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*? After completing this Concept, you'll be able to apply important properties about equilateral triangles to help you solve problems like this one.

### Watch This

Watch this video first.

James Sousa: Constructing an Equilateral Triangle

Now watch this video.

James Sousa: Equilateral Triangles Theorem

Finally, watch this video.

James Sousa: Using the Properties of Equilateral Triangles

### Guidance

All sides in an **equilateral triangle** have the same length. One important property of equilateral triangles is that all of their angles are congruent (and thus \begin{align*}60^\circ\end{align*} each). This is called the **Equilateral Triangle Theorem** and can be derived from the **Base Angles 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*}, then \begin{align*}\angle A \cong \angle B \cong \angle C\end{align*}. Conversely, if \begin{align*}\angle A \cong \angle B \cong \angle C\end{align*}, then \begin{align*}\overline{AB} \cong \overline{BC} \cong \overline{AC}\end{align*}.

#### Example A

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

**Solution:** 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 B

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 C

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.

-->

### Guided Practice

1. Find the measure of \begin{align*}y\end{align*}.

2. 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. |

3. True or false: All equilateral triangles are isosceles triangles.

**Answers:**

1. The markings show that all angles are congruent. Since all three angles must add up to \begin{align*}180^\circ\end{align*} this means that each angle must equal \begin{align*}60^\circ\end{align*}. Write and solve an equation:

\begin{align*}8y +4 & = 60\\ 8y & = 56 \\ y & =7\end{align*}

2.

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. |

3. 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.

### Explore More

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

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