Class 10: Triangles – Exercise 7.1

$\displaystyle \textbf{Question 1: }\text{Fill in the blanks using the correct word given in brackets:}$
$\displaystyle \text{(i) All circles are }\underline{\hspace{1.5cm}}\text{. (congruent, similar)}$
$\displaystyle \text{(ii) All squares are }\underline{\hspace{1.5cm}}\text{. (similar, congruent)}$
$\displaystyle \text{(iii) All }\underline{\hspace{1.5cm}}\text{ triangles are similar. (isosceles, equilateral)}$
$\displaystyle \text{(iv) Two triangles are similar, if their corresponding angles are }\underline{\hspace{1.5cm}}\text{.}$
$\displaystyle \text{(proportional, equal)}$
$\displaystyle \text{(v) Two triangles are similar, if their corresponding sides are }\underline{\hspace{1.5cm}}\text{.}$
$\displaystyle \text{(proportional, equal)}$
$\displaystyle \text{(vi) Two polygons of the same number of sides are similar, if (a) their corresponding}$
$\displaystyle \text{angles are }\underline{\hspace{1.5cm}}\text{ and (b) their corresponding sides are }\underline{\hspace{1.5cm}}\text{. (equal, proportional)}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) similar}$
$\displaystyle \text{All circles have the same shape; hence, all circles are similar.}$
$\displaystyle \text{(ii) similar}$
$\displaystyle \text{All squares have equal corresponding angles and proportional corresponding sides.}$
$\displaystyle \therefore \text{All squares are similar.}$
$\displaystyle \text{(iii) equilateral}$
$\displaystyle \text{All equilateral triangles have each angle equal to }60^\circ\text{; hence, they are similar.}$
$\displaystyle \text{(iv) equal}$
$\displaystyle \text{Two triangles are similar if their corresponding angles are equal.}$
$\displaystyle \text{(v) proportional}$
$\displaystyle \text{Two triangles are similar if their corresponding sides are proportional.}$
$\displaystyle \text{(vi) (a) equal, (b) proportional}$
$\displaystyle \text{Two polygons of the same number of sides are similar if their corresponding angles}$
$\displaystyle \text{are equal and their corresponding sides are proportional.}$
$\displaystyle \\$

$\displaystyle \textbf{Question 2: }\text{Write the truth value (T/F) of each of the following statements:}$
$\displaystyle \text{(i) Any two similar figures are congruent.}$
$\displaystyle \text{(ii) Any two congruent figures are similar.}$
$\displaystyle \text{(iii) Two polygons are similar, if their corresponding sides are proportional.}$
$\displaystyle \text{(iv) Two polygons are similar if their corresponding angles are proportional.}$
$\displaystyle \text{(v) Two triangles are similar if their corresponding sides are proportional.}$
$\displaystyle \text{(vi) Two triangles are similar if their corresponding angles are equal.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) False}$
$\displaystyle \text{Similar figures have the same shape but need not have the same size.}$
$\displaystyle \text{(ii) True}$
$\displaystyle \text{Congruent figures have the same shape and size; hence, they are always similar.}$
$\displaystyle \text{(iii) False}$
$\displaystyle \text{For polygons, proportional corresponding sides alone do not ensure similarity.}$
$\displaystyle \text{Their corresponding angles must also be equal.}$
$\displaystyle \text{(iv) False}$
$\displaystyle \text{For polygons, corresponding angles must be equal, not merely proportional.}$
$\displaystyle \text{Also, their corresponding sides must be proportional.}$
$\displaystyle \text{(v) True}$
$\displaystyle \text{If the corresponding sides of two triangles are proportional, the triangles are similar.}$
$\displaystyle \text{(vi) True}$
$\displaystyle \text{If the corresponding angles of two triangles are equal, the triangles are similar.}$
$\displaystyle \\$

$\displaystyle \textbf{Question 3: }\text{State whether the following rectangles are similar or not. Justify your answer.}$ $\displaystyle \text{Answer:}$
$\displaystyle \text{The dimensions of the first rectangle are }4\text{ cm and }2.6\text{ cm.}$
$\displaystyle \text{The dimensions of the second rectangle are }6\text{ cm and }3.9\text{ cm.}$
$\displaystyle \frac{4}{6}=\frac{2}{3}$
$\displaystyle \frac{2.6}{3.9}=\frac{26}{39}=\frac{2}{3}$
$\displaystyle \therefore \frac{4}{6}=\frac{2.6}{3.9}$
$\displaystyle \text{Thus, the corresponding sides of the two rectangles are proportional.}$
$\displaystyle \text{Also, all the corresponding angles are equal to }90^\circ\text{.}$
$\displaystyle \therefore \text{The given rectangles are similar.}$
$\displaystyle \\$