Question

line jl and line km intersect at point h. find the three unknown angle measures in the figure.
first, find m∠khl.
m∠khl = 129 °
now, find m∠lhm.
m∠lhm = °
(figure shows lines jl and km intersecting at h, with angles 51°, 129° marked and points j, k, h, m, l as shown)

Explanation:

Step1: Identify vertical angle pair

$\angle KHL$ and $\angle LHM$ are not vertical angles; $\angle KHL$ and $\angle JHM$ are vertical angles, while $\angle LHM$ is vertical to $\angle JHK$. Since vertical angles are congruent, $\angle LHM = \angle JHK$.

Step2: Use given vertical angle value

We know $m\angle JHK = 51^\circ$, so $m\angle LHM = 51^\circ$.
(Note: Additionally, the remaining unknown angle $\angle JHM$ is equal to $\angle KHL = 129^\circ$ as vertical angles, completing the three unknowns from the initial prompt.)

Answer:

$m\angle LHM = 51^\circ$, $m\angle JHM = 129^\circ$ (the third unknown angle, vertical to $\angle KHL$)