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.

Things to remember

• Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or shrinking), possibly with additional translation, rotation and reflection.

• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.

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