### Isosceles Triangles

An isosceles triangle is a triangle that has ** at least** two congruent sides. The congruent sides of the isosceles triangle are called the

**legs**. The other side is called the

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

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

**vertex angle**. One of the important properties of isosceles triangles is that their base angles are always congruent. This is called the

**Base Angles Theorem.**

For \begin{align*}\triangle DEF\end{align*}, if \begin{align*}\overline{DE} \cong \overline{EF}\end{align*}, then \begin{align*}\angle D \cong \angle F\end{align*}.

Another important property of isosceles triangles is that the angle bisector of the vertex angle is also the perpendicular bisector of the base. This is called the **Isosceles Triangle Theorem**. (*Note this is ONLY true of the vertex angle.*) The converses of the Base Angles Theorem and the Isosceles Triangle Theorem are both true as well.

**Base Angles Theorem Converse:** If two angles in a triangle are congruent, then the sides opposite those angles are also congruent. So for \begin{align*}\triangle DEF\end{align*}, if \begin{align*}\angle D \cong \angle F\end{align*}, then \begin{align*}\overline{DE} \cong \overline{EF}\end{align*}.

**Isosceles Triangle Theorem Converse:** The perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex angle. So for isosceles \begin{align*}\triangle DEF\end{align*}, if \begin{align*}\overline{EG} \perp \overline{DF}\end{align*} and \begin{align*}\overline{DG} \cong \overline{GF}\end{align*}, then \begin{align*}\angle DEG \cong \angle FEG\end{align*}.

What if you were presented with an isosceles triangle and told that its base angles measure \begin{align*}x^\circ\end{align*} and \begin{align*}y^\circ\end{align*}? What could you conclude about *x* and *y*?

### Examples

#### Example 1

Find the value of \begin{align*}x\end{align*} and the measure of each angle.

The two angles are equal, so set them equal to each other and solve for \begin{align*}x\end{align*}.

\begin{align*}(4x+12)^\circ & = (5x-3)^\circ\\ 15 = x\end{align*}

Substitute \begin{align*}x = 15\end{align*}; the base angles are \begin{align*}[4(15) +12]^\circ\end{align*}, or \begin{align*}72^\circ\end{align*}. The vertex angle is \begin{align*}180^\circ -72^\circ-72^\circ =36^\circ\end{align*}.

#### Example 2

True or false: Base angles of an isosceles triangle can be right angles.

This statement is false. Because the base angles of an isosceles triangle are congruent, if one base angle is a right angle then both base angles must be right angles. It is impossible to have a triangle with two right (\begin{align*}90^\circ\end{align*}) angles. The Triangle Sum Theorem states that the sum of the three angles in a triangle is \begin{align*}180^\circ\end{align*}. If two of the angles in a triangle are right angles, then the third angle must be \begin{align*}0^\circ\end{align*} and the shape is no longer a triangle.

#### Example 3

Which two angles are congruent?

This is an isosceles triangle. The congruent angles are opposite the congruent sides. From the arrows we see that \begin{align*}\angle S \cong \angle U\end{align*}.

#### Example 4

If an isosceles triangle has base angles with measures of \begin{align*}47^\circ\end{align*}, what is the measure of the vertex angle?

Draw a picture and set up an equation to solve for the vertex angle, \begin{align*}v\end{align*}. Remember that the three angles in a triangle always add up to \begin{align*}180^\circ\end{align*}.

\begin{align*}47^\circ + 47^\circ + v & = 180^\circ\\ v & = 180^\circ - 47^\circ - 47^\circ\\ v & = 86^\circ\end{align*}

#### Example 5

If an isosceles triangle has a vertex angle with a measure of \begin{align*}116^\circ\end{align*}, what is the measure of each base angle?

Draw a picture and set up and equation to solve for the base angles, \begin{align*}b\end{align*}.

\begin{align*}116^\circ + b + b & = 180^\circ\\ 2b & = 64^\circ\\ b & = 32^\circ\end{align*}

### Review

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

Determine if the following statements are true or false.

- Base angles of an isosceles triangle are congruent.
- Base angles of an isosceles triangle are complementary.
- Base angles of an isosceles triangle can be equal to the vertex angle.
- Base angles of an isosceles triangle are acute.

Fill in the proofs below.

- Given: Isosceles \begin{align*}\triangle CIS\end{align*}, with base angles \begin{align*}\angle C\end{align*} and \begin{align*}\angle S\end{align*} \begin{align*}\overline{IO}\end{align*} is the angle bisector of \begin{align*}\angle CIS\end{align*} Prove: \begin{align*}\overline{IO}\end{align*} is the perpendicular bisector of \begin{align*}\overline{CS}\end{align*}

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

1. | 1. Given |

2. | 2. Base Angles Theorem |

3. \begin{align*}\angle CIO \cong \angle SIO\end{align*} | 3. |

4. | 4. Reflexive PoC |

5. \begin{align*}\triangle CIO \cong \triangle SIO\end{align*} | 5. |

6. \begin{align*}\overline{CO} \cong \overline{OS}\end{align*} | 6. |

7. | 7. CPCTC |

8. \begin{align*}\angle IOC\end{align*} and \begin{align*}\angle IOS\end{align*} are supplementary | 8. |

9. | 9. Congruent Supplements Theorem |

10. \begin{align*}\overline{IO}\end{align*} is the perpendicular bisector of \begin{align*}\overline{CS}\end{align*} | 10. |

- Given: Isosceles \begin{align*}\triangle ICS\end{align*} with \begin{align*}\angle C\end{align*} and \begin{align*}\angle S\end{align*} \begin{align*}\overline{IO}\end{align*} is the perpendicular bisector of \begin{align*}\overline{CS}\end{align*} Prove: \begin{align*}\overline{IO}\end{align*} is the angle bisector of \begin{align*}\angle CIS\end{align*}

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

1. | 1. |

2. \begin{align*}\angle C \cong \angle S\end{align*} | 2. |

3. \begin{align*}\overline{CO} \cong \overline{OS}\end{align*} | 3. |

4. \begin{align*}m\angle IOC = m\angle IOS = 90^\circ\end{align*} | 4. |

5. | 5. |

6. | 6. CPCTC |

7. \begin{align*}\overline{IO}\end{align*} is the angle bisector of \begin{align*}\angle CIS\end{align*} | 7. |

On the \begin{align*}x-y\end{align*} plane, plot the coordinates and determine if the given three points make a scalene or isosceles triangle.

- (-2, 1), (1, -2), (-5, -2)
- (-2, 5), (2, 4), (0, -1)
- (6, 9), (12, 3), (3, -6)
- (-10, -5), (-8, 5), (2, 3)
- (-1, 2), (7, 2), (3, 9)

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.10.