If one of the base angles of an isosceles triangle measures `30^(@)`, find the measures of the remaining two angles ?

Options

A. `40^(@),110^(@)`.
B. `30^(@),120^(@)`.
C. `80^(@),40^(@)`.
D. `30^(@),100^(@)`.

Answer

To solve the problem of finding the measures of the remaining two angles in an isosceles triangle where one of the base angles measures \(30^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- We know that one of the base angles of the isosceles triangle is \(30^\circ\).

2. **Understand the Properties of Isosceles Triangle:**
- In an isosceles triangle, the two base angles are equal. Therefore, if one base angle is \(30^\circ\), the other base angle must also be \(30^\circ\).

3. **Label the Angles:**
- Let the triangle be named \(ABC\).
- Let angle \(B\) be \(30^\circ\) (the given angle).
- Since angle \(A\) (the other base angle) is also equal to angle \(B\), we have:
\[ \text{Angle } A = 30^\circ \]

4. **Use the Angle Sum Property of a Triangle:**
- The sum of all angles in a triangle is \(180^\circ\). Therefore, we can write:
\[ \text{Angle } A + \text{Angle } B + \text{Angle } C = 180^\circ \]
- Substituting the known values:
\[ 30^\circ + 30^\circ + \text{Angle } C = 180^\circ \]

5. **Calculate the Remaining Angle:**
- Simplifying the equation:
\[ 60^\circ + \text{Angle } C = 180^\circ \]
- To find angle \(C\), subtract \(60^\circ\) from \(180^\circ\):
\[ \text{Angle } C = 180^\circ - 60^\circ = 120^\circ \]

6. **Conclusion:**
- Therefore, the measures of the remaining two angles in the triangle are:
- Angle \(A = 30^\circ\)
- Angle \(C = 120^\circ\)

### Final Answer:
The measures of the remaining two angles are \(30^\circ\) and \(120^\circ\).