Question

two angles are complementary. the measure of the larger angle is twelve less than twice the measure of the smaller angle. find the measures of both angles.

Explanation:

Step1: Define variables

Let the measure of the smaller angle be $x$ degrees and the measure of the larger angle be $y$ degrees.

Step2: Set up equations based on the problem

Since the two angles are complementary, $x + y=90$. Also, given that the larger angle is twelve less than twice the measure of the smaller angle, so $y = 2x-12$.

Step3: Substitute the second - equation into the first

Substitute $y = 2x - 12$ into $x + y=90$, we get $x+(2x - 12)=90$.

Step4: Simplify the equation

Combine like - terms: $x+2x-12 = 90$, which simplifies to $3x-12 = 90$.
Add 12 to both sides: $3x=90 + 12$, so $3x=102$.

Step5: Solve for $x$

Divide both sides by 3: $x=\frac{102}{3}=34$.

Step6: Solve for $y$

Substitute $x = 34$ into $y = 2x-12$, then $y=2\times34-12=68 - 12=56$.

Answer:

The measure of the smaller angle is 34 degrees and the measure of the larger angle is 56 degrees.