Question

the difference between the measures of two complementary angles is 72°. determine the measures of the two angles. the larger angle has a measure of °. the smaller angle has a measure of °.

Explanation:

Step1: Set up equations

Let the larger angle be $x$ and the smaller angle be $y$. Since they are complementary, $x + y=90^{\circ}$, and the difference is $72^{\circ}$, so $x - y=72^{\circ}$.

Step2: Solve the system of equations

Add the two - equations together: $(x + y)+(x - y)=90^{\circ}+72^{\circ}$. Simplifying gives $2x = 162^{\circ}$, so $x=\frac{162^{\circ}}{2}=81^{\circ}$.

Step3: Find the value of the other angle

Substitute $x = 81^{\circ}$ into $x + y=90^{\circ}$, we get $81^{\circ}+y = 90^{\circ}$, then $y=90^{\circ}-81^{\circ}=9^{\circ}$.

Answer:

The larger angle has a measure of $81^{\circ}$.
The smaller angle has a measure of $9^{\circ}$.