M8MATH-8.3

Congruence Through Rigid Motions

Learn how rigid motions (translations, reflections, and rotations) prove two figures are congruent and how to describe the sequence of moves between them.

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What this lesson covers

Congruence means two shapes are exactly the same—same size, same shape. But how do you prove it? This lesson shows you the key idea: two figures are congruent if and only if you can slide, flip, or turn one onto the other using rigid motions. These are moves that don't change distances or angles, so the shape stays identical. You'll learn to describe sequences of these moves, apply them to coordinates, and use coordinates to decide whether two figures are really congruent.

What Are Rigid Motions?

A rigid motion is a transformation that preserves distance and angle measures. The three rigid motions are translation (sliding), reflection (flipping across a line), and rotation (turning around a point). Each one moves every point on a figure in a specific way, but crucially, distances between points stay the same and angles stay the same. For example, if you translate a triangle 5 units right and 3 units up, the side lengths and angles don't change. The triangle just moves to a new location. Dilations are NOT rigid motions because they change size. That's why congruence relies only on translations, reflections, and rotations—they are the only moves that preserve shape and size exactly.

The Definition of Congruence

Two figures are congruent if and only if one can be mapped onto the other by a sequence of rigid motions. This is a precise definition. It means: if you can perform a series of translations, reflections, and rotations to move one figure so that it lands exactly on top of the other, then they are congruent. The order of the moves can matter for efficiency, but the result is the same. You might translate first, then reflect, then rotate—or any other order. The key is that such a sequence exists. Conversely, if two figures are congruent, then a sequence of rigid motions definitely exists to map one onto the other, even if you haven't found it yet. This definition connects geometry intuition (same shape and size) to concrete, describable moves.

Describing a Sequence of Rigid Motions

To show two figures are congruent, you describe which rigid motions map one onto the other. Start by identifying a pair of corresponding points or sides. Then work backward (or forward) to figure out what moves are needed. For instance, if one triangle is to the right and upside-down compared to another, you might translate it left first to align one vertex, then reflect it across a line to flip it over. Write your sequence clearly: 'Translate 4 units left and 2 units down, then reflect across the xx-axis.' You don't have to find the shortest sequence, but your moves must actually map one figure onto the other. A useful strategy is to move one vertex at a time: translate so one vertex of the first figure lands on the corresponding vertex of the second, then rotate or reflect to align the rest.

Using Coordinates to Apply Rigid Motions

When figures are drawn on a coordinate plane, you can apply rigid motions algebraically. A translation by aa units horizontally and bb units vertically maps (x,y)(x, y) to (x+a,y+b)(x + a, y + b). A reflection across the xx-axis maps (x,y)(x, y) to (x,−y)(x, -y); across the yy-axis maps (x,y)(x, y) to (−x,y)(-x, y). A rotation of 90°90° counterclockwise about the origin maps (x,y)(x, y) to (−y,x)(-y, x); a 180°180° rotation maps (x,y)(x, y) to (−x,−y)(-x, -y); a 270°270° counterclockwise (or 90°90° clockwise) rotation maps (x,y)(x, y) to (y,−x)(y, -x). Write the coordinates of all vertices of the first figure, apply the transformations in order, and compute the new coordinates. If they match the vertices of the second figure exactly, the sequence is correct and the figures are congruent. This method is precise and leaves no room for guessing.

Using Coordinates to Decide Congruence

You can also decide whether two figures are congruent by comparing their side lengths and angles using coordinates, without explicitly finding the rigid-motion sequence. Calculate the distance between each pair of vertices using the distance formula d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. If all corresponding side lengths are equal and all corresponding angles are equal, the figures are congruent. This approach confirms congruence without describing the moves. However, the definition says congruence is proved by rigid motions, so you should be ready to describe a sequence too. Remember: if figures are on a grid or have convenient coordinates, rigid motions are often faster. If you need certainty about matching side and angle measures, use the distance formula and angle calculations. Both approaches are valid and show deep understanding.

