# Congruent (geometry)

**Congruency** is a property of multiple geometric figures.

## Contents

## Intuitive Definition

Two geometric figures are congruent if one of them can be turned and/or flipped and placed exactly on top of the other, with all parts lining up perfectly with no parts on either figure left over. In plain language, two objects are congruent if they have the same size and shape.

## Technical Definition

Two geometric objects are **congruent** if one can be transformed into the other by an isometry, such as a translation, rotation, reflection or some combination thereof.

## Axioms

- If , are two points on a straight line , and if is a point upon the same or another straight line , then, upon a given side of on the straight line , we can always find one and only one point so that the segment (or ) is congruent to the segment . We indicate this relation by writing

Every segment is congruent to itself; that is, we always have

- If a segment is congruent to the segment and also to the segment , then the segment is congruent to the segment ; that is, if and , then .

- Let and be two segments of a straight line which have no points in common aside from the point , and, furthermore, let and be two segments of the same or of another straight line having, likewise, no point other than in common. Then, if and , we have .

- Let an angle be given in the plane and let a straight line be given in a plane . Suppose also that, in the plane , a definite side of the straight line be assigned. Denote by a half-ray of the straight line emanating from a point of this line. Then in the plane there is one and only one half-ray such that the angle , or , is congruent to the angle and at the same time all interior points of the angle lie upon the given side of . We express this relation by means of the notation

Every angle is congruent to itself; that is, or

- If the angle is congruent to the angle and to the angle , then the angle is congruent to the angle ; that is to say, if and

, then .

- If, in the two triangles and the congruences

hold, then the congruences also hold.

## Triangle Congruence

### SSS Congruence

If the three sides of one triangle are congruent to the corresponding sides of of another triangle, then congruence between the two triangles is established.

We start with and shown in the diagram below where , , and supposing that . We construct point on the opposite side of from such that . Since is on the opposite side of from , segment must intersect the line at some point . Updating our diagram gives us We now have several cases depending on the location of point relative to and .

** is strictly between and , as shown in the diagram above.**
Therefore, must be in the interiors of both and , but since and , we have and . From this, we find that and are isosceles, and therefore and , so . We have by SAS congruence.

** is strictly between and .**
Without loss of generality, we can derive from the fact that is strictly between and .

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### SAS Congruence

The SAS Congruence theorem is derived from the sixth axiom of congruence. In short, the sixth axiom states that when given two triangles, if two corresponding side congruences hold and the angle between the two sides is equal on both triangles, then the other two angles of the triangle are equal.

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### ASA Congruence

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### AAS Congruence

Since the third angle of a triangle is always the difference of and the other two angles, two triangles with two pairs of congruent angles would give another pair of congruent angles. When we have a pair of congruent sides and two pairs of congruent angles adjacent to the side, it is ASA congruence. However, if we're given a pair of congruent angles opposite from the side that is congruent and a pair of congruent angles adjacent to that side, everything is congruent. Therefore, AAS congruence can be proven with ASA congruence.

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### HL Congruence

If the both the hypotenuse and leg of one right triangle are congruent to that of another, the two triangles are congruent.

Consider right and right shown in the diagram below. We are given that and . Since , the Pythagorean theorem gives us . Similarly, using the Pythagorean theorem on gives us . However, since and , substitution in the second equation gives . Subtracting this from our first equation plus a little manipiulation gives us . Therefore, since all lengths are positive, taking the square root of both sides gives , so, by the SSS congruence theorem, we have .

Since the congruent angle given is not between the two equivalent sides, this may be seen as SSA congruence, which is not necessarily correct. However, this form of SSA congruence holds true for right triangles.

### LL Congruence

LL Congruence is the basically the same as SAS congruence since we are given a leg, a right angle, and the other leg.