Congruence

Understanding Congruence in Triangles

When we talk about congruence in geometry, we mean “exactly the same size and shape.” Two geometric figures are congruent if one can be moved (by sliding, turning, or flipping) to fit exactly on top of the other.

For triangles, congruence is especially important: knowing that two triangles are congruent allows you to transfer information (like side lengths and angle sizes) from one triangle to another.

In this chapter we focus specifically on:

• What it means for two triangles to be congruent.
• Standard triangle congruence criteria.
• How to use congruent triangles to draw conclusions about sides and angles.

General ideas about triangles (like their angle sum) are handled in other chapters; here we assume you already know what a triangle is and basic triangle terminology (sides, vertices, angles, etc.).

What It Means for Two Triangles to Be Congruent

Two triangles are congruent if:

• Their corresponding sides are equal in length.
• Their corresponding angles are equal in measure.

“Corresponding” means that you match each vertex, side, and angle of one triangle with exactly one vertex, side, and angle of the other triangle, in order.

For example, if triangle $ABC$ is congruent to triangle $DEF$, we might write
$$
\triangle ABC \cong \triangle DEF.
$$

This statement means:

• $AB$ corresponds to $DE$,
• $BC$ corresponds to $EF$,
• $CA$ corresponds to $FD$,

and

• $\angle A$ corresponds to $\angle D$,
• $\angle B$ corresponds to $\angle E$,
• $\angle C$ corresponds to $\angle F$.

If the triangles are congruent, then
$$
AB = DE,\quad BC = EF,\quad CA = FD,
$$
and
$$
\angle A = \angle D,\quad \angle B = \angle E,\quad \angle C = \angle F.
$$

The order of the letters in the congruence statement matters. It tells you which vertices (and thus which sides and angles) correspond.

Transformations and Congruence

One way to think about congruence is through rigid motions (also called isometries):

• Translations (slides),
• Rotations (turns),
• Reflections (flips).

If one triangle can be moved onto another using only these motions—without stretching or shrinking—then the triangles are congruent.

You do not need to calculate these motions in detail for basic congruence problems, but it is useful to remember: congruence is about shape and size, not about position or orientation.

Triangle Congruence Criteria

Checking all three sides and all three angles directly can be slow. Instead, geometry uses a small number of efficient criteria. These tell you when certain combinations of equal sides and angles guarantee that two triangles are congruent.

In school geometry, the standard triangle congruence criteria are:

• SSS (Side–Side–Side)
• SAS (Side–Angle–Side)
• ASA (Angle–Side–Angle)
• AAS (Angle–Angle–Side)
• RHS / HL (Right angle–Hypotenuse–Side or Hypotenuse–Leg) for right triangles

Certain apparent patterns—like SSA and AAA—do not in general guarantee congruence. We will discuss why briefly.

SSS: Side–Side–Side

SSS says:

If three sides of one triangle are respectively equal to three sides of another triangle, then the triangles are congruent.

So if you know:
$$
AB = DE,\quad BC = EF,\quad CA = FD,
$$
then
$$
\triangle ABC \cong \triangle DEF
$$
by SSS.

The idea: once three side lengths are fixed, there is only one possible triangle (up to rigid motions). You cannot arrange those three lengths to form two different non-congruent triangles.

SAS: Side–Angle–Side

SAS says:

If two sides and the included angle of one triangle are respectively equal to two sides and the included angle of another triangle, then the triangles are congruent.

The “included angle” is the angle between the two known sides.

For example, if
$$
AB = DE,\quad AC = DF,\quad \angle A = \angle D,
$$
and angle $A$ is between sides $AB$ and $AC$ (and angle $D$ is between $DE$ and $DF$), then
$$
\triangle ABC \cong \triangle DEF
$$
by SAS.

The angle being included is essential. If the known angle is not between the known sides, the SAS rule does not apply.

ASA: Angle–Side–Angle

ASA says:

If two angles and the included side of one triangle are respectively equal to two angles and the included side of another triangle, then the triangles are congruent.

Here the “included side” is the side between the two known angles.

For example, if
$$
\angle A = \angle D,\quad \angle B = \angle E,\quad AB = DE,
$$
and side $AB$ lies between angles $A$ and $B$ (and similarly for $DE$ between $D$ and $E$), then
$$
\triangle ABC \cong \triangle DEF
$$
by ASA.

Because the sum of angles in a triangle is fixed, knowing two angles essentially fixes the third, so ASA is quite powerful.

AAS: Angle–Angle–Side

AAS says:

If two angles and a non-included side of one triangle are respectively equal to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.

“Non-included” means the known side is not between the two known angles.

