Step-by-step explanation:
10)
any triangle inscribed into a circle with the baseline being a diameter of the circle must be a right-angled triangle.
the arc angle of 84° means that also the angle at Q is 84°, because the arc angle is the same as the segment angle at the center of the circle (that is how arc sngles are defined).
that means the supplementary angle (together they have 180°) of Q at the inner top angle of the small left triangle is 180 - 84 = 96°.
as the legs of this small left triangle are both the radius of the circle, this triangle is an isoceles triangle (both legs are equally long, and both angles at its baseline are equally large).
remember, the sum of all angles in a triangle is always 180°.
so,
y = (180 - 96)/2 = 42°
x is a complimentary angle (together they are 90°, remember, the large triangle is inscribed and is right-angled) to the second 42° angle
x = 90 - 42 = 48°
11)
y is the supplementary angle to 126°
y = 180 - 126 = 54°
this inner segment angle is the mean value between the 2 outer angles.
54 = (57 + 2nd outer angle of y)/2
108 = 57 + 2nd outer angle of y
2nd outer angle of y = 108 - 57 = 51°
all arc angles around a circle must be 360°.
x = 360 - 57 - 104 - 51 = 148°
12)
again, all arc angles together must be 360°.
y = 360 - 148 = 212°
x is the same angle as the top angle of a triangle inscribed to the right of the chord.
and that angle is half of the opposing arc angle (148°).
it is therefore 148/2 = 74°
and so,
x = 74°
