সর্বসম বহুভুজ

Congruent polygons can be very powerful in geometry. We call two polygons congruent if all of their corresponding angles and sides are equal in measure/length. Thus, if we can prove two polygons are congruent, we can determine lengths/angles of one polygon based on information we know about the other polygon.

The most common of example of polygon congruence is with triangles. There are several ways in which we can say two triangles are congruent.

1. If the two triangles have the property that all of their sides are congruent, the triangles themselves are congruent. We call this Side-Side-Side (SSS) congruence).

2. If the two triangles have two equal angles with a congruent included side in between the angles, the triangles are congruent. This is Angle-Side-Angle (ASA) congruence.

1. Our final case that always works is Side-Angle-Side (SAS). If two triangles have two sides with an included angle that are all congruent, the triangles are congruent.

Note that AAA does not prove two triangles are congruent, since they can have completely different side lengths.

The SSA case does not always work, since the included angle between the two congruent sides can either be obtuse or acute.

However, there is one special case of SSA that always works. This is when we have two right triangles with congruent hypotenuses and one congruent leg. There is no ambiguity for the angle being obtuse or acute here, since we know it is less than $90^\circ$ and therefore acute. We call this case the Hypotenuse-Leg congruence (HL).

There is one another way of proving the congruence of triangles. This is the Angle-Angle-Side Theorem (AAS). This is when two angles and a side opposite those angles (not in between) in one triangle are congruent to the corresponding parts in another triangle which therefore, results in two triangles being congruent to each other.

Polygons are congruent if they have the same side length, same number of sides, and same angle measurements on the inside as the first (or mother) polygon.

Polygons are similar when they are the same shape but different sizes. They will always be proportional to each other along with congruent interior angles.