Which of following indicates that `Delta ABC` is a right triangle?
(I) The angle of `Delta ABC` are in the ratio 1 : 2 : 3.
(II) One of the angles of `DeltaABC` equals the sum of the other two angles.
(III) `DeltaABC` is similar to the right triangle `DletaDEF`.

Options

A. I only
B. II only
C. III only
D. I,II and III

Answer

To determine which of the given statements indicates that triangle ABC is a right triangle, we will analyze each statement step by step.

### Step 1: Analyze Statement (I)
**Statement (I): The angles of triangle ABC are in the ratio 1 : 2 : 3.**

1. Let the angles of triangle ABC be represented as:
- Angle A = K
- Angle B = 2K
- Angle C = 3K

2. According to the triangle angle sum property, the sum of the angles in a triangle is 180 degrees:
\[ K + 2K + 3K = 180 \]
\[ 6K = 180 \]
\[ K = 30 \text{ degrees} \]

3. Now substituting K back into the angles:
- Angle A = 30 degrees
- Angle B = 60 degrees
- Angle C = 90 degrees

4. Since Angle C is 90 degrees, triangle ABC is a right triangle.

### Step 2: Analyze Statement (II)
**Statement (II): One of the angles of triangle ABC equals the sum of the other two angles.**

1. Let's assume that Angle C is the angle that equals the sum of the other two angles:
\[ \text{Angle C} = \text{Angle A} + \text{Angle B} \]

2. According to the triangle angle sum property:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180 \]

3. Substituting the equation from statement (II) into the angle sum property:
\[ \text{Angle A} + \text{Angle B} + (\text{Angle A} + \text{Angle B}) = 180 \]
\[ 2(\text{Angle A} + \text{Angle B}) = 180 \]
\[ \text{Angle A} + \text{Angle B} = 90 \]

4. Therefore, Angle C must be 90 degrees, confirming that triangle ABC is a right triangle.

### Step 3: Analyze Statement (III)
**Statement (III): Triangle ABC is similar to the right triangle DEF.**

1. If triangle ABC is similar to triangle DEF, then the corresponding angles of both triangles are equal.

2. Since triangle DEF is a right triangle, it has one angle equal to 90 degrees.

3. Therefore, if triangle ABC is similar to triangle DEF, it must also have one angle that is 90 degrees.

4. This means that triangle ABC is a right triangle.

### Conclusion
All three statements indicate that triangle ABC is a right triangle. Therefore, the answer is that all statements (I), (II), and (III) indicate that triangle ABC is a right triangle.

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