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Chapter 8: Problem 6

State the characteristics of each triangle. Equilateral triangle

Short Answer

Expert verified
An equilateral triangle has all sides equal, all angles 60 degrees, three lines of symmetry, and congruent sides and angles.

Step by step solution

01

Definition

An equilateral triangle is a type of triangle where all three sides are of equal length.
02

Internal Angles

In an equilateral triangle, each interior angle is always 60 degrees because the angles are equally divided.
03

Symmetry

This type of triangle has three lines of symmetry, meaning it can be folded along these lines and produce two congruent halves.
04

Congruence

All three sides and all three angles are congruent (equal) in an equilateral triangle.
05

Altitudes, Medians, and Angle Bisectors

In an equilateral triangle, the altitude (height), median (line segment from vertex to midpoint of opposite side), and angle bisector (line segment dividing the angle) are all the same for each vertex.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equilateral Triangle
An equilateral triangle is a special type of triangle where all three sides are of equal length. This equality gives the equilateral triangle unique properties.
Because each side is the same, all its properties and characteristics are consistent and predictable, making it a key concept in geometry.
It's important to recognize an equilateral triangle by identifying that all sides are the same.
Internal Angles
In an equilateral triangle, each interior angle is exactly 60 degrees. This is because the sum of the internal angles in any triangle is always 180 degrees.
With three equal angles, you divide 180 degrees by 3, resulting in 60 degrees for each angle.
Thus, all internal angles in an equilateral triangle are congruent and equal to 60 degrees.
Symmetry in Triangles
Equilateral triangles exhibit a high degree of symmetry.
Specifically, they have three lines of symmetry.
These lines of symmetry are drawn from each vertex (corner) to the midpoint of the opposite side.
Each line of symmetry creates two mirror-image halves, showing the triangle's balanced nature and geometric harmony.
Congruence in Triangles
In an equilateral triangle, congruence is a major characteristic.
All three sides and all three angles are congruent, meaning they are identical in measure.
Congruence implies that each side matches with every other side in length, and each angle matches with every other angle in measure.
This uniformity is what makes equilateral triangles particularly special.
Altitudes and Medians
In an equilateral triangle, the altitude, median, and angle bisector coincide at every vertex.
The altitude is the perpendicular line from a vertex to the opposite side, the median is the line from a vertex to the midpoint of the opposite side, and the angle bisector divides the angle into two equal parts.
In equilateral triangles, these three lines are the same for each vertex, which helps in dividing the triangle into smaller, equal sections, further showcasing its symmetry and congruence.

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