# Triangles

## Similar Figures are the figures which have the same shape but not necessarily the same size. Two polygons with same number of sides are said to be similar if their corresponding angles are equal and their corresponding sides are in same ratio or proportion. Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. Conversely, if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. Following three criterions are used to prove the similarity of two triangles: 1. AAA Similarity: If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar. 2. SSS Similarity: If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar. 3. SAS Similarity: If one angle of a triangle is equal to corresponding angle of the other triangle and the sides including these angles are proportional, the two triangles are similar. The ratio of the areas of two similar triangles is equal to the squares of the ratio of their corresponding sides. When triangles ABC and DEF are similar, Area of triangle DEF = k2 × area of triangle ABC, where k is the scale factor (the ratio of the sides of triangle DEF to the corresponding sides of triangle ABC.) Pythagoras Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If triangle ABC is right-angled at B, then AC2 = AB2 + BC2. Conversely, in a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle. Keywords: congruence, scale factor, equiangular triangles, AA similarity

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