Math Core

Two machine parts stamped from the same mold are interchangeable: one fits exactly where the other was. Geometry has a precise word for "same size and same shape," and in the Transformations unit you already met the tools that define it. In this lesson you'll use rigid motions to say exactly what congruent means, and you'll learn to read and write congruence statements for triangles.

Congruence through rigid motions

A rigid motion is a transformation that keeps every length and every angle measure the same. Translations (slides), reflections (flips) and rotations (turns) are rigid motions, and so is any sequence of them. A dilation is not a rigid motion, because it changes lengths (unless the scale factor is $1$ ).

Definition

Congruent figures

Two figures are congruent if some sequence of rigid motions maps one figure exactly onto the other. We write $\triangle ABC \cong \triangle DEF$ , read "triangle $ABC$ is congruent to triangle $DEF$ ."

Because rigid motions preserve distance and angle measure, the pieces that land on top of each other must match. When one triangle is moved onto another, each side lands on a side of the same length and each angle lands on an angle of the same measure. These matching pieces are called corresponding parts.

The reverse is also true. If all three pairs of sides and all three pairs of angles of two triangles are congruent, you can always find rigid motions that carry one triangle onto the other. So for triangles, "congruent" means both of these things at once:

one triangle can be moved onto the other by rigid motions, and
all six pairs of corresponding parts (three sides, three angles) are congruent.

Reading a congruence statement

The order of the letters in a congruence statement is not decoration. It tells you exactly which vertex matches which.

Order tells the correspondence

$\triangle ABC \cong \triangle DEF$ means $A \leftrightarrow D$ , $B \leftrightarrow E$ and $C \leftrightarrow F$ . From that you can read off all six congruences:

Angles	Sides
$\angle A \cong \angle D$	$\overline{AB} \cong \overline{DE}$
$\angle B \cong \angle E$	$\overline{BC} \cong \overline{EF}$
$\angle C \cong \angle F$	$\overline{AC} \cong \overline{DF}$

In diagrams, matching tick marks show congruent sides and matching arcs show congruent angles. One tick matches one tick, two ticks match two ticks, and so on.

Triangle ABC is congruent to triangle DEF. Matching marks show corresponding parts.

Common mistake

Don't scramble the order. If $\triangle ABC \cong \triangle DEF$ , it is wrong to write $\triangle ABC \cong \triangle EDF$ , because that would claim $A$ matches $E$ . You may start at a different vertex as long as you keep the pairs together: $\triangle BCA \cong \triangle EFD$ says the same thing as $\triangle ABC \cong \triangle DEF$ .

Worked example: Using a congruence statement

Given $\triangle PQR \cong \triangle XYZ$ , $m\angle Q = 72^\circ$ and $QR = 9$ . Find $m\angle Y$ and $YZ$ .

The second letters match, so $Q \leftrightarrow Y$ and $\angle Y \cong \angle Q$ . That gives $m\angle Y = 72^\circ$ .

Side $\overline{QR}$ uses the second and third letters of the first triangle, so it matches the second and third letters of the other: $\overline{YZ}$ . That gives $YZ = 9$ .

Writing a congruence statement from a diagram

When the triangles are drawn in different positions, trust the marks, not the picture's orientation. Match each vertex by finding where its marked sides or angles meet.

Worked example: Writing the statement

Write a congruence statement for the two triangles.

Triangles JKL and RST. Tick marks show which sides are congruent.

Look at the vertices one at a time.

$J$ is where the one-tick side and the three-tick side meet. In the other triangle, those sides meet at $S$ . So $J \leftrightarrow S$ .
$K$ is where the one-tick and two-tick sides meet. That's $T$ . So $K \leftrightarrow T$ .
$L$ is where the two-tick and three-tick sides meet. That's $R$ . So $L \leftrightarrow R$ .

The statement is $\triangle JKL \cong \triangle STR$ . Notice that the second triangle is a mirror image of the first, so a reflection (followed by a translation) maps $\triangle JKL$ onto $\triangle STR$ .

