*work backward.*

Try an example:

Here's the proof diagram.

Let's say you manage to complete five lines of the proof, but you get stumped at "angle 3 is congruent to angle 4."

Where to now? Going forward from here may be a bit tricky, so work backward. You know that the final line of the proof has to be the *prove* statement:

Now, if you think about what the final reason has to be or what the second-to-last statement should be, it shouldn't be hard to see that you need to have two congruent angles (the two half-angles) to conclude that a larger angle is bisected.

Here's what the end of the proof looks like.

Note the if-then logic bubbles (*if* clauses in reasons connect to statements above; *then* clauses in reasons connect to statements on the same line).

Okay, so picking up where you left off on the proof: You've completed five lines of the proof, and you're up to

Now go to the end, and try working your way backward to the third-to-last statement, the fourth-to-last statement, and so on. (Working backward through a proof always involves some guesswork, but don't let that stop you.) Why might angle 7 be congruent to angle 8? Well, you probably don't have to look too hard to spot the pair of congruent vertical angles, 5 and 8, and the other pair, 6 and 7.

Okay, so you want to show that angle 7 is congruent to angle 8, and you know that angle 6 equals angle 7 and angle 5 equals angle 8. So if you were to know that angles 5 and 6 are congruent, you'd be home free.

Now that you've worked backward a number of steps, here's the argument in the forward direction: The proof could end by stating in the fourth-to-last statement that angles 5 and 6 are congruent, then in the third-to-last that angle 5 is congruent to angle 8 and angle 6 is congruent to angle 7 (because vertical angles are congruent), and then in the second-to-last that angle 7 is congruent to angle 8 by the Transitive Property (for four angles). The next figure shows how this all looks written out in the two-column format.

You can now finish by bridging the gap between statement 5

and the 4th-to-last statement

Say that congruent angles 3 and 4 are each 55 degrees. Angle 5 is complementary to angle 3, so if angle 3 were 55 degrees, angle 5 would have to be 35 degrees. Angle 6 is complementary to angle 4, so angle 6 also ends up being 35 degrees. That does it: angles 5 and 6 are congruent, and you've connected the loose ends!