# Congruent (geometry)

Congruency is a property of multiple geometric figures.

## 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.

A collection of isometries.

## 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

IV, I. If $A$, $B$ are two points on a straight line $a$, and if $A'$ is a point upon the same or another straight line $a'$, then, upon a given side of $A'$ on the straight line $a'$, we can always find one and only one point $B'$ so that the segment $AB$ (or $BA$) is congruent to the segment $A'B'$. We indicate this relation by writing $$\overline{AB}\cong\overline{A'B'}.$$ Every segment is congruent to itself; that is, we always have $$\overline{AB}\cong\overline{AB}.$$

IV, 2. If a segment $\overline{AB}$ is congruent to the segment $\overline{A'B'}$ and also to the segment $\overline{A''B''}$, then the segment $\overline{A'B'}$ is congruent to the segment $\overline{A''B''}$; that is, if $\overline{AB}\cong\overline{A'B}$ and $\overline{AB}\cong\overline{A''B''}$, then $\overline{A'B'}\cong\overline{A''B''}$.

IV, 3. Let $\overline{AB}$ and $\overline{BC}$ be two segments of a straight line $a$ which have no points in common aside from the point $B$, and, furthermore, let $\overline{A'B'}$ and $\overline{B'C'}$ be two segments of the same or of another straight line $a'$ having, likewise, no point other than $B'$ in common. Then, if $\overline{AB}\cong\overline{A'B'}$ and $\overline{BC}\cong\overline{B'C'}$, we have $\overline{AC}\cong\overline{A'C'}$.

IV, 4. Let an angle $(h,k)$ be given in the plane $\alpha$ and let a straight line $a'$ be given in a plane $\alpha'$. Suppose also that, in the plane $\alpha$, a definite side of the straight line $a'$ be assigned. Denote by $h'$ a half-ray of the straight line $a'$ emanating from a point $O'$ of this line. Then in the plane $\alpha'$ there is one and only one half-ray $k'$ such that the angle $(h,k)$, or $(k,h)$, is congruent to the angle $(h',k')$ and at the same time all interior points of the angle $(h',k')$ lie upon the given side of $a'$. We express this relation by means of the notation $$\angle (h,k) \cong \angle (h',k')$$ Every angle is congruent to itself; that is, $$\angle (h,k) \cong \angle (h,k)$$ or $$\angle (h,k) \cong \angle (k,h)$$

IV, 5. If the angle $(h,k)$ is congruent to the angle $(h',k')$ and to the angle $(h'',k'')$, then the angle $(h',k')$ is congruent to the angle $(h'',k'')$; that is to say, if $\angle (h, k) \cong \angle (h', k')$ and $\angle (h, k) \equiv \angle (h'',k'')$, then $\angle (h',k') \cong \angle (h'',k'')$.

IV, 6. If, in the two triangles $ABC$ and $A'B'C'$ the congruences $$\overline{AB}\cong\overline{A'B'}, \: \overline{AC}\cong\overline{A'C'}, \: \angle BAC\cong\angle B'A'C'$$ hold, then the congruences $$\angle ABC\cong\angle A'B'C' \:\mbox{and}\; \angle ACB\cong\angle A'C'B'$$ also hold.

stub

stub

stub

stub

## 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 $\triangle ABC$ and right $\triangle XYZ$ shown in the diagram below. We are given that $AB=XY$ and $AC=XZ$. $[asy] size(500); pair A,B,C; A=(0,0); B=(9/5,12/5); C=(5,0); draw(A--B--C--cycle); label("A",A,SW); label("B",B,N); label("C",C,SE); draw(rightanglemark(A,B,C,5)); pair X,Y,Z; X=(20,0); Y=(109/5,12/5); Z=(25,0); draw(X--Y--Z--cycle); label("X",X,SW); label("Y",Y,N); label("Z",Z,SE); draw(rightanglemark(X,Y,Z,5)); [/asy]$ Since $\angle B=90^\circ$, the Pythagorean theorem gives us $AB^2+AC^2=AC^2$. Similarly, using the Pythagorean theorem on $\triangle XYZ$ gives us $XY^2+YZ^2=XZ^2$. However, since $AB=XY$ and $AC=XZ$, substitution in the second equation gives $AB^2+YZ^2=AC^2$. Subtracting this from our first equation plus a little manipiulation gives us $AC^2=YZ^2$. Therefore, since all lengths are positive, taking the square root of both sides gives $AC=YZ$, so, by the SSS congruence theorem, we have $\triangle ABC\cong\triangle XYZ$.

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.