Proving Triangle Congruence and Similarity

In geometry, we are sometimes concerned with proving things.

Unlike the real world, where nothing could ever be proven, only shown to be consistent beyond reasonable doubt, in the perfect realm of mathematics, things can indeed be proven to be inarguably true, and our construction of proofs, as meticulous as they may seem, are incredible demonstrations of logic.

One of the things we will want to prove is the relationship between two shapes, like two triangles.

Two triangles might be unrelated, or they may have some special relationship.

They could be similar, meaning they have precisely the same shape, just different sizes, or they could be congruent, meaning they are absolutely identical.

When trying to demonstrate similarity, we have to look at the corresponding sides and angles.

Here are two triangles that look like they might be similar.

We can prove they are similar if the corresponding sides have lengths that are proportional, or if the corresponding angles are identical.

This means that if this triangle has sides equal to two, four, and six, while this one has sides equal to four, eight, and twelve, the two triangles must be similar.

Also, if this triangle has angles of 30, 60, and 90, and so does this one, these triangles must be similar no matter what the side lengths.

In geometry, we are sometimes concerned with proving things.

Unlike the real world, where nothing could ever be proven, only shown to be consistent beyond reasonable doubt, in the perfect realm of mathematics, things can indeed be proven to be inarguably true, and our construction of proofs, as meticulous as they may seem, are incredible demonstrations of logic.

One of the things we will want to prove is the relationship between two shapes, like two triangles.

Two triangles might be unrelated, or they may have some special relationship.

They could be similar, meaning they have precisely the same shape, just different sizes, or they could be congruent, meaning they are absolutely identical.

When trying to demonstrate similarity, we have to look at the corresponding sides and angles.

Here are two triangles that look like they might be similar.

We can prove they are similar if the corresponding sides have lengths that are proportional, or if the corresponding angles are identical.

This means that if this triangle has sides equal to two, four, and six, while this one has sides equal to four, eight, and twelve, the two triangles must be similar.

Also, if this triangle has angles of 30, 60, and 90, and so does this one, these triangles must be similar no matter what the side lengths.