Understanding the Isosceles Triangle Problem

The question asks us to find the measure of the third angle in an isosceles triangle. We are given a specific relationship between the angles: the two equal angles are each twice the measure of the third angle.

An isosceles triangle is a triangle that has two sides of equal length. The angles opposite these two equal sides are also equal. These are often referred to as the base angles, while the third angle is called the vertex angle. In this problem, the two equal angles are the ones that are twice the third angle.

Setting Up the Angles Based on the Relationship

Let's denote the measure of the third angle as $\theta$.

According to the problem, each of the two equal angles is twice the third angle. So, the measure of each of the two equal angles is $2 \times \theta = 2\theta$.

Thus, the three angles of the isosceles triangle are $\theta$, $2\theta$, and $2\theta$.

Using the Angle Sum Property of a Triangle

A fundamental property of all triangles is that the sum of their interior angles is always $180^\circ$.

For our isosceles triangle, the sum of the three angles is:

$\theta + 2\theta + 2\theta = 180^\circ$

Solving the Equation for the Third Angle

Now we need to solve the equation to find the value of $\theta$.

Combine the terms on the left side of the equation:

$(1 + 2 + 2)\theta = 180^\circ$

$5\theta = 180^\circ$

To find $\theta$, divide both sides of the equation by 5:

$\theta = \frac{180^\circ}{5}$

$\theta = 36^\circ$

So, the measure of the third angle is $36^\circ$.

Determining the Measures of All Angles

The third angle is $\theta = 36^\circ$.

Each of the two equal angles is $2\theta = 2 \times 36^\circ = 72^\circ$.

The three angles are $36^\circ$, $72^\circ$, and $72^\circ$.

Verifying the Solution

Let's check if these angles satisfy the conditions given in the problem and the angle sum property:

• Are two angles equal? Yes, $72^\circ = 72^\circ$.
• Is each equal angle twice the third angle? Yes, $72^\circ = 2 \times 36^\circ$.
• Do the angles sum up to $180^\circ$? Yes, $36^\circ + 72^\circ + 72^\circ = 108^\circ + 72^\circ = 180^\circ$.

All conditions are met, confirming our solution is correct.

Final Answer Determination

The measure of the third angle is $36^\circ$. Comparing this to the given options, we find that $36^\circ$ matches one of the choices.

Revision Table: Isosceles Triangle Angles

Additional Information: Types of Triangles

Triangles can be classified based on their sides and angles:

• Based on Sides:
  • Equilateral Triangle: All three sides are equal length. All three angles are equal ($60^\circ$ each).
  • Isosceles Triangle: Two sides are equal length. The angles opposite these sides are equal.
  • Scalene Triangle: All three sides have different lengths. All three angles have different measures.
• Based on Angles:
  • Acute Triangle: All three angles are acute (less than $90^\circ$).
  • Right Triangle: One angle is a right angle ($90^\circ$).
  • Obtuse Triangle: One angle is obtuse (greater than $90^\circ$).

Our problem involved an isosceles triangle. The calculated angles ($36^\circ, 72^\circ, 72^\circ$) show that this specific isosceles triangle is also an acute triangle, as all its angles are less than $90^\circ$.