Problem Statement

Find the angle which is 4 times its complement and is 10 degrees less than twice its complement.

Solution

Let's assume the angle as 'x' degrees.

Complementary angles
Complementary angles are two angles that add up to 90 degrees. If 'x' is the angle, then its complement will be (90 - x) degrees.

Given Conditions
According to the problem, we have two conditions:

1. The angle is 4 times its complement:
x = 4(90 - x)

2. The angle is 10 degrees less than twice its complement:
x = 2(90 - x) - 10

Solving the Equations
Let's solve the given equations to find the value of 'x'.

1. x = 4(90 - x)
Distribute the 4: x = 360 - 4x
Add 4x to both sides: 5x = 360
Divide by 5: x = 72

2. x = 2(90 - x) - 10
Distribute the 2: x = 180 - 2x - 10
Combine like terms: x = 170 - 2x
Add 2x to both sides: 3x = 170
Divide by 3: x = 56.67

Conclusion
By solving the equations, we found two possible values for 'x': 72 degrees and 56.67 degrees. Therefore, the angle can be either 72 degrees or 56.67 degrees.