[tex]m \angle 1 + m \angle 4 = 180^{\circ}[/tex] is always true because angle 1 and angle 4 form a straight line. They are supplementary angles.
[tex]m \angle 5 - m \angle 7 = 0[/tex] is the same as [tex]m \angle 5 = m \angle 7[/tex] which is always true since these two angles are vertical angles.
[tex]z \perp y[/tex] is sometimes true. There are many cases where it is not true, but it's not impossible for lines z and y to be perpendicular.
[tex]\angle 2 \cong \angle 8[/tex] is always true as these are alternate exterior angles. You can use a few options to prove this, but one way is to use angle 2 = angle 4 (vertical angles) and then angle 4 = angle 8 (corresponding angles). By the transitive property, angle 2 = angle 8.
Angles 1 and 3 are congruent, so we can replace [tex]m \angle 3[/tex] with [tex]m \angle 1[/tex], and we can replace angle 4 with angle 2 as well, going from this [tex]m \angle 2 - m \angle 3 = m \angle 1 - m \angle 4[/tex] to this [tex]m \angle 2 - m \angle 1 = m \angle 1 - m \angle 2[/tex] . Next, rearrange things to get [tex]2(m \angle 2 - m \angle 1) = 0[/tex] and that solves to [tex]m \angle 1 = m \angle 2[/tex]. So the original claim is sometimes true. It all depends on if angles 1 and 2 are the same measure or not.
