Similarity

Conditions for similarity of triangles

There are three conditions for similarity of triangles:

i) Angle, Angle similarity test

^{Fig:Angle Angle}

If two angles of one triangles are respectively equal to two angles of another triangle, then two triangles are similar.

For example:

Here,∠B =∠Y and∠C =∠Z. The remaining angles∠A and∠X are also equal.

∴ \(\triangle\)ABC∼ \(\triangle\)XYZ

ii) Side, Side, Side similarity test

^{Fig: SSS}

If the corresponding sides of two triangles are proportional, then the triangles are similar.

For example:

Here, PQ/XY = QR/YZ = PR/XZ

∴ \(\triangle\)PQR∼ \(\triangle\)XYZ

iii) Side, Angle, Side similarity test

^{Fig: SAS}

If two corresponding sides of two triangles are proportional and the angle contained by these sides are equal, then the triangles are similar.

For example:

Here, XY/AB = YZ/BC and∠Y =∠B

∴ \(\triangle\)XYZ∼ \(\triangle\)ABC

Similar polygons

Two polygons are similar under following conditions:

i) When two or more polygons are equiangular, they are similar.

In the figure, ∠A =∠P,∠B =∠Q,∠C =∠R,∠D =∠S

∴ quad ABCD∼ quad PQRS

ii) When the corresponding sides of two polygons are proportional, they are similar.

In the figure, AB/PQ = BC/QR = CD/RS = DA/SP

∴ quad ABCD∼ quad PQRS

iii) When the corresponding diagonals of the polygons are proportional to their corresponding sies, they are similar.

In the figure, AC/PR = BD/QS = AB/PQ

∴ quad ABCD∼ quad PQRS

iv) When the corresponding diagonals divide the polygons into the equal number of similar triangles, the polygons are equal.

\(\triangle\)ABC∼ \(\triangle\)PQR, \(\triangle\)ACD∼ \(\triangle\)PRS, \(\triangle\)ADE∼ \(\triangle\)PSV

∴ polygon ABCDE∼ polygon PQRS

Note: Theorem with '*' in similarity chapter do not need proof or experimental verification but the problems related to them are included in the curriculum.