Problem: In an isosceles triangle, the base angle is twice as large as the vertex angle. Find the angles of the triangle.

Solution:

To solve this problem, we will use the properties of an isosceles triangle and set up an equation to find the angles.

Properties of an Isosceles Triangle:

1. An isosceles triangle has two sides of equal length.
2. The base angles of an isosceles triangle are equal.

Let's assume:
Let the vertex angle be x degrees.
Since the base angle is twice as large as the vertex angle, the base angle can be represented as 2x degrees.

Sum of Angles in a Triangle:

The sum of the three angles in any triangle is always 180 degrees. Using this property, we can set up an equation to find the angles of the isosceles triangle.

The equation becomes:
x + 2x + 2x = 180

Solving the equation:

Combining like terms, we get:
5x = 180

To isolate x, we divide both sides of the equation by 5:
x = 36

Now, we can find the values of the base angles:
Base angle = 2x = 2 * 36 = 72 degrees

Angles of the Isosceles Triangle:

The angles of the isosceles triangle are:
Vertex angle: x = 36 degrees
Base angles: 2x = 2 * 36 = 72 degrees

Therefore, the angles of the isosceles triangle are:
Vertex angle: 36 degrees
Base angles: 72 degrees

Summary:

In an isosceles triangle, the base angle is twice as large as the vertex angle. By setting up an equation using the sum of angles in a triangle, we can find the values of the angles. In this case, the vertex angle was found to be 36 degrees, and the base angles were found to be 72 degrees.