Two angles are complementary if the sum of their measures is equal to \( 90^{\circ} \). Georgia draws two complementary angles. One of the angles measures \( 15^{\circ} \) more than 2 times the other angle's measure. A. Model with Mathematics Write a system of equations to represent the situation. B. What are the measures of the two angles?

To model the situation with mathematics, we can represent the measures of the two complementary angles as variables. Let's denote the measure of the smaller angle as \( x \) and the measure of the larger angle as \( y \). Given that one angle measures \( 15^{\circ} \) more than 2 times the other angle's measure, we can write the equation: \[ y = 2x + 15 \] Since the two angles are complementary, their sum is equal to \( 90^{\circ} \), so we can write the equation: \[ x + y = 90 \] Now, we have a system of equations: \[ \begin{cases} y = 2x + 15 \\ x + y = 90 \end{cases} \] To find the measures of the two angles, we need to solve this system of equations. Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}y=2x+15\\x+y=90\end{array}\right.\) - step1: Substitute the value of \(y:\) \(x+2x+15=90\) - step2: Add the terms: \(3x+15=90\) - step3: Move the constant to the right side: \(3x=90-15\) - step4: Subtract the numbers: \(3x=75\) - step5: Divide both sides: \(\frac{3x}{3}=\frac{75}{3}\) - step6: Divide the numbers: \(x=25\) - step7: Substitute the value of \(x:\) \(y=2\times 25+15\) - step8: Simplify: \(y=65\) - step9: Calculate: \(\left\{ \begin{array}{l}x=25\\y=65\end{array}\right.\) - step10: Check the solution: \(\left\{ \begin{array}{l}x=25\\y=65\end{array}\right.\) - step11: Rewrite: \(\left(x,y\right) = \left(25,65\right)\) The solution to the system of equations is \( (x, y) = (25, 65) \). Therefore, the measures of the two angles are \( 25^{\circ} \) and \( 65^{\circ} \).