Testing Polygons for congruence

How do you tell if two polygons are congruent? The basic rule of thumb is:

If you know enough about the polygons to draw them, then you have enough information to tell if they are congruent or not.
In all cases below, we assume the polygons have the same number of sides.

## The obvious case

The polygons are congruent if they have the same number of sides, and:
1. All the corresponding sides are congruent
2. All corresponding interior angles are congruent
This is the obvious test based on the definition of congruence, but you can get away with less information:

## For regular polygons

Regular polygons are congruent if they have the same number of sides, and:
1. Their sides are congruent, or:
2. Their apothems are congruent, or:
If you have enough information to draw one unique regular polygon, that is also enough to determine if they are congruent or not.

## All but one side and their included angles (SASASAS...)

You can show congruence if you know the lengths of all but one side and the included angles between those sides. The information given is enough to actually draw the polygons by simply filling in the missing side, therefore it is enough to show congruence, provided of course that all these corresponding sides and angles are in fact equal.

## All but one interior angle, and their included sides (ASASASA...)

You can show congruence if you know the measure of all but one angle, and the included sides between them. This information is enough to actually draw the polygon because we can extend the two unknown sides until they intersect, locating the missing vertex. Additionally, we can calculate the missing angle because the interior angles of any polygon with n sides add up to 180(n-2) degrees. This is therefore enough to show congruence, provided of course that all these corresponding sides and angles are in fact equal.

## What does this mean?

If you have shown that two polygons are congruent, then you know that every property of the polygons is also identical. For example they will have the same area, perimeter, exterior angles, apothem etc.

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