Refined and Comprehensive Class 10 Mathematics Notes
Similarity Criteria & Core Concepts
Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio.
Similarity Criteria
AAA (Angle-Angle-Angle): If corresponding angles of two triangles are equal, their corresponding sides are proportional and the triangles are similar. (AA criterion also applies)
SSS (Side-Side-Side): If corresponding sides of two triangles are in the same ratio, their corresponding angles are equal and the triangles are similar.
SAS (Side-Angle-Side): 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.
Basic Proportionality Theorem (Thales Theorem)
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Important Results for Similar Triangles
If ΔABC ~ ΔDEF, then corresponding sides are proportional:
AB / DE = BC / EF = AC / DF
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides:
Converse of BPT: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Area Property: The ratio of areas of two similar triangles is also equal to the ratio of the squares of their corresponding medians, altitudes, or angle bisectors.
Important Practice Questions
State and prove the Basic Proportionality Theorem (BPT).
Prove that two triangles are similar using SAS or AAA similarity criterion.
If two similar triangles have a side ratio of 3:5, find the ratio of their areas.
Quick Revision
BPT (Thales Theorem)
Line parallel to one side divides the other two sides proportionally.
Similarity Criteria
AAA, SSS, and SAS establish similarity between triangles.