Why are these angles the same?

Question

I'm doing a problem for an hour now because I'm stuck. I decided to look at the solutions and saw this picture. I can't seem to figure out how the theta angles are the same? Can somebody please explain?

Answer 1

If the tangents at $A$ and $B$ intersects at point $X$ (just below where the $d$ is on the diagram), then consider the quadrilateral $OAXB$:

$\angle A = \angle B = 90^\circ$ and $\angle O = \theta$, and $\angle O + \angle A + \angle B + \angle X = 360^\circ$, so the external angle of $\angle X$ would also be $\theta$.

Answer 2

Hint :

• Construct the height from point $B$ , can you see which angle is equal to $\theta$?

• To conclude, remember that the triangle is a right one.

Answer 3

Turning the entire figure counterclockwise by $\theta$ moves $OA$ to $OB$ ($O$ is the center of the earth), because the angle between the two lines is $\theta$.

Turning the entire figure counterclockwise by $\theta$ moves "to first sunset" to "to second sunset". Thus the angle between those two lines is also $\theta$.