Answer:
see below
Explanation:
There are a few relevant relations involved:
- an inscribed angle is half the measure of the arc it intercepts
- an arc has the same measure as the central angle that intercepts it
- the angle exterior to a circle where secants meet is half the difference of the intercepted arcs (near and far)
- the angle interior to a circle where secants meet is half the sum of the intercepted arcs
- the angle where tangents meet is the supplement of the (near) arc intercepted
- an exterior angle of a triangle is equal to the sum of the remote interior angles
- the angle between a tangent and a radius is 90°
- the angle sum theorem
AB is a diameter, so arcs AB are 180°.
a) BC is the supplement to arc AC: 180° -140° = 40°
b) BG is the supplement to AG: 180° -64° -38° = 78°
c) ∠1 has the measure of BC: 40°
d) ∠2 is inscribed in a semicircle, so has measure 180°/2 = 90°
e) ∠3 is half the measure of arc AE: 64°/2 = 32°
f) ∠4 is half the sum of arcs AG and BC: ((64°+38°) +40°)/2 = 71°
g) ∠5 is half the difference of arcs AC and EG: (140° -38°)/2 = 51°
h) ∠6 is half the sum of arcs EAC and BG: ((140°+64°) +78°)/2 = 141°
i) ∠7 is the difference of exterior angle 4 and interior angle 1: 71° -40° = 31°
j) ∠8 is the measure of arc AC: 140°
k) ∠9 is the supplement to arc AC: 180° -140° = 40°
l) ∠10 is the complement of angle 7: 90° -31° = 59°