This problem doesn't really belong in this group but okay, I'll answer it.
Given that arm L3 is parallel to the ground, call the joints starting with the one on the ground A, B, C, and D. Extend the first green line, the one through B, to the right. Draw a line through C parallel to the L1 arm, the one though AB, and extend it in both directions from C up to some point E and down to a point F where it intersects this first green line. (I am assuming from your diagram that the first green line is perpendicular to L1 and the second to L2.) Extend the second green line, the one through C, to the left to some point G.
Arguments:
1. Angle DCE equals theta1: two parallel lines make equal angles to another pair of parallel line.
2. Angle FCB equals pi/2-theta2: triangle FCB is a right triangle and angle FCB must be the complement of angle FBC.
3. Angle BCG equals pi/2: the second green line is assumed to be at right angles to L2
4. The sum of angle FCB, angle BCG and angle GCE is pi: Line ECF is a straight line.
5, Angle GCE equals theta2: The total angle at a straight line is pi and
angle GCE = pi - angle BCG - angle FCB = pi - pi/2 - (pi/2-theta2) = theta2
6. Angle DCG equals angle DCE plus angle GCE. Hence theta3 = theta1+theta2
Q.E.D.
