Question

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

Explanation:

Step1: Analyze the first option

An isosceles triangle has two equal sides and two equal angles. An equilateral triangle (a special case) has all three angles congruent, but not all isosceles triangles do. So "All three angles are congruent" is incorrect.

Step2: Analyze the second option

By the property of isosceles triangles, the bisector of the vertex angle is the perpendicular bisector of the base. This is a valid property.

Step3: Analyze the third option

An equilateral triangle has all three sides congruent, but not all isosceles triangles do. So "All three sides are congruent" is incorrect.

Step4: Analyze the fourth option

By the base - angle theorem of isosceles triangles, the base angles are congruent. This is a valid property.

Step5: Analyze the fifth option

If the base angles are congruent, then by the converse of the base - angle theorem (if two angles of a triangle are congruent, then the sides opposite those angles are congruent), the two sides opposite the base angles are congruent. This is a valid property.

Answer:

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