Question

which statement is always true, based on the venn diagram?
if a triangle is equilateral, then the triangle must also be isosceles.
if a triangle is isosceles, then the triangle must also be equilateral.
if a triangle is isosceles, then the triangle will never also be equilateral.
if a triangle is equilateral, then the triangle will not always be isosceles.

Explanation:

Step1: Analyze the Venn diagram

The Venn diagram shows that the set of equilateral triangles is completely inside the set of isosceles triangles.

Step2: Evaluate each statement

• For the statement "If a triangle is equilateral, then the triangle must also be isosceles": Since all equilateral triangles (\(3\) equal sides) satisfy the condition of isosceles triangles (\(at least 2\) equal sides), this is true.
• For the statement "If a triangle is isosceles, then the triangle must also be equilateral": There are isosceles triangles (e.g., with side lengths \(2,2,3\)) that are not equilateral. So this is false.
• For the statement "If a triangle is isosceles, then the triangle will never also be equilateral": Since equilateral triangles are a subset of isosceles triangles, this is false.
• For the statement "If a triangle is equilateral, then the triangle will not always be isosceles": As equilateral triangles are always isosceles (by definition), this is false.

Answer:

If a triangle is equilateral, then the triangle must also be isosceles.