Q1. 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, equilateral) (iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are _____________________ and (b) their corresponding sides are ______________________ .(equal, proportional)

Solution:

1. All circles are similar (congruent, similar).
   
   Explanation: Circles are all similar because they have the same shape and their radii differ by a scale factor, but they are not necessarily congruent unless the radii are equal.
2. All squares are similar (similar, congruent).
   
   Explanation: All squares are similar as all have equal angles and their sides are proportional, but they are congruent only when their sides are exactly equal.
3. All triangles are similar if their corresponding angles are equal (isosceles, equilateral).
   
   Explanation: Triangles are similar when their corresponding angles are equal, regardless of side lengths. The types of triangles like isosceles or equilateral relate to side lengths and angles but similarity depends on angle equality.
4. Two polygons of the same number of sides are similar, if (a) their corresponding angles are equal and (b) their corresponding sides are proportional (equal, proportional).
   
   Explanation: For two polygons with the same number of sides to be similar, each pair of corresponding angles must be equal, and the lengths of corresponding sides must be proportional to each other.

Q2. Give two different examples of pair of
(i) similar figures.
(ii) non-similar figures.

Solution:

• Q2. Give two different examples of pair of
  1. similar figures:
     
     • A rectangle measuring 6 cm by 4 cm and another rectangle measuring 9 cm by 6 cm (both have opposite sides equal and all angles \(90^\circ\), with sides proportional by scale factor \(\frac{3}{2}\)).
     • An equilateral triangle with side 5 cm and another equilateral triangle with side 8 cm (all angles \(60^\circ\) in both, sides proportional by scale factor \(\frac{8}{5}\)).
  2. non-similar figures:
     
     • A square with side 5 cm and a rectangle with sides 6 cm by 4 cm (square has all sides equal and angles \(90^\circ\), but rectangle's unequal adjacent sides make proportions differ).
     • A right-angled triangle with sides 3 cm, 4 cm, 5 cm and an equilateral triangle with side 4 cm (right triangle has \(90^\circ\) angle, while equilateral has all \(60^\circ\) angles).

Q3. State whether the following quadrilaterals are similar or not:

Solution:

• The first quadrilateral PQRS is a rhombus (all sides 1.5 cm but angles are not \(90^\circ\)).
• The second quadrilateral ABCD is a square (all sides 3 cm and all angles \(90^\circ\)).

Although their corresponding sides are in the same ratio \(\frac{1.5}{3}=\frac{1}{2}\), their corresponding angles are not equal, so the condition for similarity of polygons (equal angles and proportional sides) is not satisfied.