Please enable scripting (or JavaScript) in your web browser, and then reload this page.
Two geometrical figures are said to be congruent if there is some rigid motion (some combination of translations, rotations, and reflections) which makes them identical to each other. Intuitively, this means that the two figures are the same size and shape as each other. We will look at some ways to see that two figures are congruent, with a specific focus on triangles.
The controls to the left control the motion of the red sailboat. The blue, green, and purple sailboats remain in the same place.
Say you have two triangles $\cl"red"{▵ABC}$ and $\cl"blue"{▵A'B'C'}$, and you know some of their measurements (side lengths or angle measures). (A symbol like $\cl"blue"{A'}$ — pronounced “$A$ prime” — is another way to name a point, like $A$ or $B$.) Sometimes it’s possible to decide that the triangles are congruent just by looking at those measurements, without actually finding a rigid motion which makes them identical.
Two triangles $\cl"red"{▵ABC}$ and $\cl"blue"{▵A'B'C'}$ (pronounced “triangle $A$ prime $B$ prime $C$ prime”) are drawn to the left, along with rays from $\cl"red"A$ through $\cl"red"C$ and from $\cl"blue"{A'}$ through $\cl"blue"{C'}$. There are also controls which move the red triangle $▵ABC$. Notice that the length of $\cl"red"\ov{AB}$ is equal to the length of $\cl"blue"\ov{A'B'}$, and the length of $\cl"red"\ov{AC}$ is equal to the length of $\cl"blue"\ov{A'C'}$. Also, the angles $\cl"red"{∠A}$ and $\cl"blue"{∠A'}$ between those two sides are equally large.
If you have two triangles with corresponding measurements equal for two sides and the angle between them, as in $\cl"red"{▵ABC}$ and $\cl"blue"{▵A'B'C'}$, then you can always make the triangles coincide with a rigid motion in this way:
This is called the Side-Angle-Side or SAS rule for congruence of triangles. It gives you one way to know that two triangles are congruent without actually finding a rigid motion connecting them.
Now you can see a different pair of triangles $\cl"red"{▵ABC}$ and $\cl"blue"{▵A'B'C'}$, along with rays from $\cl"red"A$ through $\cl"red"C$ and from $\cl"blue"{A'}$ through $\cl"blue"{C'}$, and controls which move the red triangle $▵ABC$.
A triangle $\cl"red"{▵ABC}$ is drawn to the left, along with the ray from $\cl"red"A$ through $\cl"red"C$. By sliding the sliders, you can change the lengths of the sides $\cl"red"\ov{AB}$ and $\cl"red"\ov{AC}$, and the measure of the angle $\cl"red"{∠A}$ between them. The ray shows all the possible locations of $\cl"red"C$ for a given measure of $\cl"red"{∠A}$. The approximate length of each side and measure of each angle of the triangle are also shown.