Question

what is $m\angle mhj$? \bigcirc $35^{\circ}$ \bigcirc $50^{\circ}$ \bigcirc $72.5^{\circ}$ \bigcirc $92.5^{\circ}$

Explanation:

Step1: Identify supplementary angles

Angles \(\angle MHJ = (2x - 20)^\circ\) and \(\angle LHK=(x + 15)^\circ\) are supplementary? No, wait, actually \(\angle MHJ\) and \(\angle LHK\) are vertical angles? No, wait, \(MK\) is a straight line, so \(\angle MHJ\) and \(\angle LHK\) are actually... Wait, no, \(\angle MHJ=(2x - 20)^\circ\) and \(\angle LHK=(x + 15)^\circ\) are adjacent to a straight line? Wait, no, \(MK\) is a straight line, so \(\angle MHJ\) and \(\angle KHL\) (which is \((x + 15)^\circ\)) are supplementary? Wait, no, \(\angle MHJ\) and \(\angle KHJ\)? Wait, no, looking at the diagram, \(JL\) is a line intersecting \(MK\) at \(H\). So \(\angle MHJ=(2x - 20)^\circ\) and \(\angle LHK=(x + 15)^\circ\) are vertical angles? No, vertical angles are equal. Wait, no, \(\angle MHJ\) and \(\angle LHK\) are actually supplementary? Wait, no, \(MK\) is a straight line, so the sum of \(\angle MHJ\) and \(\angle KHJ\) is \(180^\circ\), but \(\angle KHJ\) is adjacent to \(\angle LHK\)? Wait, no, \(JL\) is a straight line, so \(\angle MHJ\) and \(\angle LHK\) are vertical angles? Wait, no, vertical angles are formed by two intersecting lines. So \(MK\) and \(JL\) intersect at \(H\), so \(\angle MHJ\) and \(\angle LHK\) are vertical angles? No, \(\angle MHJ\) and \(\angle LHK\) are actually supplementary? Wait, no, let's think again. When two lines intersect, the adjacent angles are supplementary. So \(MK\) is a straight line, so \(\angle MHJ + \angle JHK = 180^\circ\), but \(JL\) is a straight line, so \(\angle JHK + \angle LHK = 180^\circ\), so \(\angle MHJ = \angle LHK\)? No, that can't be. Wait, no, \(\angle MHJ\) and \(\angle LHK\) are vertical angles, so they should be equal. Wait, the problem is that \(\angle MHJ=(2x - 20)^\circ\) and \(\angle LHK=(x + 15)^\circ\) are vertical angles, so they are equal. So:

\(2x - 20 = x + 15\)

Step2: Solve for x

Subtract \(x\) from both sides:

\(2x - x - 20 = 15\)

\(x - 20 = 15\)

Add 20 to both sides:

\(x = 15 + 20 = 35\)

Step3: Find \(m\angle MHJ\)

Now, substitute \(x = 35\) into \(\angle MHJ=(2x - 20)^\circ\):

\(2(35) - 20 = 70 - 20 = 50^\circ\)? Wait, no, wait, that's not right. Wait, no, maybe I made a mistake. Wait, maybe \(\angle MHJ\) and \(\angle LHK\) are supplementary, not vertical. Let's check again. Wait, \(MK\) is a straight line, so the sum of \(\angle MHJ\) and \(\angle KHJ\) is \(180^\circ\), and \(JL\) is a straight line, so the sum of \(\angle KHJ\) and \(\angle LHK\) is \(180^\circ\), so \(\angle MHJ + \angle KHJ = 180^\circ\) and \(\angle KHJ + \angle LHK = 180^\circ\), so \(\angle MHJ = \angle LHK\) (vertical angles). Wait, but if that's the case, then \(2x - 20 = x + 15\), so \(x = 35\). Then \(\angle MHJ = 2(35) - 20 = 70 - 20 = 50^\circ\)? But wait, the options include 50°, but let's check again. Wait, maybe I messed up the angles. Wait, no, maybe \(\angle MHJ\) and \(\angle LHK\) are supplementary. Wait, no, let's look at the diagram again. The angle \(\angle MHJ=(2x - 20)^\circ\) and \(\angle LHK=(x + 15)^\circ\) are actually adjacent to the straight line \(MK\), so they are supplementary. Wait, that makes more sense. Because \(MK\) is a straight line, so the sum of \(\angle MHJ\) and \(\angle LHK\) is \(180^\circ\)? No, that can't be. Wait, no, \(MK\) is a straight line, so the angle on one side of \(H\) should sum to \(180^\circ\). So \(\angle MHJ + \angle JHK = 180^\circ\), and \(JL\) is a straight line, so \(\angle JHK + \angle LHK = 180^\circ\), so \(\angle MHJ = \angle LHK\) (vertical angles). But that would mean \(2x - 20 = x + 15\), so \(x = 35\), then \(\angle MHJ = 2(35) - 20 = 50^\circ\), which is one of the options. Wait, but let's check the other angle. \(\angle LHK = 35 + 15 = 50^\circ\), so they are equal, which makes sense as vertical angles. Then \(m\angle MHJ = 50^\circ\)? But wait, the options have 50° as an option. Wait, but let's check again. Wait, maybe I made a mistake in identifying the angles. Let's look at the diagram again. \(M\)---\(H\)---\(K\) is a straight line, and \(J\)---\(H\)---\(L\) is another straight line. So the angles at \(H\): \(\angle MHJ\) and \(\angle LHK\) are vertical angles, so they are equal. So \(2x - 20 = x + 15\), solve for \(x\):

\(2x - x = 15 + 20\)

\(x = 35\)

Then \(\angle MHJ = 2(35) - 20 = 70 - 20 = 50^\circ\). So that's the answer.

Answer:

\(50^\circ\) (Option: 50°)