Class 10 Mathematics · Chapter 6 NotesTriangles

Two figures that have the same shape but not necessarily the same size are called similar figures. All congruent figures are similar, but similar figures need not be congruent. For example, all circles are similar, all squares are similar, and all equilateral triangles are similar. However, a circle and a square are not similar because their shapes differ. For two polygons with the same number of sides to be similar, two conditions must hold: their corresponding angles must be equal, and their corresponding sides must be in the same ratio. This common ratio is called the scale factor. If either condition fails, the polygons are not similar. For instance, a square and a rectangle have equal corresponding angles but their sides are not in the same ratio, so they are not similar.

• Similar figuressame shape, not necessarily the same size
• All congruent figures are similar, but not all similar figures are congruent
• Two polygons are similar if corresponding angles are equal and corresponding sides are proportional
• Scale factorthe common ratio of corresponding sides

The Basic Proportionality Theorem, also known as the Thales Theorem, is a key result about triangles. It 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. In triangle ABC, if DE is parallel to BC and D lies on AB, E lies on AC, then AD/DB = AE/EC. The converse is also true: if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. These two results are used to prove many other properties, such as the midpoint theorem and results about trapeziums.

• Theorem 6.1If DE || BC, then AD/DB = AE/EC
• Theorem 6.2 (converse)If AD/DB = AE/EC, then DE || BC
• Also, if DE || BC, then AD/AB = AE/AC

Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. However, you do not always need to check all six conditions. There are four criteria that require fewer conditions. The AAA (Angle-Angle-Angle) criterion says that if corresponding angles of two triangles are equal, then their corresponding sides are proportional and the triangles are similar. Since the third angle is automatically equal when two angles are equal, this is often stated as the AA criterion. The SSS (Side-Side-Side) criterion says that if the sides of one triangle are proportional to the sides of another, then their corresponding angles are equal and the triangles are similar. The SAS (Side-Angle-Side) criterion says that if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the triangles are similar. For right triangles, the RHS similarity criterion states that if the hypotenuse and one side of one right triangle are proportional to the hypotenuse and one side of another right triangle, then the triangles are similar.

• AAAcorresponding angles equal
• AAtwo angles of one triangle equal to two angles of another
• SSScorresponding sides in the same ratio
• SASone angle equal and the including sides proportional
• RHSright triangles with hypotenuse and one side proportional

Although the chapter summary does not explicitly state the area theorem, the proof of the Basic Proportionality Theorem uses areas of triangles. The chapter shows that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. This result follows from the fact that similar triangles have proportional sides and equal corresponding angles. For example, if two triangles are similar and the ratio of their corresponding sides is k, then the ratio of their areas is k². This is useful in problems involving areas and also in the proof of the Pythagoras Theorem.

• Ratio of areas of similar triangles = (ratio of corresponding sides)²
• If sides are in ratio a:b, areas are in ratio a²:b²

The 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. In this chapter, you see a simple proof of this theorem using similarity. The proof involves drawing an altitude from the right angle to the hypotenuse, which creates two smaller triangles similar to the original triangle and to each other. Using the proportionality of sides, you can derive the theorem. Similarity is also used to solve real-life problems such as finding the height of a tower from the length of its shadow, or finding distances that cannot be measured directly. The chapter includes examples like finding the length of a girl's shadow as she walks away from a lamp-post.

• Pythagoras TheoremIn a right triangle, hypotenuse² = base² + perpendicular²
• Proof using similar triangles formed by the altitude to the hypotenuse
• Applications include indirect measurement of heights and distances

The chapter does not explicitly cover the triangle inequality theorem, but it builds on properties of triangles studied earlier. The focus is on similarity and its consequences. You use the angle sum property of triangles, the fact that corresponding angles of similar triangles are equal, and the property that sides opposite equal angles are equal. These properties are used in examples and exercises. For instance, in Example 3, you prove that a triangle is isosceles by showing two angles are equal. The chapter also uses the property that lines parallel to the same line are parallel to each other, which is helpful in problems involving trapeziums.

• Angle sum propertysum of angles of a triangle is 180°
• Sides opposite equal angles are equal
• Lines parallel to the same line are parallel to each other