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In ΔXYZ, M lies on XY and N lies on XZ such that ∠XNM = ∠XYZ. Which similarity statement is correct?

Class 10Triangles (Similarity)Geometry — Similarity
  • AΔXMN ~ ΔXYZ
  • BΔMNX ~ ΔXYZ
  • CΔXNM ~ ΔXZY
  • DΔXMN ~ ΔXZY

Step-by-Step Solution

∠X is common and ∠XNM = ∠XYZ, so by AA the correspondence is X ↔ X, N ↔ Y, M ↔ Z, i.e. ΔXNM ~ ΔXYZ. Written with M before N, the same statement reads ΔXMN ~ ΔXZY. "ΔXMN ~ ΔXYZ" is the tempting wrong answer that matches M with Y and N with Z — the opposite of what the equal angles say; "ΔXNM ~ ΔXZY" swaps only one triangle; "ΔMNX ~ ΔXYZ" misplaces the common vertex.
Answer: D) ΔXMN ~ ΔXZY

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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