01 · DEFINITION
Similarity vs Congruence
Two figures having the same shape (not necessarily the same size) are called similar figures (denoted by ~).
Definition of Similar TrianglesTwo triangles △ABC and △DEF are similar (△ABC ~ △DEF) if and only if:1. Corresponding angles are equal: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F2. Corresponding sides are proportional: AB / DE = BC / EF = AC / DF
02 · SIMILARITY CRITERIA
The 3 Criteria for Similarity of Triangles
AAA / AA
Angle-Angle Similarity Criterion
If two angles of one triangle are respectively equal to two angles of another triangle, the triangles are similar.SSS
Side-Side-Side Similarity Criterion
If the three corresponding sides of two triangles are in the same ratio (proportional), their corresponding angles are equal and the triangles are similar.SAS
Side-Angle-Side Similarity Criterion
If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the triangles are similar.03 · SOLVED EXAMPLE
Step-by-Step Problem Solving
EXAMPLE 1
A vertical pole of length 6 m casts a shadow 4 m long. At the same time a tower casts a shadow 28 m long. Find height of the tower.
Let height of pole = 6 m, shadow = 4 m. Height of tower = h m, shadow = 28 m.
At the same time, the angle of elevation of the Sun is equal, so by AA similarity, the pole triangle and tower triangle are similar!
Height of Pole / Height of Tower = Shadow of Pole / Shadow of Tower
6 / h = 4 / 28
4h = 6 × 28 = 168
h = 42 m
6 / h = 4 / 28
4h = 6 × 28 = 168
h = 42 m
Conclusion: Height of the tower is 42 metres.
04 · QUICK SUMMARY
Key Takeaways
01
All congruent figures are similar, but similar figures need not be congruent.
02
AA criterion requires only 2 corresponding equal angles to prove similarity.
03
Ratio of areas of similar triangles = ratio of squares of corresponding sides.