Once you have gathered some supporting evidence for a conjecture, you can improve your argument by writing a proof. Some statements may be very difficult to prove in this way, though. In fact, you will encounter some conjectures that you are not able to prove--either because the conjecture is not true or because you haven't yet learned the math concepts you would need to use as evidence. One famous conjecture, called Fermat's last theorem, took mathematicians over 350 years to prove!

You will be asked to prove some conjectures that fall within the knowledge and mathematical skill that you're expected to have at your current level of schooling. Let's start with this one.

First, let's talk about what vertical angles are. In this image, they are the two angles opposite each other, or \(\small\mathsf{\angle{x} }\) and \(\small\mathsf{\angle{y} }\). To show these two angles are congruent, we need to show that they have the same measure. We will also use the properties of straight angles to write the proof.  A straight angle has a measure of 180°, while supplementary angles are two angles whose measures sum to 180°.

If \(\small\mathsf{\angle{x} }\) is supplementary to the angle at the top, then the angle can be written 180° - x°. The same can be said for the the top angle and \(\small\mathsf{\angle{y} }\), so we can write it as 180° - y°

Because anything is congruent to itself, the two different measures for the top angle both have the same measure--they are equal to each other.

Since x and y are the two measures of the vertical angles, and we have proved that they are equal, then we can say that the two angles are congruent.