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A carpenter wants to check whether a cross-bar is parallel to the base of a triangular frame. The bar meets the two sloping side pieces at 26 cm and 39 cm from the apex, and the remaining lengths of those pieces below the bar are 34 cm and 51 cm. Is the bar parallel to the base?

Class 10Triangles (Similarity)Geometry — Similarity
  • AYes — both side pieces are cut in the ratio 13 : 17, so the converse of BPT applies
  • BNo — the two side pieces have different total lengths
  • CYes — because both lower pieces (34 cm and 51 cm) are longer than the upper pieces
  • DIt cannot be decided without measuring the base

Step-by-Step Solution

and . The bar divides the two sides in the same ratio, so by the converse of BPT it is parallel to the base. The sides having different total lengths (60 cm and 90 cm) is irrelevant — BPT is about RATIOS, not lengths — and merely being longer below than above proves nothing. The base is never needed for the test.
Answer: A) Yes — both side pieces are cut in the ratio 13 : 17, so the converse of BPT applies

Key idea: Three similarity criteria

**AA (Angle-Angle)**: Two angles equal → similar. (Third angle is automatic by 180° angle sum.)

**SSS (Side-Side-Side proportionality)**: All three pairs of corresponding sides in same ratio → similar.

**SAS (Side-Angle-Side)**: Two pairs of sides proportional AND included angles equal → similar.

**Important**: SSA does NOT prove similarity (just like for congruence).

AA is by far the most-used in problems: spot two equal angles (often via parallel lines or shared angle) and declare similarity.

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