State 'T' for true and 'F' for false and select the correct option.
(i) A triangle whose sides measures 8 cm, 4 cm and 12 cm can be possible.
(ii) It is possible to construct an angle of `57.5^(@)` using ruler and compass only.
(iii) It is possible to construct a `Delta XYZ` in which `angleX=60^(@), angleY = 100^(@) and angleZ = 20^(@)`.

Options

A. `{:(i,ii,iii),(T,F,T):}`
B. `{:(i,ii,iii),(F,F,T):}`
C. `{:(i,ii,iii),(F,T,T):}`
D. `{:(i,ii,iii),(T,T,F):}`

Answer

To solve the question, we need to evaluate three statements regarding triangles and angle constructions. Let's analyze each statement step by step.

### Step-by-Step Solution:

1. **Statement (i)**: A triangle whose sides measure 8 cm, 4 cm, and 12 cm can be possible.
- To determine if these sides can form a triangle, we use the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.
- Check the conditions:
- \(8 + 4 > 12\) (12 is not greater than 12, so this condition fails)
- \(8 + 12 > 4\) (20 > 4, this condition holds)
- \(4 + 12 > 8\) (16 > 8, this condition holds)
- Since one of the conditions fails, it is **not possible** to form a triangle with these sides.
- **Conclusion**: Statement (i) is **False (F)**.

2. **Statement (ii)**: It is possible to construct an angle of \(57.5^\circ\) using a ruler and compass only.
- Angles that can be constructed using a ruler and compass must be in the form of \(n \times 15^\circ\) or \(n \times 30^\circ\) or can be derived from these using bisectors.
- \(57.5^\circ\) cannot be expressed as a combination of \(15^\circ\) or \(30^\circ\) angles, nor can it be constructed by bisecting angles that result in \(57.5^\circ\).
- **Conclusion**: Statement (ii) is **False (F)**.

3. **Statement (iii)**: It is possible to construct a triangle \(XYZ\) in which \(angle X = 60^\circ\), \(angle Y = 100^\circ\), and \(angle Z = 20^\circ\).
- According to the angle sum property of triangles, the sum of the angles in a triangle must equal \(180^\circ\).
- Check the sum:
- \(60^\circ + 100^\circ + 20^\circ = 180^\circ\)
- Since the sum of the angles equals \(180^\circ\), it is possible to construct such a triangle.
- **Conclusion**: Statement (iii) is **True (T)**.

### Final Answers:
- (i) F
- (ii) F
- (iii) T

### Summary of Results:
- The correct options are:
- (i) False
- (ii) False
- (iii) True