To solve this, let’s define the angles.

Let:

• $x$ be the measure of the first angle.
• $y$ be the measure of the second angle.

Step 1: Set up the equations

We know two things:

1. The angles are complementary, meaning their sum is $90^\circ$. This gives us the equation: $x + y = 90$
2. The measure of one angle is six more than half the measure of the other angle. So, we also know: $x = \frac{1}{2}y + 6$

Step 2: Solve the system of equations

We have two equations:

1. $x + y = 90$
2. $x = \frac{1}{2}y + 6$

Substitute the second equation into the first: $\left( \frac{1}{2}y + 6 \right) + y = 90$ Now, solve for $y$.

Step 3: Solve for $y$

Combine like terms: $\frac{1}{2}y + y + 6 = 90$ $\frac{3}{2}y + 6 = 90$ Subtract 6 from both sides: $\frac{3}{2}y = 84$ Multiply both sides by $\frac{2}{3}$ to isolate $y$: $y = 56$

Step 4: Solve for $x$

Now that we know $y = 56$, substitute this back into the equation for $x$: $x = \frac{1}{2}(56) + 6$ $x = 28 + 6 = 34$

Final Answer

The two angles are:

• $x = 34^\circ$
• $y = 56^\circ$

These angles are complementary since $34 + 56 = 90^\circ$.

Would you like more details on the solution?

Here are 5 related questions:

1. How do we solve systems of linear equations in general?
2. What are complementary angles, and why do they sum to 90 degrees?
3. How can you verify a solution to an angle-related problem?
4. What are some real-world applications of complementary angles?
5. How do you approach problems with unknowns expressed in terms of other variables?

Tip: When solving systems of equations, substituting one equation into another can help eliminate a variable, making the problem easier to solve.