Question

the lines on a tennis court form adjacent angles with measures of $(3a - 27)^circ$ and $(2b + 14)^circ$. find the values for $a$ and $b$ so that the lines are perpendicular. round to the nearest whole number if needed.

a=

b=

Explanation:

Step1: Recall perpendicular angles property

Perpendicular lines form right angles, so adjacent angles sum to \(90^\circ\). Thus, \((3a - 27) + (2b + 14) = 90\)? Wait, no—wait, actually, if two lines are perpendicular, the adjacent angles (they should be complementary? Wait, no, when two lines are perpendicular, the adjacent angles formed are right angles? Wait, no, maybe the two angles are the angles between the lines, so if the lines are perpendicular, each angle is \(90^\circ\)? Wait, no, maybe the two angles are adjacent and form a right angle, so each angle is \(90^\circ\)? Wait, no, the problem says "the lines on a tennis court form adjacent angles"—maybe the two angles are the angles between the lines, and if the lines are perpendicular, then each angle is \(90^\circ\)? Wait, no, maybe the two angles are adjacent and their sum is \(90^\circ\)? Wait, no, when two lines are perpendicular, the adjacent angles (like, if two lines intersect at right angles, the adjacent angles are each \(90^\circ\))? Wait, no, if two lines are perpendicular, the angles between them are \(90^\circ\). Wait, maybe the two angles given are the angles formed by the intersection, and since the lines are perpendicular, each angle is \(90^\circ\). Wait, that makes sense. So \(3a - 27 = 90\) and \(2b + 14 = 90\)? Wait, no, maybe the two angles are adjacent and form a right angle, so their sum is \(90^\circ\)? Wait, no, if two lines are perpendicular, the adjacent angles (linear pair? No, perpendicular lines intersect at right angles, so the four angles are all \(90^\circ\). Wait, maybe the two angles are the angles between the lines, so each is \(90^\circ\). Let's re-read the problem: "The lines on a tennis court form adjacent angles with measures of \((3a - 27)^\circ\) and \((2b + 14)^\circ\). Find the values for \(a\) and \(b\) so that the lines are perpendicular." So if the lines are perpendicular, the adjacent angles (formed by their intersection) should be right angles, so each angle is \(90^\circ\). So we set each angle equal to \(90^\circ\).

Step2: Solve for \(a\)

Set \(3a - 27 = 90\).
Add 27 to both sides: \(3a = 90 + 27 = 117\).
Divide by 3: \(a = \frac{117}{3} = 39\).

Step3: Solve for \(b\)

Set \(2b + 14 = 90\).
Subtract 14 from both sides: \(2b = 90 - 14 = 76\).
Divide by 2: \(b = \frac{76}{2} = 38\).

Answer:

\(a = 39\)
\(b = 38\)