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?

### Equilateral Triangles

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*}

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

#### Solving for Variables

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

Because this is an equilateral triangle \begin{align*}3x-1=11\end{align*}

\begin{align*}3x-1 &= 11\\
3x &= 12\\
x &= 4\end{align*}

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

Let’s start with \begin{align*}y\end{align*}

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

For \begin{align*}x\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*}

#### Measuring the Sides of an Equilateral Triangle

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

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*}

**Bathroom Tile 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*}

### Examples

#### Example 1

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

The marking show that all angles are congruent. Since all three angles must add up to \begin{align*}180^\circ\end{align*}

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

#### Example 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. |

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 3

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.

### Review

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*}.

### Review (Answers)

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