Question

m∠1 + m∠2 + m∠3 = 180
m∠2 + m∠4 = 180
∠2 ≅ ∠3
∠2 ≅ ∠4
m∠3 = m∠4
the statement m∠2 + m∠4
the statement ∠2 ≅ ∠3 is
the statement ∠2 ≅ ∠4 is

Explanation:

Step1: Recall vertical - angle and supplementary - angle properties

Vertical angles are congruent. $\angle2$ and $\angle4$ are vertical angles, so $\angle2\cong\angle4$ always. $\angle2$ and $\angle3$ are adjacent angles formed by two intersecting lines. They are supplementary ($\angle2+\angle3 = 180^{\circ}$) but not congruent in general (except when the lines are perpendicular, so $\angle2\cong\angle3$ is sometimes true). Also, given $m\angle2 + m\angle4=180^{\circ}$ is true when the lines are perpendicular, so it is sometimes true.

Step2: Analyze each statement

For $m\angle2 + m\angle4 = 180^{\circ}$, this is true when the intersecting lines are perpendicular, so sometimes true. For $\angle2\cong\angle3$, this is true only when the intersecting lines are perpendicular, so sometimes true. For $\angle2\cong\angle4$, vertical angles are always congruent, so always true.

Answer:

The statement $m\angle2 + m\angle4$ is sometimes true.
The statement $\angle2\cong\angle3$ is sometimes true.
The statement $\angle2\cong\angle4$ is always true.