Question

problem 19: lesson 20) the exterior sides of two adjacent angles make a right angle. the first angle has a measure that is 3 more than half of the second. what are the measures of both angles?

Explanation:

Step1: Set up equations

Let the second angle be \(x\). The first angle is \(\frac{x}{2}+ 3\). Since the exterior sides make a right - angle (\(90^{\circ}\)), the sum of the two adjacent angles is \(90^{\circ}\). So the equation is \((\frac{x}{2}+3)+x = 90\).

Step2: Simplify the equation

Combine like terms: \(\frac{x}{2}+x+3=90\). \(\frac{x + 2x}{2}+3=90\), \(\frac{3x}{2}+3 = 90\).
Subtract 3 from both sides: \(\frac{3x}{2}=90 - 3=87\).

Step3: Solve for \(x\)

Multiply both sides by \(\frac{2}{3}\): \(x=\frac{2\times87}{3}\). \(x = 58\).

Step4: Find the first angle

Substitute \(x = 58\) into \(\frac{x}{2}+3\). \(\frac{58}{2}+3=29 + 3=32\).

Answer:

The measures of the angles are \(32^{\circ}\) and \(58^{\circ}\).