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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 \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*}.

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*}.

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.

CK-12 Equilateral Triangles

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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.

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:

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.

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

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