Step-by-step explanation:
the essential thing to remember here (and please, burn this into your brain for life) : the sum of all angles in a triangle is always (really always) 180 degrees.
therefore, in 12 and 13 we only need to add all 3 angles, and their sum must be 180.
from there we get really all 4 answers.
the next thing to consider is when a triangle had two equal angles, then it is an isoceles triangle (the 2 sides leading away from these 2 angles are equally long). that gives us the answers for 14 and 15.
12.
3x + 18 + 7x - 58 + 2x - 8 = 180
12x - 48 = 180
12x = 228
x = 19
angle A = 3x+18 = 3×19 + 18 = 57+18 = 75 degrees
angle B = 7x-58 = 7×19 - 58 = 133-58 = 75 degrees
angle C = 2x-8 = 2×19 - 8 = 38 - 8 = 30 degrees
13.
8x - 17 + 19x + 4 + 5x + 1 = 180
32x - 12 = 180
32x = 192
x = 6
angle Q = 8x-17 = 8×6 - 17 = 48-17 = 31 degrees
angle R = 19x+4 = 19×6 + 4 = 114+4 = 118 degrees
angle S = 5x+1 = 5×6 + 1 = 30+1 = 31 degrees
14.
KL = 9x - 14
LM = 7x - 37
KM = 3x + 23
angle K = angle L
so, the sides LM and KM must be equally long. that gives us
7x - 37 = 3x + 23
4x = 60
x = 15
KL = 9x-14 = 9×15 - 14 = 135-14 = 121
LM = 7x-37 = 7×15 - 37 = 105-37 = 68
KM = 3x+23 = 3×15 + 23 = 45+23 = 68
15.
XY = 13x - 21
YZ = 8x - 6
XZ = x + 4
angle X = angle Z
so, the sides XY and ZY (or YZ) must be equally long. that gives us
13x - 21 = 8x - 6
5x = 15
x = 3
XY = 13x-21 = 13×3 - 21 = 39-21 = 18
YZ = 8x-6 = 8×3 - 6 = 24-6 = 18
XZ = x + 4 = 3+4 = 7
