Question

given $m \parallel n$, find the value of $x$ and $y$.

(the image shows two parallel lines m and n cut by a transversal. angles formed are $(9x - 4)\degree$, $(8x + 1)\degree$, and $y\degree$.)

Explanation:

Step1: Identify Vertical Angles

The angles \((9x - 4)^\circ\) and \((8x + 1)^\circ\) are vertical angles? Wait, no, actually, since \(m\parallel n\), but first, notice that \((9x - 4)^\circ\) and \((8x + 1)^\circ\) are adjacent and form a linear pair? Wait, no, looking at the diagram, \((9x - 4)^\circ\) and \((8x + 1)^\circ\) are actually vertical angles? Wait, no, maybe they are supplementary? Wait, no, let's correct. Wait, the angle \((9x - 4)^\circ\) and \((8x + 1)^\circ\): wait, actually, since \(m\) is a straight line, the sum of \((9x - 4)^\circ\), \((8x + 1)^\circ\), and \(y^\circ\)? No, wait, \((9x - 4)^\circ\) and \((8x + 1)^\circ\) are vertical angles? Wait, no, vertical angles are equal. Wait, maybe I made a mistake. Wait, the two angles \((9x - 4)^\circ\) and \((8x + 1)^\circ\) are vertical angles? Wait, no, looking at the diagram, the angle \((9x - 4)^\circ\) and \((8x + 1)^\circ\) are actually equal because they are vertical angles? Wait, no, vertical angles are opposite each other when two lines intersect. Wait, the transversal intersects line \(m\), so the angles \((9x - 4)^\circ\) and \((8x + 1)^\circ\) are adjacent? Wait, no, let's re-examine. Wait, the angle \((9x - 4)^\circ\) and \((8x + 1)^\circ\) are actually equal because they are vertical angles? Wait, no, maybe they are supplementary? Wait, no, let's check the problem again. Wait, the two angles \((9x - 4)^\circ\) and \((8x + 1)^\circ\) are vertical angles, so they should be equal. So set \(9x - 4 = 8x + 1\).

$$9x - 4 = 8x + 1$$

Step2: Solve for \(x\)

Subtract \(8x\) from both sides:

$$x - 4 = 1$$

Add 4 to both sides:

$$x = 5$$

Step3: Find the Measure of the Angle

Now substitute \(x = 5\) into \((9x - 4)^\circ\):

$$9(5) - 4 = 45 - 4 = 41^\circ$$

And into \((8x + 1)^\circ\):

$$8(5) + 1 = 40 + 1 = 41^\circ$$

, so that checks out. Now, the angle \(y^\circ\) and the angle \((9x - 4)^\circ\) are supplementary (since they form a linear pair on line \(m\)). So:

$$y + (9x - 4) = 180$$

Substitute \(x = 5\):

$$y + 41 = 180$$

Step4: Solve for \(y\)

Subtract 41 from both sides:

$$y = 180 - 41 = 139$$

Answer:

\(x = 5\), \(y = 139\)