Answer:
1. a = 64°, b = 296°, c = 148°
2. a = 90°, b = 32°, c = 58°, d = 58°, e = 29°, f = 29°, g = 29°, h = 29°, i = 122°, j = 122°
3. a = 35°, b = 35°, c = 70°, d = 35°, e = 35°, f = 110°, g = 22°
4. a = 27°, b = 63°, c = 27°, d = 54°, e = 90°, f = 36°, g = 63°
5. t = 40°, u = 40°, v = 80°, w = 21°, x = 42°, y = 19°, z = 19°
Step-by-step explanation:
On the same arc angle subtended at center is twice the angle subtended at circumference
same arc, same angle on any point on the circumference
1.
a = 2 * 32° = 64°
b = 360° - a = 360° - 64° = 296°
c = b/2 = 296°/2 = 148°
a = 64°, b = 296°, c = 148°
2.
a = 180 - 90 = 90°
triangle OMT and triangle ONT are congruent
b = 32°
c = 90 - 32 = 58°
d = c = 58°
f = c /2 = 58/2 = 29°
exterior angle, c = h + f
h = c - f = 58 - 29 = 29°
j = 180 - c = 180 - 58 = 122°
similarly, e = g = 29°, i = 122°
a = 90°, b = 32°, c = 58°, d = 58°, e = 29°, f = 29°, g = 29°, h = 29°, i = 122°, j = 122°
3.
a = 70/2 = 35°
As TR = VT, b = a = 35°
arc VTR, c = a + b = 35° + 35° = 70°
arc VT, d = a = 35°
TR = VT, e = d = 35°
cyclic quadrilateral, f + a + b = 180, f = 180 - a - b = 180 - 35 - 35 = 110°
arc PQ, g = angle PRQ = 22°
a = 35°, b = 35°, c = 70°, d = 35°, e = 35°, f = 110°, g = 22°
4.
arc PQ, a = 27°
b = 90 - 27 = 63°
MP is diameter, b + c = 90°, c = 90 - b = 90 - 63 = 27°
g = 90 - c = 90 - 27 = 63°
e = 180 - 90 = 90°
angle NPR = 90 - 27 = 63°
ON + OP, a + f = angle NPR = 63°, f = 63 - a = 63 - 27 = 36°
d = 90 - f = 90 - 36 = 54°
a = 27°, b = 63°, c = 27°, d = 54°, e = 90°, f = 36°, g = 63°
5.
arc BC, t = u = 40°
arc AB, w = 21°
arc BC, v = 2u = 2 * 40 = 80°
arc AB, x = 2 * 21 = 42°
OA = OC, y = z = ( 180 - x - v )/2 = ( 160 - 42 - 80 )/2 = 19°
t = 40°, u = 40°, v = 80°, w = 21°, x = 42°, y = 19°, z = 19°
