Finding Isosceles Triangle Angles when One Angle is 120°

The problem provides a triangle ABC where ∠A = 120° and the sides AB and AC are equal (AB = AC). We need to find the values of the other two angles, ∠B and ∠C. This is a typical geometry problem involving an isosceles triangle.

Understanding Isosceles Triangles

An isosceles triangle is a triangle that has two sides of equal length. A key property of isosceles triangle angles is that the angles opposite the two equal sides are also equal. In Δ ABC, since AB = AC, the angles opposite these sides must be equal. The angle opposite side AB is ∠C, and the angle opposite side AC is ∠B. Therefore, we have:

\(\angle B = \angle C\)

Using the Angle Sum Property of a Triangle

The sum of the interior angles in any triangle is always 180 degrees. For Δ ABC, this means:

\(\angle A + \angle B + \angle C = 180^\circ\)

Calculating the Unknown Angles

We are given that ∠A = 120°, and we know that ∠B = ∠C. We can substitute these values into the angle sum equation:

\(120^\circ + \angle B + \angle B = 180^\circ\)

Combine the ∠B terms:

\(120^\circ + 2\angle B = 180^\circ\)

Now, subtract 120° from both sides of the equation:

\(2\angle B = 180^\circ - 120^\circ\)

\(2\angle B = 60^\circ\)

Finally, divide by 2 to find the value of ∠B:

\(\angle B = \frac{60^\circ}{2}\)

\(\angle B = 30^\circ\)

Since ∠C = ∠B, the value of ∠C is also 30°.

\(\angle C = 30^\circ\)

Thus, the values of ∠B and ∠C are 30° and 30°, respectively. Understanding these Isosceles Triangle Angles calculations is crucial for solving such problems.

Let's verify the sum of angles:

\(\angle A + \angle B + \angle C = 120^\circ + 30^\circ + 30^\circ = 180^\circ\)

The sum is 180°, which confirms our calculations for the Isosceles Triangle Angles are correct.

Conclusion

For the given isosceles triangle ABC with ∠A = 120° and AB = AC, the other two angles, ∠B and ∠C, are both 30°.