8th Grade · 8.G.A.2

Congruent figures in 8th grade

Congruent figures are shapes with exactly the same size and shape, even when one is turned, flipped or moved somewhere else. In 8th grade, congruence (being congruent) gets a precise meaning. Two figures are congruent when slides, flips and turns can move one exactly on top of the other. Your child learns to find those moves and to list them in order.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

8.G.A.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

What it means, in plain words

Before 8th grade, children decide that two shapes match by looking at them. Now they show it with moves. A figure is any flat shape, such as a triangle or a rectangle. Two figures are congruent if a sequence of rigid transformations carries the first figure exactly onto the second. Rigid transformations are the moves that keep size and shape: translations (slides), reflections (flips) and rotations (turns). A sequence is a list of moves done one after another.

Some teachers call these moves congruence transformations. A typical answer reads: reflect the triangle across the y-axis, then translate it 2 units down. Often more than one sequence works. Any list of moves that puts the first figure exactly on the second is correct. The order matters: the same moves in a different order can send the shape somewhere else.

When one figure sits on the other, each corner lands on a matching corner, called a corresponding vertex. The matching corners show which sides and angles are equal. If corner $A$ lands on corner $R$ , the angle at $A$ is the same size as the angle at $R$ . Much of the work with congruent figures in 8th grade is listing these pairs correctly.

Your child should be able to:

Decide whether two figures are congruent by finding moves that carry one onto the other
Describe a sequence of translations, reflections and rotations in the right order
Pair each corner of one figure with its corresponding corner on the other
Explain why two shapes of different sizes cannot be congruent

Example

Triangle $ABC$ has corners $A(1, 1)$ , $B(4, 1)$ and $C(1, 3)$ . Triangle $DEF$ has corners $D(-1, -1)$ , $E(-4, -1)$ and $F(-1, 1)$ . Are the triangles congruent, and which moves show it?

Triangle $DEF$ is on the other side of the y-axis and faces the other way, so start with a reflection across the y-axis.
The reflection changes the sign of each x-coordinate: $A$ goes to $(-1, 1)$ , $B$ to $(-4, 1)$ and $C$ to $(-1, 3)$ .
Each of those points is 2 units above a corner of $DEF$ , so translate the triangle 2 units down: $(-1, -1)$ , $(-4, -1)$ and $(-1, 1)$ .
Those are exactly $D$ , $E$ and $F$ . The triangles are congruent: reflect across the y-axis, then translate 2 units down.

Example

One rectangle is 2 cm by 3 cm. Another rectangle is 4 cm by 6 cm. Are they congruent?

Slides, flips and turns never change a length.
The 2 cm side would have to become 4 cm, and no rigid move can do that.
The rectangles are not congruent. They have the same shape but not the same size.

