Tiêu đề 6. TRIANGLES.pdf ✅

CLASS - X MATHEMATICS 113 FOUNDATION 6. TRIANGLES VOL - I Introduction Similar Figures Similarity of Triangles Theorems on Similarity Criterion Basic Proportionality Theorem and its converse Baudhayan / Pythagoras Theorem and its converse A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted triangle ∆ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it. Looking around you will see many objects which are of the same shape but of same or different sizes. For examples, leaves of a tree have almost the same shape but same or different sizes. Similarly, photographs of different sizes developed from the same negative are of same shape but different sizes, the miniature model of a building and the building itself are of same shape but different sizes. All those objects which have the same shape but not necessarily the same size are called similar objects SYNOPSIS-1 1.1 SIMILAR FIGURES Two figures are said to be similar if they are the same shape. In more mathematical language, two figures are similar if their corresponding angles are congruent, and the ratios of the lengths of their corresponding sides are equal. This common ratio is called the scale factor. The symbol ~ is used to indicate similarity. For example: 1. In the figure given below ABCDE ~ pentagon VWXYZ. CHAPTER 6 TRIANGLES


116 MATHEMATICS VOL - I CLASS - X 6. TRIANGLES FOUNDATION Corresponding angles are equal. Corresponding sides are in the same ratio. a p b q c r = = To test for similar triangles: AA - If 2 corresponding angles are equal. SSS - If 3 corresponding sides are in the same ratio. SAS - Ratio of 2 pairs of corresponding sides equal and their included angles are equal. Let us understand these similar triangles theorems with their proofs. THEOREM 1: AA (OR AAA) OR ANGLE-ANGLE SIMILARITY CRITERION AA similarity criterion states that if any two angles in a triangle are respectively equal to any two angles of another triangle, then they must be similar triangles. AA similarity rule is easily applied when we only know the measure of the angles and have no idea about the length of the sides of the triangle. In the image given below, if it is known that ∠B = ∠G, and ∠C = ∠F. And we can say that by the AA similarity criterion, ∆ABC and ∆EGF are similar or ∆ABC ~∆EGF. AB EG BC GF AC EF = = and ∠A = ∠E. THEOREM 2: SAS OR SIDE-ANGLE-SIDE SIMILARITY CRITERION According to the SAS similarity theorem, if any two sides of the first triangle are in exact proportion to the two sides of the second triangle along with the angle formed by these two sides of the individual triangles are equal, then they must be similar triangles. This rule is generally applied when we only know the measure of two sides and the angle formed between those two sides in both the triangles respectively. In the image given below, if it is known that AB DE AC DF = , and ∠A = ∠D And we can say that by the SAS similarity criterion, ∆ABC and ∆DEF are similar or ∆ABC ~ ∆DEF. THEOREM 3: SSS OR SIDE-SIDE-SIDE SIMILARITY CRITERION According to the SSS similarity theorem, two triangles will similar to each other if the corresponding ratio of all the sides of the two triangles are equal. This criterion is commonly used when we only have the measure of the sides of the triangle and have less information about the angles of the triangle.