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.

Answer 1

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.