The vertical angle of an isosceles triangle is `100^(@)`. Find its base angles

Options

A. `100^(@)`
B. `40^(@)`
C. `80^(@)`
D. `90^(@)`

Answer

To find the base angles of an isosceles triangle with a vertical angle of \(100^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Properties of the Triangle**:
An isosceles triangle has two equal sides and two equal angles. The angle opposite the equal sides is called the vertical angle.

2. **Identify the Angles**:
Let's denote the triangle as \(ABC\) where \(A\) is the vertex angle (the vertical angle), and \(B\) and \(C\) are the base angles. Given that the vertical angle \(A = 100^\circ\).

3. **Use the Triangle Sum Property**:
The sum of all angles in a triangle is \(180^\circ\). Therefore, we can write the equation:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \]
Substituting the known value:
\[ 100^\circ + \text{Angle B} + \text{Angle C} = 180^\circ \]

4. **Set Base Angles Equal**:
Since \(B\) and \(C\) are the base angles of the isosceles triangle, we have:
\[ \text{Angle B} = \text{Angle C} = x \]
Thus, we can rewrite the equation as:
\[ 100^\circ + x + x = 180^\circ \]

5. **Simplify the Equation**:
Combine like terms:
\[ 100^\circ + 2x = 180^\circ \]

6. **Isolate \(x\)**:
Subtract \(100^\circ\) from both sides:
\[ 2x = 180^\circ - 100^\circ \]
\[ 2x = 80^\circ \]

7. **Solve for \(x\)**:
Divide both sides by \(2\):
\[ x = \frac{80^\circ}{2} = 40^\circ \]

8. **Conclusion**:
Therefore, the base angles \(B\) and \(C\) are:
\[ \text{Angle B} = 40^\circ \quad \text{and} \quad \text{Angle C} = 40^\circ \]

### Final Answer:
The base angles of the isosceles triangle are \(40^\circ\) each.

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