CHAPTER AT A GLANCE
Similar Figures
Two figures having the same shape but not necessarily the same size are called similar figures/
Two figures having the same shape as well as same size are called congruent figures.
Note : that all congruent figures are similar but the similar figures need not be congruent.
Similarity of Triangles
Two triangles are similar, if
(i) their corresponding angles are equal and
(ii) their corresponding sides are in the same ratio (or proportion)
NOTE : If corresponding angles of two triangles are equal, then they are known as
Equiangular Triangles.
Criteria for Similarity of Triangles
(i) AAA Similarity Criterion : If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.
(ii) AA Similarity Criterion : If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar.
(iii) SSS Similarity Criterion : If in two triangles, corresponding sides are in the same ratio then their correspoding angles are equal and hence the triangles are similar.
(iv) SAS Similarity Criterion : If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportion), then the two triangles are similar.
Theorems and some Important Results
Theorem : Basic Proportionality Theorem or Thale’s Theorem
Statement: “If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio”
In ABC, a line parallel to BC intersects AB at D and AC at E. then
AD/DB = AE/EC
Theorem: Converse of B.P. Theorem
Statement: “If a line divides any two sides of a triangle in the same ratio,”
Areas of Two Similar Triangles
Theorem 1: Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
Theorem 2: Prove that the areas of two similar triangles are in the ratio of the square of corresponding altitudes.
Theorem: Pythagoras Theorem or Bodhayan Theorem
Statement: Prove that,”In a right angled triangle, the square of the hypotenuse is equal to the sum of the other two sides”.
Theorem: Converse of Pythagoras Theorem
Statement: “In a triangle, the square of the longer side is equal to the sum of the squares of the other two sides then the triangle is right angled triangle”.
ABC is a triangle such that BC2 = AB2 + AC2 then LA = 90°
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