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# Equilateral Triangles

## Properties of triangles with three equal sides.

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Equilateral Triangles

### Equilateral Triangle Theorem

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

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

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?

### Examples

#### 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.

### Review

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*}.

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

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