Question

find the value of x using the figure, shown to the right. the value of x is \boxed{}. (simplify your answer. do not include the degree symbol in your answer.)

Explanation:

Step1: Identify angle relationship

The two angles \(60^\circ\) and \(20x^\circ\) are complementary (they form a right angle, so their sum is \(90^\circ\)). So, we set up the equation: \(60 + 20x = 90\).

Step2: Solve for x

Subtract 60 from both sides: \(20x = 90 - 60\)
\(20x = 30\)
Divide both sides by 20: \(x = \frac{30}{20}\)
Simplify: \(x = \frac{3}{2} = 1.5\)? Wait, no, wait—wait, maybe they are supplementary? Wait, no, the figure: if the two pink arrows are parallel, and the transversal, but wait, the angles given: 60 and 20x. Wait, maybe they are complementary? Wait, no, maybe I misread. Wait, the problem: the two angles, 60 and 20x, are they adjacent and form a right angle? Wait, no, maybe they are supplementary? Wait, no, let's check again. Wait, the sum of 60 and 20x: if they are complementary, sum to 90. If supplementary, sum to 180. Wait, 60 + 20x = 90? Then 20x=30, x=1.5? But that seems odd. Wait, maybe the angles are vertical or alternate? Wait, no, the figure: two parallel lines (pink arrows) cut by a transversal, and another line. Wait, maybe the 60 and 20x are complementary (since they form a right angle). Wait, let's re-express.

Wait, maybe the correct equation is \(60 + 20x = 90\)? Wait, no, maybe 60 and 20x are supplementary? Wait, 60 + 20x = 180? Then 20x=120, x=6. Wait, that makes more sense. Wait, maybe I misjudged the angle relationship. Let's think again. If the two lines with pink arrows are parallel, and the transversal, but the other line: maybe the 60 and 20x are supplementary (they form a linear pair). Wait, the problem says "using the figure", but the figure shows two angles: 60 and 20x, adjacent, forming a right angle? No, wait, the user's figure: the two angles, 60 and 20x, are they adding up to 90 or 180? Wait, maybe the correct approach is: if the two angles are complementary (sum to 90), then 60 + 20x = 90. Solving: 20x = 30 → x = 1.5. But that seems small. Wait, maybe the angles are supplementary (sum to 180). Then 60 + 20x = 180 → 20x = 120 → x = 6. Wait, which is it?

Wait, looking at the figure: the two pink arrows are parallel, and the transversal creates angles. Wait, maybe the 60 and 20x are complementary (right angle). Wait, the problem says "the value of x". Let's check the equations again.

Wait, maybe the correct angle relationship is that 60 and 20x are complementary (they form a right angle, so their sum is 90 degrees). So:

\(60 + 20x = 90\)

Subtract 60 from both sides:

\(20x = 90 - 60\)

\(20x = 30\)

Divide both sides by 20:

\(x = \frac{30}{20} = \frac{3}{2} = 1.5\)? No, that's 1.5, but maybe I made a mistake. Wait, maybe the angles are supplementary. Let's try that:

\(60 + 20x = 180\)

\(20x = 120\)

\(x = 6\)

Ah, that's more likely. Maybe the two angles are supplementary (they form a linear pair, so sum to 180 degrees). So the correct equation is 60 + 20x = 180. Then solving:

20x = 180 - 60 = 120

x = 120 / 20 = 6.

Yes, that makes sense. So the correct step is:

Step1: Set up supplementary angles equation

The angles \(60^\circ\) and \(20x^\circ\) are supplementary (sum to \(180^\circ\)), so:
\(60 + 20x = 180\)

Step2: Solve for \(x\)

Subtract 60 from both sides:
\(20x = 180 - 60\)
\(20x = 120\)

Divide both sides by 20:
\(x = \frac{120}{20}\)
\(x = 6\)

Answer:

6