Congruence, Equality, and Geometry Examples

Prove that if two angles are supplementary to the same angle, then they're congruent.

We'll say that ∠1 and ∠2 are both supplementary to ∠3. Our goal is to prove that ∠1 and ∠2 are congruent.

Supplementary angles are just like complementary angles, except their sum is 90°, not 180°. That means m∠1 + m∠3 = 180 and m∠2 + m∠3 = 180. Subtract m∠3 from both sides of both equations we have both m∠1 and m∠2 equaling 180 – m∠3. If that's not congruent, we don't know what is.

Identify all the pairs of congruent angles and segments in the following figure.

The most obvious pairs of congruent segments are those with tick marks on them. Only three segments have the same style tick mark, so all three of them are congruent. In other words, PQ ≅ ML ≅ LN. Easy peasy.

While there aren't any marked angles in the image, that doesn't mean there aren't any congruent angles. For instance, we can see straight from the figure that ∠MON and ∠POQ are vertical angles. Since we know that vertical angles are congruent, we can conclude that ∠MON ≅ ∠POQ.

But it doesn't stop there! What about ∠MOP and ∠NOQ? Even though they're horizontal, they fall under the vertical angles category too. That means ∠MOP ≅ ∠NOQ as well.

In the figure, M is the midpoint of AC. Prove that M is the midpoint of BD.

From the image, M being the midpoint of BD looks really intuitive. Proving it, on the other hand, might take a bit of thinking through.

If M is the midpoint of BD, we need to prove that BM and MD are congruent. The picture tells us that AM is congruent to BM and MC is congruent to MD. If M is the midpoint of AC, that means AM and MC are congruent and equal in measure. Using the transitive property, we can see that AM, BM, MC, and MD are all congruent. In that case, BM = MD, which means that M is the midpoint. Booyah!