Key terms

Rigid motion.
A transformation that preserves all distances and angle measures; the three types are translation, reflection, and rotation.
Translation.
A rigid motion that slides a figure a fixed distance in a fixed direction; also called a slide.
Reflection.
A rigid motion that flips a figure across a line of reflection, creating a mirror image.
Rotation.
A rigid motion that turns a figure around a fixed point (the center of rotation) by a fixed angle.
Congruent figures.
Two figures are congruent if one can be mapped onto the other by a sequence of rigid motions; they have the same shape and size.
Correspondence.
The pairing between vertices, sides, or angles of two figures that are being compared for congruence.
Sequence of transformations.
Two or more rigid motions applied one after another to map one figure onto another.

Worked example

Triangle ABCABC has vertices at A=(1,1)A = (1, 1), B=(4,1)B = (4, 1), and C=(4,3)C = (4, 3). Triangle DEFDEF has vertices at D=(−3,−2)D = (-3, -2), E=(−3,−5)E = (-3, -5), and F=(−1,−5)F = (-1, -5). Describe a sequence of rigid motions that maps triangle ABCABC onto triangle DEFDEF, and verify that the figures are congruent.
Start by identifying corresponding vertices. Look at the side lengths: In triangle ABCABC, side ABAB goes from (1,1)(1, 1) to (4,1)(4, 1), a horizontal segment of length 3. Side BCBC goes from (4,1)(4, 1) to (4,3)(4, 3), a vertical segment of length 2. In triangle DEFDEF, side DEDE goes from (−3,−2)(-3, -2) to (−3,−5)(-3, -5), a vertical segment of length 3. Side EFEF goes from (−3,−5)(-3, -5) to (−1,−5)(-1, -5), a horizontal segment of length 2. This suggests that triangle ABCABC has been rotated. Let's check if a 90°90° clockwise rotation about the origin maps ABCABC to DEFDEF. A 90°90° clockwise rotation maps (x,y)(x, y) to (y,−x)(y, -x). Apply it to each vertex: A=(1,1)→(1,−1)A = (1, 1) \to (1, -1); B=(4,1)→(1,−4)B = (4, 1) \to (1, -4); C=(4,3)→(3,−4)C = (4, 3) \to (3, -4). Now we have triangle with vertices at (1,−1)(1, -1), (1,−4)(1, -4), (3,−4)(3, -4). This is not yet at DD, EE, FF. We need to translate. The new A′A' is at (1,−1)(1, -1) but should land on D=(−3,−2)D = (-3, -2). Translate −4-4 units horizontally and −1-1 unit vertically: (1,−1)+(−4,−1)=(−3,−2)(1, -1) + (-4, -1) = (-3, -2) ✓; (1,−4)+(−4,−1)=(−3,−5)(1, -4) + (-4, -1) = (-3, -5) ✓; (3,−4)+(−4,−1)=(−1,−5)(3, -4) + (-4, -1) = (-1, -5) ✓. Sequence: Rotate 90°90° clockwise about the origin, then translate 44 units left and 11 unit down. Verify congruence by checking side lengths: AB=3AB = 3, BC=2BC = 2, AC=(4−1)2+(3−1)2=13AC = \sqrt{(4-1)^2 + (3-1)^2} = \sqrt{13}. Similarly, DE=3DE = 3, EF=2EF = 2, DF=(−1−(−3))2+(−5−(−2))2=13DF = \sqrt{(-1-(-3))^2 + (-5-(-2))^2} = \sqrt{13}. All sides match, confirming the triangles are congruent.

Practice questions

Figure PP is a rectangle with vertices at (0,0)(0, 0), (5,0)(5, 0), (5,3)(5, 3), and (0,3)(0, 3). Figure QQ is a rectangle with vertices at (−5,0)(-5, 0), (−5,−3)(-5, -3), (0,−3)(0, -3), and (0,0)(0, 0). Which sequence of rigid motions maps figure PP onto figure QQ?
  1. Reflect across the xx-axis, then reflect across the yy-axis.
  2. Translate left 5 units, then rotate 180°180° about the origin.
  3. Rotate 180°180° about the origin, then translate right 5 units.
  4. Reflect across the line y=xy = x.

