# 4.5: Equilateral Triangles

- Page ID
- 2174

Properties of triangles with three equal sides.

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

If \(\overline{AB}\cong \overline{BC}\cong \overline{AC}\), then \(\angle A\cong \angle B\cong \angle C\). Conversely, if \(\angle A\cong \angle B\cong \angle C\), then \(\overline{AB}\cong \overline{BC}\cong \overline{AC}\).

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 \(y\)?

Example \(\PageIndex{1}\)

Fill in the proof:

__Given__: Equilateral \(\Delta RST\) with

\(\overline{RT}\cong \overline{ST}\cong \overline{RS}\)

__Prove__: \(\Delta RST\) is equiangular

**Solution**

Statement |
Reason |
---|---|

1. | 1. Given |

2. | 2. Base Angles Theorem |

3. | 3. Base Angles Theorem |

4. | 4. Transitive \(PoC\) |

5. \(\Delta RST\) is equiangular | 5. |

Statement |
Reason |
---|---|

1. \(RT\overline{AB}\cong ST\overline{AB}\cong RS\overline{AB}\) | 1. Given |

2. \(\angle R\cong \angle S\) | 2. Base Angles Theorem |

3.\(\angle T\cong \angle R\) | 3. Base Angles Theorem |

4. \(\angle T\cong \angle S\) | 4. Transitive \(PoC\) |

5. \(\Delta RST\) is equiangular | 5. Definition of equiangular. |

Example \(\PageIndex{2}\)

True or false: All equilateral triangles are isosceles triangles.

**Solution**

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 \(\PageIndex{3}\)

Find the value of \(x\).

**Solution**

Because this is an equilateral triangle \(3x−1=11\). Solve for \(x\).

\(\bgin{align*} 3x−1&=11 \\3x&=12 \\ x&=4 \end{align*}\)

Example \(\PageIndex{4}\)

Find the values of \(x\) and \(y\).

**Solution**

The markings show that this is an equilateral triangle since all sides are congruent. This means all sides must equal \(10\). We have \(x=10\) and \(y+3=10\) which means that \(y=7\).

Example \(\PageIndex{5}\)

Two sides of an equilateral triangle are \(2x+5\) units and \(x+13\) units. How long is each side of this triangle?

**Solution**

The two given sides must be equal because this is an equilateral triangle. Write and solve the equation for \(x\).

\(\egin{align*}2x+5 &=x+13 \\ x&=8 \end{align*}\)

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

### Review

The following triangles are equilateral triangles. Solve for the unknown variables.

- Find the measures of \(x\) and \(y\).

## Review (Answers)

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

## Additional Resources

Interactive Element

Video: Equilateral Triangles Principles - Basic

Activities: Equilateral Triangles Discussion Questions

Study Aids: Equilateral Triangles Discussion Questions

Practice: Equilateral Triangles

Real World: Equilateral Triangles