For example, if
$$
\angle A = \angle D,\quad \angle B = \angle E,\quad AC = DF,
$$
with side $AC$ opposite angle $B$ (and $DF$ opposite $E$), then
$$
\triangle ABC \cong \triangle DEF
$$
by AAS.

AAS and ASA are closely related. In many proofs, you use the angle sum of a triangle to convert between them (two angles automatically determine the third).

RHS / HL: Right Angle–Hypotenuse–Side

For right triangles, there is a special congruence rule often called:

• RHS: Right angle–Hypotenuse–Side, or
• HL: Hypotenuse–Leg.

It says:

If two right triangles have equal hypotenuses and one pair of equal corresponding legs, then the triangles are congruent.

Suppose $\triangle ABC$ and $\triangle DEF$ are right triangles with right angles at $C$ and $F$:

• $AB$ and $DE$ are hypotenuses.
• $AC$ and $DF$ are legs.
• $BC$ and $EF$ are legs.

If
$$
AB = DE \quad\text{and}\quad AC = DF,
$$
then
$$
\triangle ABC \cong \triangle DEF
$$
by RHS (or HL).

This rule is specific to right triangles and relies on their special structure.

Non-Congruence Cases: AAA and SSA

Not all combinations of equal parts guarantee congruence.

AAA: Angle–Angle–Angle

AAA (three pairs of equal angles) guarantees that triangles are similar—they have the same shape—but not necessarily the same size. One triangle could be a scaled-up or scaled-down version of the other.

So AAA is a similarity condition, not a congruence condition.

SSA: Side–Side–Angle (the “Ambiguous Case”)

SSA (two sides and a non-included angle) is called the ambiguous case because:

• Sometimes it gives one triangle,
• Sometimes it gives two different triangles,
• Sometimes it gives no triangle at all.

Because of this ambiguity, SSA is not a valid triangle congruence criterion in general.

There is an exception in some right-triangle situations, but those are covered by the RHS/HL rule rather than treating SSA as a general rule.

Using Congruent Triangles to Prove Equal Sides and Angles

Congruent triangles are often used in geometric reasoning: you show two triangles are congruent, and then you can immediately conclude that their corresponding parts are equal. This principle is often abbreviated as:

$$
\text{CPCTC} \quad \text{(Corresponding Parts of Congruent Triangles are Congruent)}.
$$

The typical structure of such arguments is:

1. Identify the triangles you want to compare.
2. Show a congruence criterion holds (SSS, SAS, ASA, AAS, RHS).
3. State that the triangles are congruent, for example
   $$
   \triangle ABC \cong \triangle DEF.
   $$
4. Use CPCTC to conclude that a particular pair of sides or angles are equal, such as $BC = EF$ or $\angle C = \angle F$.

In simple problems, steps like CPCTC may be written in words as “corresponding sides of congruent triangles are equal.”

Congruent triangles are especially useful for:

• Proving that certain segments in a figure are equal.
• Proving that certain angles are equal.
• Showing that two geometric constructions produce the same result.

Congruence and Naming Correspondence

When you write a congruence statement, the order of the letters encodes the correspondence:

If you write
$$
\triangle ABC \cong \triangle DEF,
$$
you are saying:

• $A \leftrightarrow D$,
• $B \leftrightarrow E$,
• $C \leftrightarrow F$.

Thus:

• $AB \leftrightarrow DE$,
• $BC \leftrightarrow EF$,
• $CA \leftrightarrow FD$,

and so on.

If a problem gives you a diagram and a congruence statement, use the order of the letters to identify which sides and angles match. This is often needed when using CPCTC to reach a conclusion.

Congruence in Constructions (Brief Idea)

In geometric constructions (using straightedge and compass), congruence is built into many steps:

• Copying a line segment produces a segment congruent to the original.
• Copying an angle produces an angle congruent to the original.
• Constructing an equilateral triangle on a given segment gives three congruent sides.

These constructions often rely on congruent triangles implicitly: by ensuring certain sides and angles match, you guarantee that the constructed figure has the desired properties.

Summary

• Triangles are congruent if they can be matched exactly in size and shape.
• Congruence is written using the symbol $\cong$; the order of vertices tells you which parts correspond.
• Standard triangle congruence criteria:
• SSS: three pairs of equal sides.
• SAS: two pairs of equal sides and the included angle.
• ASA: two pairs of equal angles and the included side.
• AAS: two pairs of equal angles and a non-included side.
• RHS/HL: right triangles with equal hypotenuse and one equal leg.
• AAA and SSA are not general congruence criteria.
• Once triangles are known to be congruent, all corresponding sides and angles are equal (CPCTC), which is a key tool in geometric reasoning and proofs.