Question

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

Explanation:

Step1: Analyze vertical - angle relationship

Vertical angles are always congruent. $\angle2$ and $\angle3$ are vertical angles, so $\angle2\cong\angle3$ always holds.

Step2: Analyze supplementary - angle relationship

We know that $m\angle2 + m\angle4=180^{\circ}$. They are supplementary. $\angle2\cong\angle4$ only when $m\angle2 = m\angle4 = 90^{\circ}$, so it is sometimes true.

Step3: Analyze the equality of $\angle3$ and $\angle4$

Since $\angle2\cong\angle3$ and $m\angle2 + m\angle4 = 180^{\circ}$, if $m\angle3=m\angle4$, then $m\angle2 + m\angle3=180^{\circ}$. But $\angle2$ and $\angle3$ are vertical - angles and non - supplementary (except in the case of right angles). So $m\angle3=m\angle4$ is never true.

Answer:

The statement $\angle2\cong\angle3$ is always true.
The statement $\angle2\cong\angle4$ is sometimes true.
The statement $m\angle3 = m\angle4$ is never true.