Congruent figures: what congruent means in geometry | In depth

Same size, same shape

Two figures are congruent when they have exactly the same size and the same shape. One of them may be moved somewhere else, turned around or flipped over, but if you could pick it up and lay it on the other one, every edge and corner would line up.

The symbol is $\cong$ , which is a squiggle (for "same shape") sitting on top of an equals sign (for "same size"). So $\triangle ABC \cong \triangle DEF$ reads "triangle $$ABC$$ is congruent to triangle $$DEF$$ ".

That everyday description is good for intuition. Modern geometry courses, including the Common Core standards used in most of the United States, make it precise with rigid motions.

The three rigid motions

A rigid motion moves every point of a figure without stretching or shrinking anything. On the coordinate plane each one has a simple rule.

Translation (a slide): every point moves the same distance in the same direction. Sliding $$a$$ units right and $$b$$ units up sends $$(x, y)$$ to $$(x + a, y + b)$$ .
Reflection (a flip): the figure is mirrored across a line. Across the $$y$$ -axis, $$(x, y)$$ goes to $$(-x, y)$$ ; across the $$x$$ -axis, $$(x, y)$$ goes to $$(x, -y)$$ .
Rotation (a turn): the figure spins around a point. A $$90°$$ counterclockwise turn about the origin sends $$(x, y)$$ to $$(-y, x)$$ ; a $$180°$$ turn sends it to $$(-x, -y)$$ .

Every rigid motion keeps lengths and angle measures the same, so perimeter and area don't change either. That is exactly why a figure and its image are congruent.

\triangle A'B'C'

has the same angles as

\triangle ABC

and sides twice as long: the scale factor is

\(2\)

Congruent segments and angles

For segments and angles, congruence comes down to one number.

Two segments are congruent when they have the same length.
Two angles are congruent when they have the same measure.

The length of the rays you happen to draw doesn't matter for an angle: a $$35°$$ angle drawn small and a $$35°$$ angle drawn large are congruent.

On diagrams, congruent segments get matching tick marks and congruent angles get matching arcs. One tick on two sides means those two are congruent; a double tick on two other sides means those two are congruent to each other (but not necessarily to the single-tick pair).

Matching tick marks show congruent segments:

\(AM = MB\)

Equilateral, isosceles and scalene triangles: matching tick marks show congruent sides.

Some angle pairs are congruent automatically, and they show up in proofs over and over:

vertical angles (the opposite angles formed by two crossing lines);
alternate interior angles and corresponding angles when a transversal crosses parallel lines;
the base angles of an isosceles triangle;
two angles that are complements (or supplements) of the same angle.

Congruent or equal?

Textbooks are picky about this, and it costs points on tests, so here it is plainly.

Figures are congruent: $\overline{AB} \cong \overline{CD}$ , $\angle P \cong \angle Q$ .
Numbers are equal: $$AB = CD$$ (no bar on top means "the length of"), $m\angle P = m\angle Q$ ("the measure of angle $$P$$ ").

So $\overline{AB} \cong \overline{CD}$ and $$AB = CD$$ say the same thing in two languages: one about the segments, one about their lengths. What you should not write is $\overline{AB} = \overline{CD}$ , which would claim the two segments are literally the same segment.

Congruent, similar or neither

Relationship	Same shape?	Same size?	Example
congruent	yes	yes	two copies of the same puzzle piece
similar	yes	not necessarily	a photo and its enlargement
same area only	no	area only	a $2 \times 8$ rectangle and a $4 \times 4$ square

Congruent figures are always similar (with scale factor $$1$$ ) and always have the same area. The reverse fails both ways: the rectangle and the square above both have area $$16$$ but are not congruent.

Congruent polygons and congruence statements

Two polygons are congruent when all pairs of corresponding sides and all pairs of corresponding angles are congruent. Sides alone are not enough once you get past triangles: a square and a rhombus with $$3$$ -inch sides have four congruent sides each, but the rhombus has slanted angles. A picture frame made of four sticks can be pushed out of square for the same reason.

The congruence statement tells you which parts match. In $\triangle ABC \cong \triangle DEF$ , the letters pair up in order: $A \leftrightarrow D$ , $B \leftrightarrow E$ , $C \leftrightarrow F$ . From that you can read off all six pairs:

\overline{AB} \cong \overline{DE}, \quad \overline{BC} \cong \overline{EF}, \quad \overline{AC} \cong \overline{DF}, \quad \angle A \cong \angle D, \quad \angle B \cong \angle E, \quad \angle C \cong \angle F

For triangles, you don't have to check all six pairs: the shortcuts SSS, SAS, ASA, AAS and HL each need only three. They are explained in the triangle congruence lesson.

What each shortcut needs: two sides and the angle between them, two angles and the side between them, three sides.

A two-column proof

Most proofs that two segments or two angles are congruent follow the same plan: find two triangles that contain them, prove the triangles congruent, then use CPCTC (corresponding parts of congruent triangles are congruent).

ExampleA diagonal splits a parallelogram into congruent triangles

Given: $$ABCD$$ is a parallelogram with diagonal $\overline{AC}$ . Prove: $\triangle ABC \cong \triangle CDA$ , and so $\overline{AB} \cong \overline{CD}$ .

Statement	Reason
$\overline{AB} \parallel \overline{DC}$ and $\overline{AD} \parallel \overline{BC}$	definition of a parallelogram
$\angle BAC \cong \angle DCA$	alternate interior angles ( $\overline{AB} \parallel \overline{DC}$ )
$\angle BCA \cong \angle DAC$	alternate interior angles ( $\overline{AD} \parallel \overline{BC}$ )
$\overline{AC} \cong \overline{CA}$	reflexive property
$\triangle ABC \cong \triangle CDA$	ASA
$\overline{AB} \cong \overline{CD}$	CPCTC

Notice the reflexive property in the fourth line: a shared side is congruent to itself. It looks too obvious to write down, but a proof needs every reason stated.

Properties of congruence

Congruence behaves like equality:

reflexive: every figure is congruent to itself;
symmetric: if $F \cong G$ , then $G \cong F$ ;
transitive: if $F \cong G$ and $G \cong H$ , then $F \cong H$ .

Common mistakes

Frequently asked questions

Is a mirror image congruent to the original?

Yes. A reflection is a rigid motion, so the image has the same side lengths and angles. It faces the other way, like a left hand and a right hand, but it is still congruent.

Are all circles congruent?

No, only circles with the same radius. Any two circles are similar, though.

Can two figures be congruent if they are in different places?

Yes, that is the usual situation. Position and direction don't matter; only size and shape do.

What does CPCTC stand for, and when can I use it?

"Corresponding parts of congruent triangles are congruent." You can use it only after you have proved the triangles congruent, never as the reason for that step.

Practice problems

Quick check

Which transformation does not always produce a congruent figure?