UpStudy Free Solution:

For isosceles triangles, the following properties apply:

1. The base angles are congruent.

2. The two sides opposite the base angles are congruent.

3. The bisector of the vertex angle is the perpendicular bisector of the base.

So, the correct properties are:

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.

The other properties listed do not apply to all isosceles triangles:

- All three angles are congruent: This property applies only to equilateral triangles, which are a specific type of isosceles triangle.

- All three sides are congruent: This property also applies only to equilateral triangles.

Supplemental Knowledge:

An isosceles triangle is a type of triangle that has specific properties. Here are the key properties of isosceles triangles:

1. Base Angles: The angles opposite the two equal sides (base angles) are congruent.

2. Congruent Sides: An isosceles triangle has at least two sides that are congruent (the sides opposite the base angles).

3. Vertex Angle Bisector: The bisector of the vertex angle (the angle between the two congruent sides) is also the perpendicular bisector of the base.

Other types of triangles for comparison:

- Equilateral Triangle: All three sides and all three angles are congruent.

- Scalene Triangle: No sides or angles are congruent.

Understanding geometric principles such as those governing isosceles triangles can greatly expand your mathematical abilities, and UpStudy makes that happen!