In a certain isosceles triangle , the measure of the vertex angle is four times the measure of each of the base angles , what is the measure , in degrees, of the vertex angle ?

Options

A. `30^@`
B. `45^@`
C. `60^@`
D. `120^@`

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the angles
Let the measure of each base angle of the isosceles triangle be \( x \) degrees. Since the triangle is isosceles, both base angles are equal.

### Step 2: Express the vertex angle
According to the problem, the measure of the vertex angle is four times the measure of each of the base angles. Therefore, the vertex angle can be expressed as \( 4x \) degrees.

### Step 3: Set up the equation
The sum of the angles in a triangle is always \( 180 \) degrees. Thus, we can write the equation:
\[ 4x + x + x = 180 \]

### Step 4: Simplify the equation
Combine like terms:
\[ 4x + x + x = 6x \]
So, the equation becomes:
\[ 6x = 180 \]

### Step 5: Solve for \( x \)
To find \( x \), divide both sides of the equation by \( 6 \):
\[ x = \frac{180}{6} = 30 \]

### Step 6: Find the vertex angle
Now that we have the value of \( x \), we can find the measure of the vertex angle:
\[ \text{Vertex angle} = 4x = 4 \times 30 = 120 \text{ degrees} \]

### Conclusion
The measure of the vertex angle is \( 120 \) degrees.

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