What does congruent mean?
Two figures are congruent if they have exactly the same shape and the same size. We write ABCD ≅ PQRS.
A simple test: trace one figure on thin paper. If you can place the tracing exactly over the other figure (you may slide it, turn it or flip it over), they are congruent.
Same shape but a different size is not congruent; that is similar.
Rigid motions: translation, reflection and rotation
A rigid motion (also called an isometry) moves a figure without changing distances.
- Translation: every point moves the same distance in the same direction, given by a vector such as (5, 0.6). In coordinates: (x, y) → (x + a, y + b).
- Reflection: every point goes to the same distance on the other side of a mirror line. In the y-axis: (x, y) → (−x, y). In the x-axis: (x, y) → (x, −y). The figure's orientation is reversed (clockwise labels become anticlockwise).
- Rotation: every point turns through the same angle about a fixed centre. 90° anticlockwise about the origin: (x, y) → (−y, x). 180°: (x, y) → (−x, −y).
All three keep: lengths, angle sizes, parallel lines parallel, and straight lines straight. So the image of a figure is always congruent to it.
A dilation (x, y) → (kx, ky) with k ≠ 1 changes lengths, so it is not rigid.
Sequences of rigid motions and corresponding parts
Two figures are congruent exactly when a sequence of rigid motions maps one onto the other. To show it:
- Name matching vertices in order: A↔A′, B↔B′, C↔C′…
- Choose moves: often a reflection (if the orientation is reversed), then a rotation (to line up the directions), then a translation (to move into place).
- Check that every vertex lands on its partner.
Once figures are congruent, every pair of corresponding parts is equal: AB = A′B′, ∠A = ∠A′, and so on.
Why SSS, SAS and ASA work
For triangles we do not need to check all six parts. If two sides and the angle between them match (SAS), we can slide and turn one triangle so the angle and the two sides sit on top of each other; the third side then has no choice and fits too. SSS and ASA follow in a similar way. More detail is in the lesson on triangle congruence.
Congruence in daily life, design and problem solving
- Building and making: bricks, tiles, screws and spare parts must be congruent to fit.
- Design and patterns: a pattern is made by repeating one motif using slides (borders), flips (mirror symmetry) and turns (rangoli, wheels, flowers).
- Measuring the unmeasurable: to find the width of a river, build a congruent triangle on your side of the bank and measure that instead.
Try it at home
Cut a shape from cardboard. On paper, draw around it, then slide, flip or turn it and draw around it again. Measure matching sides and angles with a ruler and protractor: they are equal.
Exam-style tasks: describe a sequence of moves that maps one figure to another, find a missing length or angle using corresponding parts, and decide if a transformation keeps congruence.
Key formulas and definitions
- Translation by (a, b): (x, y) → (x + a, y + b)
- Reflection in y-axis: (x, y) → (−x, y); in x-axis: (x, y) → (x, −y)
- Reflection in y = x: (x, y) → (y, x)
- Rotation 90° anticlockwise about O: (x, y) → (−y, x); 180°: (−x, −y)
- Dilation scale k about O: (x, y) → (kx, ky), congruent only if k = 1
- Congruent ⇒ corresponding sides and angles equal
Worked examples
1. Triangle with vertices (1, 2), (4, 2), (1, 6) is translated by (−3, 1). Find the image.
Add −3 to x and 1 to y: (−2, 3), (1, 3), (−2, 7). Sides are still 3, 4 and 5.
2. Reflect the point (3, −2) in the y-axis, then rotate 90° anticlockwise about the origin.
Reflection: (−3, −2). Rotation (x, y) → (−y, x): (2, −3).
3. PQRS ≅ WXYZ with P↔W, Q↔X. If PQ = 7 cm and ∠Q = 110°, find WX and ∠X.
Corresponding parts are equal: WX = 7 cm and ∠X = 110°.
4. Square A has side 3 cm. Square B is its image under a dilation of scale 2. Are they congruent?
No. B has side 6 cm. Same shape, different size: similar, not congruent.
5. Figure F has labels clockwise; its image G has the same labels anticlockwise and the same size. Which move must be in the sequence?
A reflection (an odd number of reflections), because only reflections reverse orientation.
6. Describe a sequence mapping triangle A(0,0), B(2,0), C(0,1) onto A′(5,0), B′(5,2), C′(4,0).
Rotate 90° anticlockwise about O: (0,0), (0,2), (−1,0). Then translate by (5, 0): (5,0), (5,2), (4,0). Done.
Common mistakes
- Calling two figures congruent because they look alike, even though their sizes differ.
- Matching the wrong vertices: in ABC ≅ PQR, A goes with P, B with Q, C with R, in that order.
- Thinking a reflected figure is not congruent because it looks reversed. Flips keep size and shape.
- Using the wrong rotation rule: 90° anticlockwise is (x, y) → (−y, x), not (y, −x).