Question

Need help on number 5

Answer 1

• 
  `m \angle 1 + m \angle 4 = 180^(\circ)` is always true because angle 1 and angle 4 form a straight line. They are supplementary angles.
• 
  `m \angle 5 - m \angle 7 = 0` is the same as
  `m \angle 5 = m \angle 7` which is always true since these two angles are vertical angles.
• 
  `z \perp y` 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.
• 
  `\angle 2 \cong \angle 8` 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
  `m \angle 3` with
  `m \angle 1`, and we can replace angle 4 with angle 2 as well, going from this
  `m \angle 2 - m \angle 3 = m \angle 1 - m \angle 4` to this
  `m \angle 2 - m \angle 1 = m \angle 1 - m \angle 2` . Next, rearrange things to get
  `2(m \angle 2 - m \angle 1) = 0` and that solves to
  `m \angle 1 = m \angle 2`. So the original claim is sometimes true. It all depends on if angles 1 and 2 are the same measure or not.