Answer: Reflect across the xx-axis, then reflect across the yy-axis.

Let's test the first choice. Reflect PP across the xx-axis: (0,0)→(0,0)(0, 0) \to (0, 0), (5,0)→(5,0)(5, 0) \to (5, 0), (5,3)→(5,−3)(5, 3) \to (5, -3), (0,3)→(0,−3)(0, 3) \to (0, -3). This gives us a rectangle at (0,0)(0, 0), (5,0)(5, 0), (5,−3)(5, -3), (0,−3)(0, -3). Now reflect across the yy-axis: (0,0)→(0,0)(0, 0) \to (0, 0), (5,0)→(−5,0)(5, 0) \to (-5, 0), (5,−3)→(−5,−3)(5, -3) \to (-5, -3), (0,−3)→(0,−3)(0, -3) \to (0, -3). We get (−5,0)(-5, 0), (−5,−3)(-5, -3), (0,−3)(0, -3), (0,0)(0, 0)—exactly figure QQ. The other sequences either don't map PP to QQ exactly or leave the figure rotated incorrectly.
Two quadrilaterals ABCDABCD and EFGHEFGH are congruent. Side ABAB has length 7. Which of the following must be true?
  1. Side EFEF has length 7.
  2. Side EFEF is parallel to side ABAB.
  3. Side EFEF is perpendicular to side ABAB.
  4. Quadrilateral EFGHEFGH is a square.

Answer: Side EFEF has length 7.

If the quadrilaterals are congruent, a sequence of rigid motions maps one onto the other. Rigid motions preserve distance, so corresponding sides must have the same length. If ABAB corresponds to EFEF, then EF=7EF = 7. The other choices are not guaranteed by congruence alone. Translations, reflections, and rotations change position and direction but not length. Parallelism and perpendicularity depend on the orientation after the moves, not on congruence itself. The shape of EFGHEFGH is identical to ABCDABCD, but we don't know what shape that is—it could be a square, but it doesn't have to be.
Explain why two congruent figures must have the same perimeter and the same area. Use the definition of congruence in your answer.

Answer: Two congruent figures are related by a sequence of rigid motions. Since rigid motions preserve all distances, every side length of one figure matches the corresponding side length of the other. Therefore, the sum of all side lengths—the perimeter—must be the same. Similarly, rigid motions preserve the space enclosed by the figure, so the area must be the same. The shape and size do not change, only position and orientation.

This is a direct consequence of the definition of congruence. Students often confuse congruence with similarity and think similar figures might have different areas. This question clarifies that congruence is stronger: it guarantees identical measurements. A good answer names the rigid motions and explains that they preserve distance and area. It connects the formal definition to a concrete conclusion.

FAQ

What's the difference between congruence and similarity?
Congruence means two figures are exactly the same shape and size. Rigid motions map one onto the other. Similarity means the figures have the same shape but may be different sizes. Dilations (scaling) are used to show similarity, and dilations are not rigid motions because they change size. If two figures are congruent, they are also similar, but not every pair of similar figures is congruent.
Do I have to describe the exact sequence of rigid motions, or can I just say the figures are congruent?
To prove two figures are congruent, you should describe a sequence of rigid motions that maps one onto the other. You can also verify it by comparing all side lengths and angles using coordinates and the distance formula. Either method shows understanding. However, just saying 'they look congruent' is not enough—you need to show your work either by describing moves or by calculating measurements.
Can the order of rigid motions change the result?
Sometimes yes, sometimes no. The order can matter. For example, translating then rotating often gives a different final position than rotating then translating. However, the key point is that if a sequence of rigid motions works, the figures are congruent. Your job is to find at least one working sequence, not all possible sequences. Different sequences might achieve the same goal, and that's fine.
Why don't dilations count as rigid motions?
A dilation changes the size of a figure by multiplying all coordinates by a scale factor. This changes distances between points and side lengths. Rigid motions by definition preserve all distances and angles. Since dilations don't preserve distance, they're not rigid. Dilations are used to show similarity, not congruence. Two figures related by a dilation are similar but not congruent.

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