Question

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

Explanation:

Step1: Recall perpendicular angles property

Perpendicular lines form right angles, so adjacent angles sum to \(90^\circ\). Wait, no—wait, actually, when two lines are perpendicular, adjacent angles (like a linear pair? No, wait, on a tennis court, the lines are perpendicular, so the adjacent angles formed should be complementary? Wait, no, if two lines are perpendicular, the adjacent angles (forming a right angle) should each be \(90^\circ\)? Wait, no, maybe I misread. Wait, the problem says "adjacent angles" with measures \((3a - 27)^\circ\) and \((2b + 14)^\circ\). Wait, maybe they are complementary? No, wait, if the lines are perpendicular, the adjacent angles (like the angles formed by a transversal and the perpendicular lines) – wait, no, maybe the two angles are each \(90^\circ\)? Wait, no, that doesn't make sense. Wait, maybe the two angles are complementary? Wait, no, perpendicular lines form right angles, so if two lines are perpendicular, the adjacent angles (forming a linear pair? No, maybe the two angles are the angles between the lines, so they should be \(90^\circ\) each? Wait, no, let's re-express.

Wait, maybe the two angles are adjacent and form a right angle, so their sum is \(90^\circ\)? No, that can't be. Wait, no—wait, when two lines are perpendicular, the angle between them is \(90^\circ\). But the problem says "adjacent angles" with measures \((3a - 27)^\circ\) and \((2b + 14)^\circ\). Wait, maybe there's a typo, and actually, the two angles are equal? No, that doesn't make sense. Wait, maybe the two angles are each \(90^\circ\)? Wait, no, that would mean \(3a - 27 = 90\) and \(2b + 14 = 90\)? Wait, that might be it. Because if the lines are perpendicular, the adjacent angles (like the angles formed by the intersection of the lines) – wait, no, adjacent angles formed by perpendicular lines: if two lines are perpendicular, they intersect at \(90^\circ\), so the adjacent angles (vertical angles? No, adjacent angles) – wait, maybe the two angles are the angles of the perpendicular lines, so each is \(90^\circ\). Let's check that.

So, if the lines are perpendicular, then each adjacent angle (the angles formed by the intersection) should be \(90^\circ\). 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: \(b = \frac{76}{2} = 38\)

Wait, but let's verify. If \(a = 39\), then \(3a - 27 = 3*39 -27 = 117 -27 = 90\). If \(b = 38\), then \(2b +14 = 2*38 +14 = 76 +14 = 90\). So both angles are \(90^\circ\), which makes sense for perpendicular lines (they form right angles). So that works.

Answer:

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