Question

in the figure, m is parallel to n and m∠4 = 123°. find the measures of the other angles.
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Explanation:

Step1: Identify vertical - angle relationship

$\angle4$ and $\angle2$ are vertical angles. Vertical angles are equal. So $m\angle2 = 123^{\circ}$.

Step2: Identify linear - pair relationship

$\angle4$ and $\angle3$ form a linear - pair. Since a linear - pair of angles is supplementary ($m\angle4 + m\angle3=180^{\circ}$), then $m\angle3 = 180 - 123=57^{\circ}$. $\angle1$ and $\angle3$ are vertical angles, so $m\angle1 = 57^{\circ}$.

Step3: Use parallel - line properties

Since the two lines are parallel, corresponding angles are equal. $\angle4$ and the angle corresponding to it (let's assume it's another angle in the parallel - line setup) are equal. Also, alternate - interior angles are equal. For example, if we consider the alternate - interior angles related to $\angle3$ and its pair, they are equal.

Answer:

$m\angle1 = 57^{\circ}$, $m\angle2 = 123^{\circ}$, $m\angle3 = 57^{\circ}$ (assuming the standard naming of angles formed by two parallel lines and a transversal)