Question

1. the measure of one angle is 8 degrees more than one - third the measure of another angle. if the angles are supplementary, find both angle measures.

Explanation:

Step1: Define the angles

Let one angle be $x$ degrees. Then the other angle is $\frac{1}{3}x + 8$ degrees.

Step2: Use the supplementary - angle property

Since the two angles are supplementary, their sum is 180 degrees. So, we set up the equation $x+(\frac{1}{3}x + 8)=180$.

Step3: Simplify the left - hand side of the equation

Combine like terms: $x+\frac{1}{3}x+8=\frac{3x + x}{3}+8=\frac{4x}{3}+8$. So the equation becomes $\frac{4x}{3}+8 = 180$.

Step4: Isolate the term with $x$

Subtract 8 from both sides: $\frac{4x}{3}=180 - 8=172$.

Step5: Solve for $x$

Multiply both sides by $\frac{3}{4}$: $x = 172\times\frac{3}{4}=129$.

Step6: Find the other angle

Substitute $x = 129$ into $\frac{1}{3}x+8$. We get $\frac{1}{3}\times129 + 8=43 + 8=51$.

Answer:

The two angles are 129 degrees and 51 degrees.