Question

Which of these statements is true? (A) An isosceles triangle is always equilateral. \(\quad\) OR (B) An equilateral triangle is always isosceles.

Short answer

(B) An equilateral triangle is always isosceles.

Step-by-step solution

Understand the Terms

Before deciding which statement is true, ensure you know the definitions of the terms. An isosceles triangle has at least two equal sides. An equilateral triangle has all three sides equal and, hence, is a specific type of isosceles triangle since it fulfills the condition of having at least two equal sides.

Analyze Statement A

Statement (A) claims that an isosceles triangle is always equilateral. However, for a triangle to be equilateral, all three of its sides must be equal. An isosceles triangle may only have two equal sides, not necessarily three, so it is not necessarily equilateral. Hence, statement (A) is false.

Analyze Statement B

Statement (B) claims that an equilateral triangle is always isosceles. Since an equilateral triangle has all equal sides, it automatically satisfies the condition of an isosceles triangle (having at least two equal sides). Therefore, an equilateral triangle is indeed always isosceles. Hence, statement (B) is true.

Key concepts

Isosceles Triangle

An isosceles triangle is a fascinating geometric shape, and it's essential to understand its defining properties. One of the key characteristics of an isosceles triangle is that it has at least two sides of equal length. These equal sides are often referred to as the 'legs' of the triangle, while the third, unequal side, if present, is known as the 'base.'
Identifying an isosceles triangle becomes easy once you remember that it possesses:

• Two equal sides.
• Two equal angles across from the equal sides.

The equal angles are key when solving for unknown angles or sides in problems involving isosceles triangles. Since the two angles opposite the equal sides are themselves equal, this property is a common reason why isosceles triangles are widely used in geometric proofs and theorems.
Although every equilateral triangle is isosceles, not every isosceles triangle is equilateral. This difference is critical for distinguishing between types of triangles in geometry.

Equilateral Triangle

An equilateral triangle is one of the simplest and most symmetrical types of triangles you'll encounter in geometry. Each equilateral triangle has the following clear-cut features:

• All three sides of equal length.
• All internal angles measuring exactly 60 degrees.

This unique configuration makes equilateral triangles regular polygons, meaning they are highly symmetric and often serve as the building blocks for more complex geometric figures.
Importantly, because all sides are equal, any of the properties of isosceles triangles also apply. This means an equilateral triangle can always be considered as a special case of an isosceles triangle. It fulfills the isosceles condition by having more than just two sides equal, which naturally includes it within this classification.

Geometry Definitions

Understanding basic geometry definitions is crucial for navigating more complex geometric principles and problems. Here are some foundational terms that will help clarify concepts about triangles:

• Triangle: A polygon with three edges and three vertices. It's the simplest form of polygon that exists.
• Sides: The lines or line segments forming a triangle.
• Angles: The space (usually measured in degrees) between two intersecting lines or surfaces at or close to the point where they meet.
• Vertex: The point where two sides of a triangle meet. A triangle will have three vertices, one for each pair of sides that meet.
• Base: In triangles, this can refer to any side upon which the triangle is presumed to stand, especially in right-angled triangles or isosceles triangles.

Recognizing these definitions supports a deep understanding of how triangles function and interact in geometric problems. This knowledge forms the basis for solving the types of exercises often found in textbooks, just like the one you've worked on here.