Question

isosceles triangles
what is always true about the angles of an isosceles triangle?
the vertex angle is obtuse.
at least two of the angles are congruent.
all three angles are congruent.
the vertex angle is a right angle.

Explanation:

An isosceles triangle has at least two congruent sides (the legs). By the Isosceles Triangle Theorem, the angles opposite the congruent sides (the base angles) are congruent. So, an isosceles triangle always has at least two congruent angles.

• The vertex angle can be acute (e.g., in an acute - angled isosceles triangle with angles \(70^{\circ},70^{\circ},40^{\circ}\)), obtuse (e.g., \(30^{\circ},30^{\circ},120^{\circ}\)), or right - angled (in a right - isosceles triangle with angles \(45^{\circ},45^{\circ},90^{\circ}\)). So, the statements about the vertex angle being always obtuse or always a right angle are false.
• A triangle with all three angles congruent is an equilateral triangle. While an equilateral triangle is a special case of an isosceles triangle, not all isosceles triangles are equilateral. So, the statement “All three angles are congruent” is not always true for isosceles triangles.

Answer:

At least two of the angles are congruent.