a) the two angles are supplementary, then:
(3x - 45) + (2x + 25) = 180
(3x + 2x) + (-45 + 25) = 180
5x - 20 = 180
5x = 180 + 20
5x = 200
x = 200/5
x = 40
Then, the measure of the angles are:
3x - 45 = 3*40 - 45 = 75°
2x + 25 = 2*40 + 25 = 105°
b) the two angles are complementary, then:
(2y) + (6y - 30) = 90
(2y + 6y) - 30 = 90
8y = 90 + 30
8y = 120
y = 120/8
y = 15
Then, the measure of the angles are:
2y = 2*15 = 30°
6y - 30 = 6*15 - 30 = 60°
c) the two angles are vertical angles, then they are congruent, that is,
2z + 7 = 5z - 23
7 + 23 = 5z - 2z
30 = 3z
30/3 = z
10 = z
Then, the measure of the angles are:
2z + 7 = 2*10 + 7 = 27°
5z - 23 = 5*10 - 23 = 27°
d) the two angles are alternative exterior angles, then they are congruent, that is,
4x - 5 = 3x + 15
4x - 3x = 15 + 5
x = 20
Then, the measure of the angles are:
4x - 5 = 4*20 - 5 = 75°
3x + 15 = 3*20 + 15 = 75°