The opening exercise of Chapter-4 RD Sharma Class 10 textbook introduces you with figures having the same shape but not necessarily the same size. Such shapes are said to have similar figures. Having understood the difference between congruent and similar shapes, you shall now proceed with studying the theorems.
Fill in the blanks using the correct word given in brackets :
(i) All circles are……..(congruent, similar).
(ii) All squares are………(similar, congruent).
(iii) All……triangles are similar (isosceles, equilaterals).
(iv) Two triangles are similar, if heir corresponding angles are………(proportional, equal)
(v) Two triangles are similar, if their corresponding sides are………(proportional, equal)
(vi) Two polygons of the same number of sides are similar, if (a) their corresponding angles ae and (b) heir corresponding sides are………(equal, proportional)
Write the truth value (T/F) of each of the following statements:
(i) Any two similar figures are congruent.
(ii) Any two congruent figures are similar.
(iii) Two polygons are similar, if their corresponding sides are proportional.
(iv) Two polygons are similar if their corresponding angles are proportional.
(v) Two triangles are similar if their corresponding sides are proportional.
(vi) Two triangles are similar if their corresponding angles are proportional.
|Chapter 1 - Real Numbers|
|Chapter 2 - Polynomials|
|Chapter 3 - Pair of Linear Equations in Two Variables|
|Chapter 4 - Triangles|
|Chapter 5 - Trigonometric Ratios|
|Chapter 6 - Trigonometric Identities|
|Chapter 7 - Statistics|
|Chapter 8 - Quadratic Equations|
|Chapter 9 - Arithmetic Progressions|
|Chapter 10 - Circles|
|Chapter 11 - Constructions|
|Chapter 12 - Some Application of Trigonometry|
|Chapter 13 - Probability|
|Chapter 14 - Co-ordinate Geometry|
|Chapter 15 - Areas Related to Circles|
|Chapter 16 - Surface Areas and Volumes|
Theorem 1: Line drawn parallel to one side of a triangle to intersect the other two sides in distinct points results in the division of the other two sides in the same ratio.
Theorem 2: If a line is drawn intersecting any two sides of a triangle in the same ratio then the line will become parallel to the third side.
Theorem 3: If there are two triangles with equal corresponding angles, then their corresponding sides will have the same ratio. These two triangles will be similar to each other.
Theorem 4: If there are two triangles with sides having similar ratio then their corresponding angles will also be considered as equal. This way both the triangles will be similar.
Theorem 5: If in two triangles, one angle of the first triangle is equal to one angle of the second triangle, then the two triangles are said to be similar.
Theorem 6: The ratio of the areas of two triangles is equal to the square of the ratio of corresponding sides.
Theorem 7: If you draw a perpendicular line from the vertex to the hypotenuse of a right-angled triangle, then both the triangles separated by the perpendicular are similar to each other and the whole triangle.
Theorem 8: In a right-angled triangle, the square of the longest side is equal to the sum of the squares of the remaining two sides.
Theorem 9: If in a triangle, the square of the longest side is equal to the sum of the squares of the remaining two sides, then the angle opposite the longest side is a right angle.
This chapter is wholly based on the application of different theorems of triangles that are used in accordance with the conditions. With a total of 9 theorems, you must first develop a better understanding about them from NCERT textbook before you proceed with the RD Sharma Class 10 questions.