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 and similarity:

1. (i) All circles are similar. Circles have the same shape regardless of their radius. While they can have different sizes, their fundamental form is identical, making them similar.
2. (ii) All squares are similar. All squares have four equal sides and four right angles. This means they all have the same shape, and their sides are always proportional, hence they are similar.
3. (iii) All equilateral triangles are similar. An equilateral triangle has all sides equal and all angles equal to 60 degrees. Therefore, any two equilateral triangles will have the same shape, making them similar.
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:

• Two equilateral triangles: One with a side length of 1 cm and another with a side length of 2 cm. Both have angles of 60 degrees, and their sides are in the ratio 1:2.
• Two squares: One with a side length of 1 cm and another with a side length of 2 cm. Both have four right angles, and their sides are in the ratio 1:2.

(ii) Non-similar figures:

• A trapezium and a square. They do not have the same number of sides or the same shape.
• A triangle and a parallelogram. They have a different number of sides and different shapes.

The given quadrilaterals are a square PQRS with side length 1.5 cm and a rectangle ABCD with sides 1.5 cm and 3 cm. Let's analyze if they are similar.

For two polygons to be similar, two conditions must be met:

1. Their corresponding angles must be equal.
2. Their corresponding sides must be proportional.

Let's check the angles:

• In the square PQRS, all angles (∠P, ∠Q, ∠R, ∠S) are 90 degrees.
• In the rectangle ABCD, all angles (∠A, ∠B, ∠C, ∠D) are 90 degrees.

So, the corresponding angles are equal (all are 90 degrees).

Now let's check the sides. We need to compare the ratio of corresponding sides. Let's assume PQ corresponds to AB, QR to BC, RS to CD, and SP to DA.

• Ratio of sides: $\frac{PQ}{AB} = \frac{1.5}{3} = \frac{1}{2}$
• $\frac{QR}{BC} = \frac{1.5}{1.5} = 1$
• $\frac{RS}{CD} = \frac{1.5}{3} = \frac{1}{2}$
• $\frac{SP}{DA} = \frac{1.5}{1.5} = 1$

The ratios of the corresponding sides are not all equal ($\frac{1}{2}$ and 1). Since the corresponding sides are not proportional, the square PQRS and the rectangle ABCD are not similar.