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# Equilateral Triangles

## Properties of triangles with three equal sides.

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Equilateral Triangles

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

CK-12 Equilateral Triangles

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 60\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 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*}.

#### Example A

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

Solution: Because this is an equilateral triangle 3x1=11\begin{align*}3x-1=11\end{align*}. Solve for x\begin{align*}x\end{align*}.

3x13xx=11=12=4

#### Example B

Find the values of x\begin{align*}x\end{align*} and y\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 10\begin{align*}10\end{align*}. We have x=10\begin{align*}x=10\end{align*} and y+3=10\begin{align*}y+3=10\end{align*} which means that y=7\begin{align*}y=7\end{align*}.

#### Example C

Two sides of an equilateral triangle are 2x+5\begin{align*}2x+5\end{align*} units and x+13\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 x\begin{align*}x\end{align*}.

2x+5x=x+13=8

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

CK-12 Equilateral Triangles

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### Guided Practice

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

2. Fill in the proof:

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

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

Prove: RST\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. RST\begin{align*}\triangle RST\end{align*} is equiangular 5.

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

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

8y+48yy=60=56=7

2.

Statement Reason
1. RT¯¯¯¯¯ST¯¯¯¯RS¯¯¯¯¯\begin{align*}\overline{RT} \cong \overline{ST} \cong \overline{RS}\end{align*} 1. Given
2. RS\begin{align*}\angle{R} \cong \angle{S}\end{align*} 2. Base Angles Theorem
3. TR\begin{align*}\angle{T} \cong \angle{R}\end{align*} 3. Base Angles Theorem
4. TS\begin{align*}\angle{T} \cong \angle{S}\end{align*} 4. Transitive PoC
5. RST\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.

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

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 4.11.