Triangle tests and similar-solid scales

Enough data must determine the triangle

Two triangles are congruent with matching SSS (three sides), SAS (two sides and their included angle), ASA/AAS (two angles and a corresponding side), or RHS (a right angle, hypotenuse and one corresponding shorter side). State the matching order and which data are given.

Three equal corresponding angles establish similarity, not size equality. Two sides and a non-included angle do not generally establish congruence; there can be two distinct triangles with those data.

Similarity needs one common length scale

Use two equal corresponding angles (AA), proportional corresponding sides (SSS similarity), or proportional pairs around an equal included angle (SAS similarity). For sides 3, 4, 5 and 6, 8, 10, every larger side is twice the corresponding smaller side, so the triangles are similar with scale factor 2.

If only two side ratios match but their included angle differs, similarity is not established. A proof needs enough conditions, not merely a suggestive drawing.

Dimensions determine the powers of a scale factor

For similar plane figures, area ratio is k² when length ratio is k. For similar solids, volume ratio is k³. Lengths, areas and volumes scale in one, two and three independent directions respectively.

If similar solids have volume ratio 27:64, their length ratio is 3:4 and corresponding surface-area ratio 9:16. Taking the square root of the volume ratio would use the wrong dimension. These rules need similar solids, not merely equal volumes or a matching pair of edges.

Check your understanding 1 Recover the length scale first

Two similar solids have surface-area ratio 25:49. The smaller volume is 125 cm³. Find the larger volume.

Your answer or drawing on paper

Show answer

The length ratio is 5:7, so volume ratio is 125:343. The larger volume is 343 cm³. Applying the surface-area factor 49/25 directly to the volume would mix dimensions.

State a proof’s conclusion precisely

If two triangle angles are equal in pairs, you can conclude the third pair match and the triangles are similar. You cannot conclude corresponding sides are equal until the scale factor is known to be 1. Keep the claim no stronger than the data support.