The third angles theorem

The angles of every triangle add to $180^\circ$ . That fact gives a quick result you will use again and again.

Third angles theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

For example, if two angles of one triangle measure $50^\circ$ and $60^\circ$ , and two angles of another triangle measure $50^\circ$ and $60^\circ$ , the third angles must both be $180^\circ - 50^\circ - 60^\circ = 70^\circ$ . Be careful: this says the angles match. It does not say the triangles are congruent, because one triangle could be a larger copy of the other.

Showing congruence with coordinates

On the coordinate plane you can prove two figures are congruent by naming a rigid motion that maps one onto the other.

Worked example: Finding the rigid motion

$\triangle ABC$ has vertices $A(1, 1)$ , $B(4, 1)$ , $C(1, 3)$ . $\triangle DEF$ has vertices $D(-1, 1)$ , $E(-4, 1)$ , $F(-1, 3)$ . Show that the triangles are congruent.

Triangle ABC and its reflection DEF across the y-axis.Open in grapher →

Each vertex of $\triangle DEF$ has the same $y$ -coordinate as a vertex of $\triangle ABC$ and the opposite $x$ -coordinate. That is the rule for a reflection across the $y$ -axis: $(x, y) \to (-x, y)$ .

A(1, 1) \to (-1, 1) = D, \qquad B(4, 1) \to (-4, 1) = E, \qquad C(1, 3) \to (-1, 3) = F.

A reflection is a rigid motion, so $\triangle ABC \cong \triangle DEF$ , with $A \leftrightarrow D$ , $B \leftrightarrow E$ , $C \leftrightarrow F$ .

Congruence and algebra

Corresponding parts have equal measures, so you can set their expressions equal and solve.

Worked example: Solving for unknowns

$\triangle ABC \cong \triangle DEF$ . $AB = 3x + 1$ , $DE = 2x + 7$ , $m\angle C = (5y - 10)^\circ$ and $m\angle F = 60^\circ$ . Find $x$ , $y$ and $AB$ .

$\overline{AB}$ corresponds to $\overline{DE}$ , so

3x + 1 = 2x + 7 \quad\Rightarrow\quad x = 6.

Then $AB = 3(6) + 1 = 19$ . (Check: $DE = 2(6) + 7 = 19$ .)

$\angle C$ corresponds to $\angle F$ , so

5y - 10 = 60 \quad\Rightarrow\quad 5y = 70 \quad\Rightarrow\quad y = 14.

Tip

Before matching parts, write the correspondence under the statement: $A \to D$ , $B \to E$ , $C \to F$ . Then any side or angle is just a matter of swapping letters.

Practice

Practice 1

Given $\triangle MNP \cong \triangle RST$ , which segment is congruent to $\overline{NP}$ ?

Practice 2

Which transformation does not always produce a figure congruent to the original?

Practice 3

Given $\triangle GHK \cong \triangle LMN$ , $m\angle H = 48^\circ$ and $m\angle N = 67^\circ$ . Find $m\angle L$ in degrees.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

In $\triangle ABC$ and $\triangle XYZ$ , the correspondence is $A \leftrightarrow Y$ , $B \leftrightarrow Z$ , $C \leftrightarrow X$ . Which congruence statement is correct?

Practice 5

Given $\triangle DEF \cong \triangle QRS$ , $DE = 4x - 3$ and $QR = 2x + 11$ . Find $DE$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Given $\triangle JKL \cong \triangle TUV$ , $JK = 8$ , $KL = 12$ , and the perimeter of $\triangle TUV$ is $30$ . Find $JL$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

$\triangle ABC$ has vertices $A(2, 1)$ , $B(5, 1)$ , $C(2, 4)$ . It is rotated $90^\circ$ counterclockwise about the origin, using the rule $(x, y) \to (-y, x)$ . What are the vertices of the image, which is congruent to $\triangle ABC$ ?

Practice 8

Given $\triangle ABC \cong \triangle PQR$ , $m\angle A = (2x + 10)^\circ$ and $m\angle P = (3x - 15)^\circ$ . Find $m\angle A$ in degrees.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.