Problem 8 - Intermediate Algebra

Question

Fill in the blanks. An _______________ triangle has three sides of equal length and three angles of equal measure.

Step-by-Step Solution

Answer

The word is 'equilateral'.

Step 1: Understanding the Problem

The exercise asks for a specific type of triangle that has equal side lengths and equal angle measures. We need to identify the name of this triangle.

Step 2: Recall Triangle Properties

Recall that a triangle with all three sides of equal length is known as an equilateral triangle. Additionally, in an equilateral triangle, all three interior angles are also equal.

Step 3: Fill in the Blank

Based on our understanding, the word that describes a triangle with three equal sides and three equal angles is 'equilateral'.

Step 4: Conclusion

Fill in the blank with the word 'equilateral'. Thus, the complete sentence is: "An equilateral triangle has three sides of equal length and three angles of equal measure."

Key Concepts

Understanding Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and properties of space. One key aspect of geometry is the study of different types of triangles, like the equilateral triangle. In geometry, an equilateral triangle is a triangle where all sides are of equal length. This means it is a type of regular polygon, a shape where all sides and angles are the same. Geometry helps us understand the structure and properties of different shapes. It is essential in everyday life for designing buildings, creating art, and more. It's all about understanding the space around us.

Triangle Properties at a Glance

Triangles are three-sided polygons, and they have specific properties depending on their classification. An important type of triangle is the equilateral triangle.

• Equilateral Triangle: All sides and all interior angles are equal.
• Isosceles Triangle: Has at least two equal sides and two equal angles.
• Scalene Triangle: All sides and angles are different.

The properties of these triangles define their classification and are fundamental in solving numerous geometric problems. Understanding these properties helps in recognizing and creating precise constructions in mathematics and other fields.

Unveiling Interior Angles of Triangles

The interior angles of a triangle are the angles inside the triangle where two sides meet. The sum of the interior angles in any triangle always equals 180 degrees.In an equilateral triangle, since all angles are equal, each interior angle is \(\frac{180^\circ}{3} = 60^\circ\).Understanding the concept of interior angles is crucial. It helps in proving various theorems and solving problems related to different types of triangles. Recognizing that the angles adjust based on the triangle type allows for proper geometric reasoning.