Decide whether each statement is always true, sometimes true, or never true.

a. a square is a rectangle
b. a right triangle is an isosceles triangle
c. an acute triangle is an isosceles triangle
d. an isosceles triangle is a right triangle

Where is your answer?

To determine whether each statement is always true, sometimes true, or never true, we can rely on the definitions and properties of shapes and geometric terms.

a. "A square is a rectangle."
This statement is always true. A square is a special type of rectangle where all sides are equal in length. Since a square meets the definition of a rectangle (a quadrilateral with four right angles), it is always true that a square is a rectangle.

b. "A right triangle is an isosceles triangle."
This statement is sometimes true. An isosceles triangle is defined as a triangle with at least two equal sides. A right triangle does not necessarily have any equal sides, as it only requires one right angle. However, a right triangle can be isosceles if it happens to have two equal legs, forming a 45-45-90 degree triangle. So, while a right triangle can be isosceles, it is not always the case.

c. "An acute triangle is an isosceles triangle."
This statement is never true. An acute triangle is defined as a triangle where all three angles are acute (less than 90 degrees). An isosceles triangle, on the other hand, has at least two equal sides. These definitions are distinct, and there is no overlap between the two. Therefore, an acute triangle is not an isosceles triangle.

d. "An isosceles triangle is a right triangle."
This statement is never true. An isosceles triangle is defined as a triangle with at least two equal sides. A right triangle, on the other hand, contains a right angle. While an isosceles triangle can have a right angle, it doesn't have to. In fact, most isosceles triangles do not have a right angle, so it is not accurate to say that an isosceles triangle is always a right triangle.

Remember, when evaluating such statements, it is crucial to consider the definitions and properties of the shapes in question.