What if your parents want to redo the bathroom? Below is the tile they would like to place in the shower. The blue and green triangles are all equilateral. What type of polygon is dark blue outlined figure? Can you determine how many degrees are in each of these figures? Can you determine how many degrees are around a point? 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 Foundation: Chapter4EquilateralTrianglesA

James Sousa: Constructing an Equilateral Triangle

James Sousa: Equilateral Triangles Theorem

James Sousa: Using the Properties of Equilateral Triangles

### Guidance

By definition, all sides in an equilateral triangle have exactly the same length.

##### Investigation: Constructing an Equilateral Triangle

Tools Needed: pencil, paper, compass, ruler, protractor

1. Because all the sides of an equilateral triangle are equal, pick a length to be all the sides of the triangle. Measure this length and draw it horizontally on your paper.

2. Put the pointer of your compass on the left endpoint of the line you drew in Step 1. Open the compass to be the same width as this line. Make an arc above the line.

3. Repeat Step 2 on the right endpoint.

4. Connect each endpoint with the arc intersections to make the equilateral triangle.

Use the protractor to measure each angle of your constructed equilateral triangle. What do you notice?

From the Base Angles Theorem, the angles opposite congruent sides in an isosceles triangle are congruent. So, if all three sides of the triangle are congruent, then all of the angles are congruent or \begin{align*}60^\circ\end{align*} each.

**Equilateral Triangles Theorem:** All equilateral triangles are also equiangular. Also, all equiangular triangles are also equilateral.

#### Example A

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

Because this is an equilateral triangle \begin{align*}3x-1=11\end{align*}. Now, we have an equation, 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*}.

Let’s start with \begin{align*}y\end{align*}. Both sides are equal, so set the two expressions equal to each other and solve for \begin{align*}y\end{align*}.

\begin{align*}5y-1 &= 2y+11\\ 3y &= 12\\ y &= 4\end{align*}

\begin{align*}(2x+5)^\circ+(2x+5)^\circ+(3x-5)^\circ &= 180^\circ\\ (7x+5)^\circ &= 180^\circ\\ 7x &= 175^\circ\\ x &= 25^\circ\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.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter4EquilateralTrianglesB

#### Concept Problem Revisited

Let’s focus on one tile. First, these triangles are all equilateral, so this is an equilateral hexagon (6 sided polygon). Second, we now know that every equilateral triangle is also equiangular, so every triangle within this tile has \begin{align*}360^\circ\end{align*} angles. This makes our equilateral hexagon also equiangular, with each angle measuring \begin{align*}120^\circ\end{align*}. Because there are 6 angles, the sum of the angles in a hexagon are \begin{align*}6.120^\circ\end{align*} or \begin{align*}720^\circ\end{align*}. Finally, the point in the center of this tile, has \begin{align*}660^\circ\end{align*} angles around it. That means there are \begin{align*}360^\circ\end{align*} around a point.

### Vocabulary

An ** isosceles triangle** is a triangle that has

**at least**two congruent sides. The congruent sides of the isosceles triangle are called the

**. The other side is called the**

*legs***. The angles between the base and the legs are called**

*base***and are always congruent by the**

*base angles***. The angle made by the two legs is called the**

*Base Angles Theorem***. An**

*vertex angle***is a triangle with three congruent sides.**

*equilateral triangle***means all angles are congruent. All equilateral triangles are equiangular.**

*Equiangular*### 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.

### Interactive Practice

### Practice

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*}.