Fill in the blanks and select the correct option .
(i) The difference between the lengths of any two sides of a triangle is ________than the length of the third side.
(ii) If the three angles of a triangle are in the ratio 1 : 5 : 4, then the measure of the greatest angle is ______.
(iii) In an equilateral triangle, all angles are ________.
(iv) In a right angled triangle, the side opposite to the right angle is called __________.

Options

A. `{:(i,ii,iii,iv),("Smaller",72^(@),"obtuse","perpendicular"):}`
B. `{:(i,ii,iii,iv),("Greater",72^(@),"right","base"):}`
C. `{:(i,ii,iii,iv),("Greater",90^(@),"acute","hypotenuse"):}`
D. `{:(i,ii,iii,iv),("Smaller",90^(@),"acute","hypotenuse"):}`

Answer

Let's solve the question step by step:

### Step 1: Fill in the first blank
**Statement:** The difference between the lengths of any two sides of a triangle is ________ than the length of the third side.

**Solution:** According to the triangle inequality theorem, the difference between the lengths of any two sides of a triangle is **smaller** than the length of the third side.

### Step 2: Fill in the second blank
**Statement:** If the three angles of a triangle are in the ratio 1 : 5 : 4, then the measure of the greatest angle is ______.

**Solution:** Let the angles be represented as \( x \), \( 5x \), and \( 4x \). The sum of the angles in a triangle is \( 180^\circ \).
So, \( x + 5x + 4x = 180 \)
This simplifies to \( 10x = 180 \)
Thus, \( x = 18^\circ \).
The greatest angle is \( 5x = 5 \times 18 = 90^\circ \).

### Step 3: Fill in the third blank
**Statement:** In an equilateral triangle, all angles are ________.

**Solution:** In an equilateral triangle, all angles are **equal**. Since the total sum of angles in a triangle is \( 180^\circ \), each angle in an equilateral triangle is \( \frac{180}{3} = 60^\circ \).

### Step 4: Fill in the fourth blank
**Statement:** In a right-angled triangle, the side opposite to the right angle is called __________.

**Solution:** In a right-angled triangle, the side opposite to the right angle is called the **hypotenuse**.

### Final Answers:
1. The difference between the lengths of any two sides of a triangle is **smaller** than the length of the third side.
2. The measure of the greatest angle is **90 degrees**.
3. In an equilateral triangle, all angles are **60 degrees** (or equal).
4. The side opposite to the right angle is called the **hypotenuse**.