Source:https://en.m.wikipedia.org/wiki/Inscribed_angle
I don't understand how the green "theta" angle is claimed to be equal to "theta."
The theta angle is not 90 degrees, and if green theta were equal to theta, the yellow theta would also be equal to theta, but that's a contradiction because two angles on the same line add up to 180 degrees, and their equality would imply they are equal to 90 degrees, but that's not possible per Thale's theorem only inscribed angle 90 degrees corresponds to a right triangle with hypotenuse's length equal to the diameter.
Best Answer
The yellow theta is not equal to theta: note the bit of the theorem that says "subtends the same arc": the arc subtended by the two black thetas is the same (the minor arc below the chord) but the arc subtended by the yellow theta is the major arc above the chord: the angles subtended by the major and minor arcs cut off by a chord sum to 180 degrees - as the yellow and green thetas do. The Wikipedia quote should say "Therefore, the angle does not change as its vertex is moved to different positions on the circle, provided you don't move it to the other side of the chord".