Read over the starting statement several times and look up any new vocabulary. Reword the statement to an easier language so that you understand the exact meaning of the statement. For example, you could rephrase the above statement as "If two lines cross, the angles directly across from each other are the same size."
Make a drawing with labels that represents the statement. In this case draw two straight lines across the paper. Label the point where they cross "O" for origin. Label one of the lines "a" on the left side of the origin and "b" on the right side. Do the same for the second line, but use "c" on the left side and "d" on the right side.
Look for obvious facts in the drawing that might be related to your statement and write them down. For example, a straight line has an angle of 180 degrees (the horizon), so you can say that the angle between a and d plus the angle between d and b must equal 180 degrees. Likewise the angle between c and b plus the angle between b and d must equal 180 degrees. This can be represented as a^d + d^b = 180 and d^b + b^c = 180 where '^' is a notation for angle.
Work with your observation through trial and error. In this case there are two statements with three unknowns that both equal 180 degrees. Algebraically, this means that you can set a^d + d^b = d^b + b^c. If you take away d^b from both sides (think of these as normal variables and that you are subtracting the same value from both sides) you are left with a^d = b^c. This is the original statement where two opposite angles have the same value.
Write up your proof and remove any redundancy. Earlier it was OK to use lengthy sentences to get an understanding, but now you want to make it as simple as possible without losing any detail. Test your proof in a variety of situations to make sure it is correct.