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In trapezium ABCD, AB ∥ DC and the diagonals AC and BD cut each other at O. Which pair of triangles is similar?

Class 10Triangles (Similarity)Geometry — Similarity
  • AΔAOB ~ ΔCOD
  • BΔAOB ~ ΔDOC
  • CΔAOB ~ ΔOCD
  • DΔABC ~ ΔBCD

Step-by-Step Solution

Since AB ∥ DC, the transversal AC gives ∠OAB = ∠OCD and the transversal BD gives ∠OBA = ∠ODC (alternate angles). So A ↔ C and B ↔ D, and with the vertically opposite angles at O we get ΔAOB ~ ΔCOD (AA). "ΔAOB ~ ΔDOC" matches A with D and B with C — the right pair of triangles but the wrong vertex order; "ΔAOB ~ ΔOCD" puts O in the wrong position.
Answer: A) ΔAOB ~ ΔCOD

Key idea: Similarity vs congruence

**Congruent**: same shape AND same size. All corresponding parts equal.
**Similar**: same shape, possibly DIFFERENT size. Corresponding angles equal; corresponding sides PROPORTIONAL (in fixed ratio k).

If two triangles are similar with scale factor :
- Corresponding sides differ by factor .
- Corresponding angles are equal (always, by definition).
- Areas differ by factor — critical fact, and a common test question.

**Examples**:
- Equilateral triangles are ALL similar (same angles 60°-60°-60°).
- A photograph and its enlarged print are similar.
- Two right triangles sharing one acute angle are similar (AA).

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