Congruent Figures and Corresponding Parts | Grade 8 Math | LevelKite

You already know how to translate, reflect and rotate a figure on the coordinate plane. Each of these moves is a rigid motion. A rigid motion is a transformation that keeps every length and every angle measure the same. The figure that a move produces is called the image.

In this lesson, you will use rigid motions to decide when two figures have the same size and shape. You will also name the parts that match and find missing side lengths and angle measures.

Key concept

Congruent figures match part for part

Two figures are congruent when one or more rigid motions, done one after another, carry one figure exactly onto the other. These moves can be translations, reflections or rotations. Rigid motions keep lengths and angle measures. So each side and each angle of the first figure lands on a corresponding part of the second figure, which is its matching partner. That is why corresponding parts of congruent figures always have equal measures.

Decide with coordinates, not with how a picture looks. Lengths on the coordinate plane are in grid units. For two points on the same horizontal line, the distance is the absolute value of the difference of their $x$ -coordinates. For two points on the same vertical line, use the $y$ -coordinates. The absolute value keeps every distance nonnegative.

To show figures are congruent, name a rigid motion and apply its rule to every vertex. The images must be exactly the vertices of the other figure, in the same order around it. Then the motion carries one figure onto the other.
To show figures are not congruent, find a side length in one figure that no side of the other figure has. Rigid motions never change lengths, so no sequence of them can match the figures.

Equal side lengths alone do not prove congruence. A square and a rhombus that is not a square can both have four sides of length $3$ , but their angles are different.

In the figure below, the $x$ -axis is horizontal and the $y$ -axis is vertical. The numbers 1, 2 and 3 name the figures. They are not measurements.

Labeled corners

(4, 3)

,

(-3, 4)

and

(4, -3.5)

belong to figures 1, 2 and 3. The origin

(0, 0)

is the center of the quarter turn tested below.

Worked example

Which figures are congruent to figure 1?

In the figure above, figure 1 has vertices $(1, 1)$ , $(4, 1)$ , $(4, 3)$ and $(1, 3)$ . Figure 2 has vertices $(-1, 1)$ , $(-1, 4)$ , $(-3, 4)$ and $(-3, 1)$ . Figure 3 has vertices $(1, -1)$ , $(4, -1)$ , $(4, -3.5)$ and $(1, -3.5)$ . Which of figures 2 and 3 are congruent to figure 1?

Figure 2 looks like figure 1 after a quarter turn. A look is only a guess, so test it. The $90^\circ$ counterclockwise rotation about the origin sends $(x, y)$ to $(-y, x)$ . Apply that rule to every vertex of figure 1.
$(1, 1) \to (-1, 1),\ (4, 1) \to (-1, 4),\ (4, 3) \to (-3, 4),\ (1, 3) \to (-3, 1)$
The four images are exactly the vertices of figure 2, in the same order. A rotation is a rigid motion, so it carries figure 1 onto figure 2. The figures are congruent, even though figure 2 is turned so that it is taller than it is wide.
Now check figure 3 using lengths. Every side of figures 1 and 3 is horizontal or vertical. So subtract the coordinates that change along a side and take the absolute value. Figure 1 is $3$ units wide and $2$ units tall, so its sides measure $3$ , $2$ , $3$ and $2$ units.
$|4 - 1| = 3,\qquad |3 - 1| = 2$
Figure 3 is $3$ units wide but exactly $2.5$ units tall. So its sides measure $3$ , $2.5$ , $3$ and $2.5$ units.
$|4 - 1| = 3,\qquad |-1 - (-3.5)| = 2.5$
Figure 1 has sides of length $2$ units, and no side of figure 3 has that length. Rigid motions keep lengths, so no rigid motion or sequence of them can carry a $2$ -unit side onto a side of figure 3. Figure 3 is not congruent to figure 1, even though the two look close.
$2 \ne 3 \quad\text{and}\quad 2 \ne 2.5$

Answer

Figure 2 is congruent to figure 1, and figure 3 is not congruent to figure 1.

When two figures are congruent, a congruence statement records which vertices match. The symbol $\cong$ means is congruent to, and $\triangle$ means triangle. In $\triangle PQR \cong \triangle XYZ$ , the vertices pair up by position: $P$ with $X$ , $Q$ with $Y$ , and $R$ with $Z$ .

Those pairs name all the corresponding parts:

sides $PQ$ and $XY$ , $QR$ and $YZ$ , $RP$ and $ZX$ ;
angles $\angle P$ and $\angle X$ , $\angle Q$ and $\angle Y$ , $\angle R$ and $\angle Z$ .

Here $PQ$ also stands for the length of side $PQ$ , and $m\angle P$ means the measure of $\angle P$ in degrees. You may start at any vertex if every pair stays in matching positions. So $\triangle QRP \cong \triangle YZX$ says the same thing. The pairs come from a rigid motion that carries one figure onto the other. Each vertex is paired with the vertex it lands on.

The figure below shows a triangle and its image. As before, the $x$ -axis is horizontal and the $y$ -axis is vertical.

Point A

(1, 1)

, Point B

(5, 1)

and Point C

(1, 3)

form triangle

ABC

. Point D

(-1, 1)

, Point E

(-1, 3)

and Point F

(-5, 1)

form its reflection across the

y

-axis.

Worked example

Write a congruence statement

In the figure above, triangle $ABC$ has vertices $A(1, 1)$ , $B(5, 1)$ and $C(1, 3)$ . Reflecting it across the $y$ -axis gives the triangle with vertices $D(-1, 1)$ , $E(-1, 3)$ and $F(-5, 1)$ . Write a congruence statement and name the corresponding parts.

