Which of the following staements is correct?

Options

A. If lengths of any two sides of a triangle are 8 cm and 11 cm, then the length of its third side lies between 3 cm and 19 cm
B. It is possible to construct a unique triangle, if all its three angles are given.
C. An angle of `30^(@)` can' be constructed using compass and ruler.
D. A triangle can be constructed by taking two of its angles as `90^(@)` and `90^(@)`

Answer

To determine which of the statements is correct, we will analyze each option step by step.

### Step 1: Analyze the first statement
**Statement 1:** If the lengths of any two sides of a triangle are 8 cm and 11 cm, then the length of the third side lies between 3 cm and 19 cm.

- According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
- Therefore, if we denote the sides as \( a = 8 \) cm, \( b = 11 \) cm, and \( c \) as the third side, we have:
- \( a + b > c \) → \( 8 + 11 > c \) → \( 19 > c \) → \( c < 19 \)
- \( a + c > b \) → \( 8 + c > 11 \) → \( c > 3 \)
- \( b + c > a \) → \( 11 + c > 8 \) → This condition is always satisfied as \( c \) is positive.

Thus, the third side \( c \) must be greater than 3 cm and less than 19 cm. Therefore, this statement is **correct**.

### Step 2: Analyze the second statement
**Statement 2:** It is possible to construct a unique triangle if all its three angles are given.

- When only the angles of a triangle are given, we can construct a triangle, but it is not unique. Different triangles can have the same angles but different side lengths (similar triangles).
- Therefore, this statement is **false**.

### Step 3: Analyze the third statement
**Statement 3:** We cannot make a 30-degree angle using a ruler and compass.

- It is indeed possible to construct a 30-degree angle using a ruler and compass. One common method is to first create a 60-degree angle and then bisect it to get a 30-degree angle.
- Therefore, this statement is **false**.

### Step 4: Analyze the fourth statement
**Statement 4:** A triangle can be constructed by taking two of its angles, 90 degrees and another angle.

- The sum of angles in a triangle must equal 180 degrees. If one angle is 90 degrees, the other angle must be \( 180 - 90 = 90 \) degrees. This would imply that the triangle is not possible since it would have two right angles.
- Therefore, this statement is **false**.

### Conclusion
After analyzing all the statements, we find that only **Statement 1** is correct.