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MathidiaMathematics Academy
Similarity, congruence and scale
Lesson 47 of 75
Easy practice

Triangle similarity

Objective. Establish similarity using equal angles and proportional sides.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
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Learn: the key idea

Triangle similarity means two triangles have the same shape but possibly different sizes, with equal corresponding angles and proportional corresponding sides. To prove similarity, show either that all three pairs of angles are equal (AA is enough, since the third angle is then automatically equal), or that all three pairs of sides are in the same ratio (SSS similarity). For triangles with angles 50°, 60°, 70° and another with the same three angles, they are similar regardless of size, written triangle ABC~triangle DEF. If two sides are proportional and the included angle is equal, this also proves similarity (SAS similarity). Always match corresponding vertices carefully, since the order in the similarity statement indicates which angles and sides correspond.

Worked example

Triangle PQR has angles 40°, 75° and 65°. Triangle STU has angles 40° and 75°. Explain why the triangles are similar.

Diagram of a triangle
  1. 1Angles in a triangle sum to 180°, so STU's third angle = 180−40−75 = 65°
  2. 2Triangle STU has angles 40°, 75°, 65°, matching triangle PQR exactly
  3. 3All three corresponding angles are equal between the two triangles
  4. 4Equal corresponding angles (AA) proves the triangles are similar

Answer: Similar, by AA (equal angles 40°, 75°, 65°)

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Common mistake. Pupils assume two triangles are similar just because they look alike, without checking that angles actually match or sides are in a consistent ratio. Two triangles with sides 3,4,5 and 3,4,6 are not similar since the ratios 1,1,1.2 are not equal. Always verify proportional sides or equal angles numerically.

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