Explanation:
4a)
the principle of similar triangles : they have the same angles.
and we do not forget : the sum of all angles in a triangle is always 180 degrees.
a = 180 - 65 - 45 = 70 degrees
b = 65 degrees
c = 70 degrees
b)
same principles.
x = 58 degrees
y = 52 degrees
z = 180 - 58 - 52 = 70 degrees
c)
same principles.
plus the fact that complementary angles around a point on one side of a line are also together 180 degrees.
x = 85 degrees
y = 180 - 40 - 85 = 55 degrees
z = 180 - 85 = 95 degrees
d)
again same principles.
y = 2x (as these 2 angles correspond to each other among the 2 similar triangles).
the angle POQ is again complementary with 87 degrees,
POQ = 180 - 87 = 93 degrees
x + y + 93 = x + 2x + 93 = 180
3x = 87
x = 29 degrees
y = 2x = 2×29 = 58 degrees