Where children usually get stuck

Your child says two shapes are congruent because they look the same, and writes no moves.
Matching by eye was enough in earlier grades. In 8th grade the moves are the reason, and naming them feels pointless when the answer already seems obvious.
What helps. Ask "Which move comes first?" Then have your child trace the first shape on thin paper and move the tracing onto the second shape, naming each move out loud.
Your child pairs corners by where they sit on the page, top left with top left, and writes "B matches S" when B really matches R.
After a flip or a turn, the top-left corner can end up anywhere, so position on the page is no longer a guide.
What helps. Put a dot on one corner of the tracing, do the moves and see where the dot lands. The corner under the dot is its match.
Your child lists the right moves in the wrong order, and the shape ends up in the wrong place.
The order changes the result. A slide right followed by a flip across the y-axis lands somewhere different from the same flip followed by the same slide.
What helps. Have your child do one move at a time on graph paper and plot the corners after each step, so the place where each move leaves the shape is visible.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
The blue figure is the start, and the red figure is the target. Order the cards to map blue to red, using every card once.
Drag a card to move it, or tap a card, then tap where it should go.
1
Reflect across the y-axis
2
Translate 1 unit left and 2 units down
Show the answer and how to solve it
The correct order is Reflect across the y-axis → Translate 1 unit left and 2 units down.
1. 1Track a blue vertex through the cards. Reflect across the y-axis. The vertex at $(1, 1)$ moves to $(-1, 1)$ .
2. 2Translate 1 unit left and 2 units down. The vertex at $(-1, 1)$ moves to $(-2, -1)$ . Apply the same moves to the other vertices. Each one lands on a vertex of the red target.
Common mistake. Check the whole sequence, not just the first move. For example, try this order: Translate 1 unit left and 2 units down -> Reflect across the y-axis. The vertex starting at $(1, 1)$ follows $(1, 1)$ -> $(0, -1)$ -> $(0, -1)$ . It ends at $(0, -1)$ , where the red target has no vertex, so this order does not work.
The idea underneath. In a sequence of rigid transformations, order matters because each move changes the figure that the next move starts from.
2On level
Look at the blue figure and the red figure. Match each vertex listed on the blue figure to the corresponding vertex on the red figure.
Connect each item on the left to its match on the right.
Blue figure vertices
Red figure vertices
M
L
S
Q
R
Use Tab to navigate between items. Click or tap a card to select it, then click or tap a card on the opposite side to create a match. You can also drag a card or its connector dot to a card on the opposite side.
Show the answer and how to solve it
The matches are M with R and L with S. The red vertex Q is left over: it corresponds to J, which is not in the list.
1. 1Look at the blue figure and the red figure. They are the same size and shape, and the red figure is in a new place on the grid. Match each blue vertex in the list, M and L, to the corresponding red vertex, the one in the same spot on the shape. There are 3 red vertices to choose from and 2 blue vertices to match, so one red vertex is left over.
2. 2How to find a match. Corresponding vertices sit between sides of the same lengths, because the figures are the same size and shape. Moving a figure can change the direction a side points, but not its length. So count each side on the grid. For a horizontal or vertical side, count the grid units between its endpoints. Name a slanted side by its two counts: a side that goes 1 unit one way and 3 units the other way is a 1-by-3 side, whichever way it points.
3. 3Start with L. From L, the side to M goes 3 units left and 1 unit down: a 1-by-3 side. The side to K goes 3 units down: a straight side 3 units long. So the red vertex that corresponds to L sits between a straight side 3 units long and a 1-by-3 side.
4. 4Check each red vertex in the right column: S sits between a straight side 3 units long and a 1-by-3 side, Q sits between a straight side 4 units long and a 1-by-2 side, and R sits between a 1-by-2 side and a 1-by-3 side. Only S has both of these sides, so L corresponds to S.
5. 5Next, M. Follow the 1-by-3 side from L: it leads to M. At S, the 1-by-3 side leads to R. So M corresponds to R. Check: M and R both sit between a 1-by-2 side and a 1-by-3 side.
6. 6Last, the red vertex Q. Follow the 1-by-2 side from M: it leads to J. At R, the 1-by-2 side leads to Q. So the red vertex Q corresponds to J, which is not in the list. The red vertex Q is left over. Check: J and Q both sit between a straight side 4 units long and a 1-by-2 side.
Common mistake. One common mistake is pairing letters by their place in the alphabet. In the alphabet, M comes fourth among the blue letters and S comes fourth among the red letters, but S does not correspond to M. Name a slanted side by its two counts: a side that goes 1 unit one way and 2 units the other way is a 1-by-2 side, whichever way it points. The blue vertex M sits between a 1-by-2 side and a 1-by-3 side. The red vertex S has a 1-by-3 side like M, but no 1-by-2 side. Compare the sides at each vertex, not the letters.
The idea underneath. Figures with the same size and shape have sides of the same lengths, so a vertex and its corresponding vertex sit between sides of the same lengths. Find one match by the sides at a vertex. Then follow a side: the vertex at the other end corresponds to the vertex at the other end of the matching side.
3On level
Look at the two congruent figures. Marcus listed the matching vertices below. Select all claims that are not correct.
A
B matches S
B
A matches Q
C
D matches P
D
C matches R
Show the answer and how to solve it
Select "B matches S" and "C matches R": those claims are not correct. In fact, B matches R and C matches S. The other claims are correct, so do not select them.
1. 1Look at the two figures, ABCD and PQRS. They are congruent: the same size and shape. Marcus listed which vertex of PQRS matches each vertex of ABCD. Check each claim, then select the claims that are not correct.
