CBSE Class 10 Mathematics Chapter 6: Triangles NCERT Solutions

The correct words to fill in the blanks are based on the properties of geometric shapes:

1. (i) All circles are similar. Circles have the same shape regardless of their size. Their corresponding angles are equal (if considered as degenerate polygons) and their radii are proportional.
2. (ii) All squares are similar. Squares have four equal angles (90°) and four equal sides. This ensures that any two squares will have equal corresponding angles and proportional corresponding sides.
3. (iii) All equilateral triangles are similar. Equilateral triangles have all angles equal to 60°, which satisfies the condition for similarity (equal corresponding angles).
4. (iv) Two polygons of the same number of sides are similar if:
   1. their corresponding angles are equal, and
   2. their corresponding sides are proportional.

Here are two different examples for each category:

(i) Similar figures:

• Example 1: Two equilateral triangles. One triangle can have sides of length 1 cm, and the other can have sides of length 2 cm. Both triangles have all angles equal to 60°, and their sides are in the ratio 1:2.
• Example 2: Two squares. One square can have a side length of 1 cm, and the other can have a side length of 2 cm. Both have all angles equal to 90°, and their sides are in the ratio 1:2.

(ii) Non-similar figures:

• Example 1: A square with side length 1 cm and a rectangle with side lengths 1 cm and 2 cm. Both are quadrilaterals with equal corresponding angles (90°), but their corresponding sides are not proportional (ratio 1:1 vs 1:2).
• Example 2: A triangle and a parallelogram. These figures do not have the same number of sides and their angles and side properties are fundamentally different, so they cannot be similar.

The given quadrilaterals are a square PQRS with side length 1.5 cm and a rectangle ABCD with sides AB = CD = 3 cm and BC = DA = 1.5 cm.

To determine if they are similar, we need to check two conditions:

1. Corresponding angles must be equal:
   • In square PQRS, all angles (∠P, ∠Q, ∠R, ∠S) are 90°.
   • In rectangle ABCD, all angles (∠A, ∠B, ∠C, ∠D) are 90°.
   So, the corresponding angles are equal (90° = 90°).
2. Corresponding sides must be proportional: Let's check the ratio of corresponding sides. Assume PQ corresponds to AB, QR to BC, RS to CD, and SP to DA.
   • Ratio of PQ to AB = $\frac{1.5}{3}$ = $\frac{1}{2}$
   • Ratio of QR to BC = $\frac{1.5}{1.5}$ = 1
   • Ratio of RS to CD = $\frac{1.5}{3}$ = $\frac{1}{2}$
   • Ratio of SP to DA = $\frac{1.5}{1.5}$ = 1
   The ratios of corresponding sides are not all equal (we have ratios of 1/2 and 1). Therefore, the corresponding sides are not proportional.

Since the condition of proportional corresponding sides is not met, the square PQRS and the rectangle ABCD are not similar.