Key takeaway: Congruent figures have exactly the same size and shape, so every pair of corresponding sides and angles is equal and one figure can be moved to fit on the other.
Grade 88.G.A.2Geometry and Measurement
Definition of Congruent Figures
Congruent figures are perfect copies. If you trace one on tracing paper, you can slide, flip or turn the paper and lay it exactly over the other with no gaps and no overhang. Because of that, matching sides have the same length and matching angles have the same measure.
The order of the letters tells which parts match. In \(\triangle ABC \cong \triangle DEF\), vertex A corresponds to D, B to E and C to F, so side AB matches DE and angle B matches angle E. Slides, flips and turns produce congruent images, but a dilation by a factor other than 1 does not. Tick marks and arcs in a diagram show which parts are equal.
Formula and notation
The symbol \(\cong\) means “is congruent to”:
\[\triangle ABC \cong \triangle DEF\]
It implies \(AB = DE\), \(BC = EF\), \(CA = FD\) and \(\angle A = \angle D\), \(\angle B = \angle E\), \(\angle C = \angle F\).
Key facts
- Corresponding sides are equal and corresponding angles are equal.
- The order of letters in the statement shows which parts correspond.
- A slide, a flip or a turn gives a congruent copy.
- Congruent figures have equal perimeters and equal areas.
Congruent Figures example
Triangles \(ABC\) and \(DEF\) are congruent, with \(\triangle ABC \cong \triangle DEF\). Given \(AB = 6.5\) cm, \(BC = (3x + 4)\) cm, \(EF = 22\) cm and \(m\angle B = 72^{\circ}\), find \(x\), \(DE\) and \(m\angle E\).
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The letter order pairs BC with EF, so \(3x + 4 = 22\).
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Solve: \(3x = 18\), so \(x = 6\).
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AB corresponds to DE, so \(DE = 6.5\) cm.
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Angle B corresponds to angle E, so \(m\angle E = 72^{\circ}\).
\(x = 6\), \(DE = \) 6.5 cm and \(m\angle E = \) \(72^{\circ}\).
Common mistake
Watch out
Two triangles have side lengths 3, 4, 5 and 6, 8, 10 with equal angles. Ines says they are congruent because they have the same shape.
Why it is wrong
Same shape is not enough; the size must match too. The second triangle is twice as large, so the triangles are similar with scale factor 2, not congruent. Corresponding sides of congruent figures must be exactly equal, and 3 does not equal 6.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Geometry and Measurement