
y = -10
Case 2: 3x-26 = 90
3x = 116
x = 38.67
Substitute x back into y = 200 - 5x:
y = 200 - 5(38.67)
y = 200 - 193.33
y = 6.67
Therefore, the angles of the right triangle are as follows:
- Angle 1: 2x+6 = 2(42)+6 = 90 degrees
- Angle 2: 3x-26 = 3(42)-26 = 120 degrees
- Angle 3: y = -10 degrees or 6.67 degrees
Note: Since angles cannot have negative measures, y = 6.67 degrees is the only valid solution.