In an isosceles triangle ABC with a 130-degree angle at vertex B, what must be true about the triangle?

In an isosceles triangle, at least two sides are of equal length, and the angles opposite those sides are equal. Given that angle B measures 130 degrees, we can deduce important information about angles A and C.

Since the sum of all angles in any triangle is always 180 degrees, we can first calculate the sum of the angles A and C:

Angle A + Angle C + Angle B = 180 degrees

Substituting the value of angle B:

Angle A + Angle C + 130 degrees = 180 degrees

This simplifies to:

Angle A + Angle C = 50 degrees

Now, because triangle ABC is isosceles, angles A and C must be equal (as they are the angles opposite the equal sides). Therefore, we can set:

Angle A = Angle C

Let’s denote the measure of each angle as x:

x + x = 50 degrees

Which simplifies to:

2x = 50 degrees

Dividing by 2 gives us:

x = 25 degrees

Thus, both angle A and angle C must measure 25 degrees each.

In summary, the statement that must be true in this scenario is:

In triangle ABC, Angle A = 25 degrees and Angle C = 25 degrees.

This conclusion follows from the properties of isosceles triangles and the triangle angle sum theorem.