If one angle of a triangle is equal to the sum of the other two angles, then the triangle is (a) an isosceles triangle           (b)  an obtuse triangle (c) an equilateral triangle        (d)  a right triangle

Options

A. (a) an isosceles
B. (b)  an obtuse triangle
C. (c) an equilateral
triangle
D. (d)  a right triangle

Answer

To solve the problem, we need to analyze the given condition about the angles of a triangle.

### Step-by-Step Solution:

1. **Understanding the Triangle's Angles**:
We know that in any triangle, the sum of all three angles is always equal to 180 degrees.
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \quad \text{(Equation 1)} \]

2. **Applying the Given Condition**:
The problem states that one angle of the triangle is equal to the sum of the other two angles. Let's assume that Angle A is equal to the sum of Angle B and Angle C.
\[ \text{Angle A} = \text{Angle B} + \text{Angle C} \quad \text{(Equation 2)} \]

3. **Substituting Equation 2 into Equation 1**:
We can substitute Equation 2 into Equation 1. Replacing Angle A in Equation 1 gives us:
\[ (\text{Angle B} + \text{Angle C}) + \text{Angle B} + \text{Angle C} = 180^\circ \]
This simplifies to:
\[ 2 \cdot \text{Angle B} + 2 \cdot \text{Angle C} = 180^\circ \]

4. **Simplifying the Equation**:
Dividing the entire equation by 2, we get:
\[ \text{Angle B} + \text{Angle C} = 90^\circ \]

5. **Finding Angle A**:
Since we know from Equation 2 that Angle A is equal to the sum of Angle B and Angle C, we can substitute:
\[ \text{Angle A} = 90^\circ \]

6. **Conclusion**:
If one angle of the triangle (Angle A) is 90 degrees, then the triangle is classified as a right triangle. Therefore, the correct answer is:
\[ \text{(d) a right triangle} \]