Find where each vertex lands. A reflection across the $y$ -axis sends $(x, y)$ to $(-x, y)$ .
$A(1, 1) \to (-1, 1) = D,\ B(5, 1) \to (-5, 1) = F,\ C(1, 3) \to (-1, 3) = E$
The images are exactly the other triangle's vertices. A reflection is a rigid motion, so the triangles are congruent. Write each image vertex in the same position as the vertex it came from: $D$ goes with $A$ , $F$ with $B$ , and $E$ with $C$ .
$\triangle ABC \cong \triangle DFE$
Read the corresponding sides from letter pairs in matching positions. The third pair joins the last vertex back to the first.
$AB = DF,\qquad BC = FE,\qquad CA = ED$
Read the corresponding angles from single letters in matching positions.
$m\angle A = m\angle D,\quad m\angle B = m\angle F,\quad m\angle C = m\angle E$
Check one pair with coordinates. Sides $AB$ and $DF$ are horizontal, so use the absolute value of the difference of the $x$ -coordinates. The slanted sides $BC$ and $FE$ are equal too. The reflection keeps every length, even one you have not computed.
$AB = |5 - 1| = 4,\qquad DF = |-1 - (-5)| = 4$

Answer

The congruence statement is $\triangle ABC \cong \triangle DFE$ . The corresponding sides are $AB$ and $DF$ , $BC$ and $FE$ , and $CA$ and $ED$ . The corresponding angles are $\angle A$ and $\angle D$ , $\angle B$ and $\angle F$ , and $\angle C$ and $\angle E$ .

A congruence statement lets you find missing measures without measuring. Corresponding sides have equal lengths, and corresponding angles have equal measures. So a measure you know in one figure is also the measure of its partner in the other figure. Keep the units the same: a side in centimeters matches a side in centimeters, and an angle in degrees matches an angle in degrees. If a measure is written with a variable, the two equal measures give you an equation to solve.

Worked example

Find measures from a statement

Quadrilateral $JKLM \cong$ quadrilateral $STUV$ . In $JKLM$ , $KL = 5$ cm. In $STUV$ , $VS = 4$ cm and $m\angle V = 90^\circ$ . Find $TU$ , $MJ$ and $m\angle M$ .

Pair the vertices by position, because the statement lists matching vertices in the same positions.
$J \leftrightarrow S,\quad K \leftrightarrow T,\quad L \leftrightarrow U,\quad M \leftrightarrow V$
$TU$ joins the second and third vertices of $STUV$ , so its partner is $KL$ , which joins the second and third vertices of $JKLM$ . Corresponding sides are equal.
$TU = KL = 5\text{ cm}$
$MJ$ joins the fourth vertex back to the first, so its partner is $VS$ , which does the same in $STUV$ .
$MJ = VS = 4\text{ cm}$
$\angle M$ sits at the fourth vertex, so its partner is $\angle V$ . Corresponding angles are equal.
$m\angle M = m\angle V = 90^\circ$

Answer

Side $TU$ is $5$ cm long, side $MJ$ is $4$ cm long, and $m\angle M = 90^\circ$ .

Worked example

Use an equation to find a measure

$\triangle GHK \cong \triangle UVW$ . In $\triangle GHK$ , $GH = 8$ cm and $m\angle G = 60^\circ$ . In $\triangle UVW$ , $UV = (3x - 4)$ cm and $m\angle U = (5y + 15)^\circ$ . Here $x$ and $y$ are numbers, and $3x - 4$ must be positive because it is a length. Find $x$ and $y$ .

By position, $G$ matches $U$ and $H$ matches $V$ . So side $UV$ corresponds to side $GH$ , and $\angle U$ corresponds to $\angle G$ . Corresponding parts are equal, which gives two equations.
$3x - 4 = 8 \qquad\text{and}\qquad 5y + 15 = 60$
Solve for $x$ . Adding $4$ to both sides of the equation keeps it true. Dividing both sides by $3$ also keeps it true. Division is allowed because $3 \ne 0$ .
$3x - 4 + 4 = 8 + 4 \;\Rightarrow\; 3x = 12 \;\Rightarrow\; x = \tfrac{12}{3} = 4$
Check $x = 4$ in the length. It gives $8$ cm, which is positive and equal to $GH$ .
$UV = 3(4) - 4 = 12 - 4 = 8\text{ cm}$
Solve for $y$ the same way. Subtract $15$ from both sides of the equation, then divide both sides by $5$ , which is not zero.
$5y + 15 - 15 = 60 - 15 \;\Rightarrow\; 5y = 45 \;\Rightarrow\; y = \tfrac{45}{5} = 9$
Check $y = 9$ in the angle. It gives $60^\circ$ , which equals $m\angle G$ .
$m\angle U = 5(9) + 15 = 45 + 15 = 60^\circ$

Answer

The solution is $x = 4$ and $y = 9$ , so $UV = 8$ cm and $m\angle U = 60^\circ$ , matching $GH$ and $m\angle G$ .

Takeaway: Two figures are congruent when a sequence of rigid motions carries one exactly onto the other. You decide with coordinates and lengths, not with how the figures look. A congruence statement lists matching vertices in matching positions, and that order tells you which sides and angles are equal. Next, you will graph sequences of congruence transformations and describe a sequence that shows two figures are congruent.