Question

which properties belong to all isosceles triangles? check all that apply. the base angles are congruent. all three angles are congruent. the two sides opposite the base angles are congruent. all three sides are congruent. the bisector of the vertex angle is the perpendicular bisector of the base.

Explanation:

1. The base angles are congruent: By definition of an isosceles triangle (a triangle with at least two congruent sides), the angles opposite those congruent sides (base angles) are congruent. This is a fundamental property of isosceles triangles.
2. All three angles are congruent: This describes an equilateral triangle, which is a special case of an isosceles triangle, but not all isosceles triangles are equilateral. So this property does not belong to all isosceles triangles.
3. The two sides opposite the base angles are congruent: In an isosceles triangle, the sides opposite the base angles are the legs (the congruent sides). By definition, these sides are congruent.
4. All three sides are congruent: Again, this is true for equilateral triangles only, not all isosceles triangles (which only need at least two congruent sides).
5. The bisector of the vertex angle is the perpendicular bisector of the base: In an isosceles triangle, the angle bisector of the vertex angle also acts as the median and altitude to the base, meaning it is the perpendicular bisector of the base. This is a result of the congruence of the two triangles formed by the angle bisector (proven by SAS congruence).

Answer:

• The base angles are congruent.
• The two sides opposite the base angles are congruent.
• The bisector of the vertex angle is the perpendicular bisector of the base.