Two angles in a triangle are equal and their sum is equal to the third angle in the triangle. What are the measures of each of the three interior angles?

Let the measures of the two equal angles in the triangle be \( x \). Since the sum of the two equal angles is equal to the third angle, the third angle will be \( 2x \).

In any triangle, the sum of the interior angles is always \( 180^\circ \). Therefore, we can write the equation:

\[
x + x + 2x = 180^\circ
\]

Simplifying this equation:

\[
4x = 180^\circ
\]

Now, solving for \( x \):

\[
x = \frac{180^\circ}{4} = 45^\circ
\]

Thus, the two equal angles are \( 45^\circ \) each, and the third angle is:

\[
2x = 2 \times 45^\circ = 90^\circ
\]

Therefore, the measures of the three interior angles are:

- Two angles of \( 45^\circ \) each
- One angle of \( 90^\circ \)

This means the triangle is a right-angled triangle.

Would you like more details or have any other questions?

Here are 5 related questions:
1. How do you prove that the sum of the interior angles of a triangle is always 180 degrees?
2. Can a triangle have more than one right angle? Why or why not?
3. What are the properties of an isosceles right triangle?
4. If two angles of a triangle are equal, is the triangle always isosceles?
5. How can you classify a triangle based on its angles and sides?

**Tip:** In any triangle, the largest angle is always opposite the longest side.

Learn how to find the interior angles of a triangle where two angles are equal, and their sum is equal to the third angle. This geometry problem uses the Angle Sum Theorem to determine that the measures of the angles are 45°, 45°, and 90°, making it a right-angled triangle. Suitable for students in Grades 6-8.

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