2. 2Here is how to check a claim. Matching vertices sit between sides of the same lengths, because the figures are congruent. Moving a figure can change the direction a side points, but not its length. Use the grid to compare the sides. Each horizontal or vertical side goes along a grid line, so its length is the number of units it covers. Call a slanted side that goes 1 unit one way and 1 unit the other way a 1-by-1 side, whichever way it points. For example, at A the side to B goes 3 units right: a horizontal or vertical side 3 units long. The side to D goes 1 unit right and 1 unit up: a 1-by-1 side.
3. 3Is "A matches Q" correct? A sits between a horizontal or vertical side 3 units long and a 1-by-1 side. Q sits between a horizontal or vertical side 3 units long and a 1-by-1 side. These are the same two sides, and no other vertex of PQRS has them. So the claim "A matches Q" is correct.
4. 4Is "B matches S" correct? B sits between two horizontal or vertical sides, 2 units and 3 units long. S sits between a horizontal or vertical side 2 units long and a 1-by-2 side. S has a horizontal or vertical side 2 units long like B, but no horizontal or vertical side 3 units long. So the claim "B matches S" is not correct. B matches R: R is the only vertex of PQRS with B's two sides.
5. 5Is "C matches R" correct? C sits between a horizontal or vertical side 2 units long and a 1-by-2 side. R sits between two horizontal or vertical sides, 2 units and 3 units long. R has a horizontal or vertical side 2 units long like C, but no 1-by-2 side. So the claim "C matches R" is not correct. C matches S: S is the only vertex of PQRS with C's two sides.
6. 6Is "D matches P" correct? D sits between a 1-by-1 side and a 1-by-2 side. P sits between a 1-by-1 side and a 1-by-2 side. These are the same two sides, and no other vertex of PQRS has them. So the claim "D matches P" is correct.
Common mistake. One common mistake is pairing letters by their place in the alphabet: the third letter of ABCD with the third letter of PQRS, so C with R. Marcus's claim "C matches R" follows that pattern. But C does not match R. C sits between a horizontal or vertical side 2 units long and a 1-by-2 side (1 unit one way and 2 units the other). R has a horizontal or vertical side 2 units long like C, but no 1-by-2 side. Compare the sides at each vertex, not the letters.
The idea underneath. Congruent figures have sides of the same lengths, so matching vertices sit between sides of the same lengths. Moving a figure changes where a vertex is and which way its sides point, but not how long the sides are. To check a claim, compare the sides at the two vertices, not the letters.
4Stretch
The blue figure is the original and the orange figure is its image. Which description matches the transformation that maps the blue figure to the orange figure?
A
Rotate $180^{\circ}$ about the origin
B
Rotate $90^{\circ}$ counterclockwise about the origin
C
Rotate $90^{\circ}$ clockwise about the origin
D
Reflect across the y-axis
Show the answer and how to solve it
The answer is "Rotate $90^{\circ}$ counterclockwise about the origin." It is the only description that moves every blue corner onto an orange corner: $(x, y)$ goes to $(-y, x)$ .
1. 1Look at the two figures. The blue figure is the original and the orange figure is its image. The blue figure is in the upper right of the grid. The orange figure is in the upper left of the grid. They are the same size and shape. The matching description has to move every blue corner onto an orange corner. Test each description by moving the highest blue corner, $(2, 5)$ , the way the description says.
2. 2Does "Rotate $90^{\circ}$ clockwise about the origin" match? In a rotation, the figure turns around a fixed point, here the origin, $(0, 0)$ . A $90^{\circ}$ clockwise rotation goes the way a clock's hands move. It sends every point $(x, y)$ to $(y, -x)$ .
3. 3Test it on the highest blue corner, $(2, 5)$ . A $90^{\circ}$ clockwise rotation sends it to $(5, -2)$ . There is no orange corner at $(5, -2)$ . So rotating $90^{\circ}$ clockwise about the origin does not map the blue figure to the orange figure.
4. 4Does "Rotate $180^{\circ}$ about the origin" match? A $180^{\circ}$ rotation sends every point straight through the origin to the opposite side: $(x, y)$ goes to $(-x, -y)$ .
5. 5Test it on the same corner, $(2, 5)$ . A $180^{\circ}$ rotation sends it to $(-2, -5)$ . There is no orange corner at $(-2, -5)$ . So rotating $180^{\circ}$ about the origin does not map the blue figure to the orange figure.
6. 6Does "Reflect across the y-axis" match? In a reflection, the figure flips over a line, like folding the grid along that line. Across the y-axis, a point keeps its y-coordinate and its x-coordinate changes sign: $(x, y)$ goes to $(-x, y)$ .
7. 7Test it on the same corner, $(2, 5)$ . A reflection across the y-axis sends it to $(-2, 5)$ . There is no orange corner at $(-2, 5)$ . So reflecting across the y-axis does not map the blue figure to the orange figure.
8. 8Does "Rotate $90^{\circ}$ counterclockwise about the origin" match? A $90^{\circ}$ counterclockwise rotation goes against a clock's hands. It sends every point $(x, y)$ to $(-y, x)$ .
9. 9Test it on every blue corner. $(1, 2)$ lands on $(-2, 1)$ , $(3, 1)$ on $(-1, 3)$ , $(4, 3)$ on $(-3, 4)$ , and $(2, 5)$ on $(-5, 2)$ . All 4 corners land on orange corners. So rotating $90^{\circ}$ counterclockwise about the origin maps the blue figure to the orange figure. This description matches the transformation.
Common mistake. A common mistake is choosing the other direction, "Rotate $90^{\circ}$ clockwise about the origin." A $90^{\circ}$ clockwise rotation sends the corner $(2, 5)$ to $(5, -2)$ , where there is no orange corner. A $90^{\circ}$ counterclockwise rotation sends it to $(-5, 2)$ , an orange corner. The blue figure was rotated $90^{\circ}$ counterclockwise about the origin.
The idea underneath. To test a description, move one blue corner the way it says. If there is no orange corner where it lands, the description does not match. The matching description moves every blue corner onto an orange corner. Here every point $(x, y)$ went to $(-y, x)$ : a $90^{\circ}$ counterclockwise rotation about the origin.