Question

two angles are a linear pair. half the measure of the first plus 9 is equal to the measure of the second. what are the measures of both angles?

Explanation:

Step1: Set up equations

Let the first angle be $x$ and the second angle be $y$. Since they are a linear - pair, $x + y=180$ (linear - pair angles add up to 180 degrees), and $\frac{x}{2}+9 = y$.

Step2: Substitute $y$ in the first equation

Substitute $y=\frac{x}{2}+9$ into $x + y=180$. We get $x+(\frac{x}{2}+9)=180$.

Step3: Simplify the equation

Combine like - terms: $x+\frac{x}{2}+9 = 180$. First, rewrite $x$ as $\frac{2x}{2}$, then $\frac{2x}{2}+\frac{x}{2}+9 = 180$, which gives $\frac{2x + x}{2}+9 = 180$, or $\frac{3x}{2}+9 = 180$.

Step4: Isolate the variable term

Subtract 9 from both sides: $\frac{3x}{2}=180 - 9=171$.

Step5: Solve for $x$

Multiply both sides by $\frac{2}{3}$: $x = 171\times\frac{2}{3}=114$.

Step6: Solve for $y$

Substitute $x = 108$ into $y=\frac{x}{2}+9$. Then $y=\frac{108}{2}+9=54 + 9=72$. So the first angle $x = 108$ degrees and the second angle $y = 72$ degrees.

Answer:

The first angle is 108 degrees and the second angle is 72 degrees.