Find the measure of each angle that is labeled on the diagram shown below. tell what definition, postulate or theorem allows you to justify your answer.

Lets take it in order:

1) 90 deg because the angle right next to it is a right angle
2) 128 deg becasue of the angle directly across from it
3) x = 180 - 128 = 52 deg
4) 180 = 90 + 52 + x so x = 180 - 90 - 52 = 38
5) 180 = 38 + x so x = 180 - 38 = 142
6) Same as 4
7) Same as 5

Check my math, as I just did it in my head 

Ok by definitions, here it is.
1) its 90° since there are 4 of them and you can see from the picture they are same
2) this angle is 128° because definition says that a straight line is 180° and if angle 3) is 180 - 128= 52° (since there is a straight line pointing upwards.) now back to angle 2) there is another straight line pointing up-left and there are angles 2 and 3 on it. since you know now angle 3 is 52°, angle 2 would be 180-52=128°
first lets do 5 so you can understand 4 afterwards
5) see how angle 128 on picture is a straight line pointing up-left intersected with one pointing upwards, same down here with angle 5, that same line pointing up-left is intersected by line pointing to the right, but it is pointing directly to the right, like in previous case where line pointed directly upwards, this angle must be the same as 128°
4) now, there is a straight line for 4) and 5), remember, straight line is 180°, and since 5) is 128°, then 4) is 180-128=52°
now you can do round calculate everything, but you can freely say 7) is equal to 5) and 6) is equal to 4), ANOTHER RULE: an angle which is directly across another angle is same as that other angle, so 7) is across 5) and you see that they are similar in size from the picture, same goes for 4) and 6) who are similar.
RULES so you can learn them better:
1. if one angle is 90°, then all other angles around it are also 90° - angle 1)
2. a straight line, completely straight line, has an angle of 180°, and if you know one angle, the other will always be 180° - that angle you know - angle 3) and 4)
3.When there are 4 angles in one place, and you can see that first and third are similar on picture, and second and fourth are also similar in picture, they are always same, because first angle is across third, and second is across fourth.
hope this helped