<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Skip Navigation
Our Terms of Use (click here to view) and Privacy Policy (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use and Privacy Policy.

Equilateral Triangles

Properties of triangles with three equal sides.

Atoms Practice
Estimated6 minsto complete
Practice Equilateral Triangles
Estimated6 minsto complete
Practice Now
Equilateral Triangles

Equilateral Triangle Theorem

Equilateral Triangle Theorem: All equilateral triangles are also equiangular. Furthermore, all equiangular triangles are also equilateral.

If \begin{align*}\overline{AB} \cong \overline{BC} \cong \overline{AC}\end{align*}AB¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯AC¯¯¯¯¯¯¯¯, then \begin{align*}\angle A \cong \angle B \cong \angle C\end{align*}ABC. Conversely, if \begin{align*}\angle A \cong \angle B \cong \angle C\end{align*}ABC, then \begin{align*}\overline{AB} \cong \overline{BC} \cong \overline{AC}\end{align*}AB¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯AC¯¯¯¯¯¯¯¯.

What if you were presented with an equilateral triangle and told that its sides measure x, y, and 8? What could you conclude about x and y?




Example 1

Fill in the proof:

Given: Equilateral \begin{align*}\triangle RST\end{align*} with

\begin{align*}\overline{RT} \cong \overline{ST} \cong \overline{RS}\end{align*}

Prove: \begin{align*}\triangle RST\end{align*} is equiangular

Statement Reason
1. 1. Given
2. 2. Base Angles Theorem
3. 3. Base Angles Theorem
4. 4. Transitive PoC
5. \begin{align*}\triangle RST\end{align*} is equiangular 5.
Statement Reason
1. \begin{align*}\overline{RT} \cong \overline{ST} \cong \overline{RS}\end{align*} 1. Given
2. \begin{align*}\angle{R} \cong \angle{S}\end{align*} 2. Base Angles Theorem
3. \begin{align*}\angle{T} \cong \angle{R}\end{align*} 3. Base Angles Theorem
4. \begin{align*}\angle{T} \cong \angle{S}\end{align*} 4. Transitive PoC
5. \begin{align*}\triangle RST\end{align*} is equiangular 5. Definition of equiangular.

Example 2

True or false: All equilateral triangles are isosceles triangles.

This statement is true. The definition of an isosceles triangle is a triangle with at least two congruent sides. Since all equilateral triangles have three congruent sides, they fit the definition of an isosceles triangle.

Example 3

Find the value of \begin{align*}x\end{align*}.

Because this is an equilateral triangle \begin{align*}3x-1=11\end{align*}. Solve for \begin{align*}x\end{align*}.

\begin{align*}3x-1 & = 11\\ 3x & = 12\\ x & = 4\end{align*}

Example 4

Find the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

The markings show that this is an equilateral triangle since all sides are congruent. This means all sides must equal \begin{align*}10\end{align*}. We have \begin{align*}x=10\end{align*} and \begin{align*}y+3=10\end{align*} which means that \begin{align*}y=7\end{align*}.

Example 5

Two sides of an equilateral triangle are \begin{align*}2x+5\end{align*} units and \begin{align*}x+13\end{align*} units. How long is each side of this triangle?

The two given sides must be equal because this is an equilateral triangle. Write and solve the equation for \begin{align*}x\end{align*}.

\begin{align*} 2x+5 &= x+13 \\ x &= 8\end{align*}

To figure out how long each side is, plug in \begin{align*}8\end{align*} for \begin{align*}x\end{align*} in either of the original expressions. \begin{align*}2(8)+5=21\end{align*}. Each side is \begin{align*}21\end{align*} units.


The following triangles are equilateral triangles. Solve for the unknown variables.

  1. Find the measures of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.11. 






My Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / notes
Show More

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Equilateral Triangles.
Please wait...
Please wait...