Question

m∠klm = m∠lkn because they are alternate interior angles.
m∠lmk = m∠lkm because they are
supplementary angles

Explanation:

Step1: Use the property of supplementary angles

Since \(m\angle LMK\) and \(m\angle LKM\) are supplementary angles, we know that \(m\angle LMK + m\angle LKM=180^{\circ}\). Given \(m\angle LMK = 83^{\circ}\), then \(m\angle LKM=180^{\circ}- 83^{\circ}=97^{\circ}\).

Step2: Use the angle - sum property of a triangle

In \(\triangle LK\) (assuming the triangle formed by the relevant angles), if we consider the angles \(29^{\circ}\), \(x^{\circ}\) and \(m\angle LKM - 29^{\circ}\) (because of the alternate - interior - angle relation \(m\angle KLM=m\angle LKN\)).

We know that the sum of angles in a triangle is \(180^{\circ}\). But another way: Since \(m\angle LKM=x + 29\) (by the angle - addition postulate, as \(m\angle LKM\) is composed of \(x\) and \(29^{\circ}\) angles).

Since \(m\angle LKM = 97^{\circ}\), then \(x=97 - 29\).

Answer:

\(x = 54\)