Question

line jl and line km intersect at point h. find the three unknown angle measures in the figure.
first, find ( mangle khl ).
( mangle khl = 129^circ )
now, find ( mangle lhm ).
( mangle lhm = 51^circ )
finally, find ( mangle mhj ).
( mangle mhj = square^circ )

Explanation:

Step1: Identify angle relationship

∠MHJ and ∠KHL are vertical angles (or ∠MHJ + ∠LHM = 180°? Wait, no. Wait, line JL and KM intersect at H. So ∠MHJ and ∠KHL: Wait, looking at the diagram, ∠KHL is 129°, and ∠MHJ should be equal to ∠KHL? Wait, no, wait. Wait, ∠LHM is 51°, and ∠MHJ: let's see, ∠MHJ and ∠KHL? Wait, no, let's use linear pair. ∠LHM is 51°, so ∠MHJ + ∠LHM = 180°? Wait, no, ∠JHL is a straight line? Wait, JL is a straight line, so ∠JHM is a straight line? Wait, no, JL and KM intersect at H. So ∠MHJ and ∠KHL: Wait, ∠KHL is 129°, and ∠MHJ should be equal to ∠KHL because they are vertical angles? Wait, no, wait. Wait, the angle between JH and MH: let's see, ∠LHM is 51°, so ∠MHJ = 180° - 51°? No, wait, ∠KHL is 129°, and ∠MHJ is vertical to ∠KHL? Wait, no, let's check the diagram again. The given angles: 51°, 129°, 51°. So ∠MHJ: since JL is a straight line, the angle on a straight line is 180°. So ∠MHJ + ∠LHM = 180°? Wait, ∠LHM is 51°, so ∠MHJ = 180 - 51 = 129? Wait, no, wait. Wait, ∠KHL is 129°, and ∠MHJ is vertical to ∠KHL? Wait, maybe I made a mistake. Wait, the problem: first, m∠KHL is 129, m∠LHM is 51. Now, ∠MHJ: since JL is a straight line, ∠MHJ + ∠LHM = 180°? Wait, no, ∠JHL is 180°, so ∠MHJ + ∠LHM = 180°? Wait, ∠LHM is 51°, so ∠MHJ = 180 - 51 = 129? Wait, but also, ∠MHJ and ∠KHL are vertical angles, so they should be equal. ∠KHL is 129°, so ∠MHJ is 129°? Wait, but let's check the linear pair. ∠LHM is 51°, so ∠MHJ + 51° = 180°? 180 - 51 = 129. Yes, that's correct. So m∠MHJ = 180 - 51 = 129, or since it's vertical to ∠KHL (129°), so it's 129°.

Step2: Calculate m∠MHJ

Using linear pair: ∠MHJ + ∠LHM = 180° (since JL is a straight line). Given m∠LHM = 51°, so m∠MHJ = 180 - 51 = 129°.

Answer:

129