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# Isosceles Triangles

## Properties of triangles with two equal sides.

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Isosceles Triangles

### 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 DEF\begin{align*}\triangle DEF\end{align*}, if DE¯¯¯¯¯¯¯¯EF¯¯¯¯¯¯¯¯\begin{align*}\overline{DE} \cong \overline{EF}\end{align*}, then DF\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 DEF\begin{align*}\triangle DEF\end{align*}, if DF\begin{align*}\angle D \cong \angle F\end{align*}, then DE¯¯¯¯¯¯¯¯EF¯¯¯¯¯¯¯¯\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 DEF\begin{align*}\triangle DEF\end{align*}, if EG¯¯¯¯¯¯¯¯DF¯¯¯¯¯¯¯¯\begin{align*}\overline{EG} \perp \overline{DF}\end{align*} and DG¯¯¯¯¯¯¯¯GF¯¯¯¯¯¯¯¯\begin{align*}\overline{DG} \cong \overline{GF}\end{align*}, then DEGFEG\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 x\begin{align*}x^\circ\end{align*} and y\begin{align*}y^\circ\end{align*}? What could you conclude about x and y?

### Examples

#### Example 1

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

(4x+12)15=x=(5x3)\begin{align*}(4x+12)^\circ & = (5x-3)^\circ\\ 15 = x\end{align*}

Substitute x=15\begin{align*}x = 15\end{align*}; the base angles are [4(15)+12]\begin{align*}[4(15) +12]^\circ\end{align*}, or 72\begin{align*}72^\circ\end{align*}. The vertex angle is 1807272=36\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 (90\begin{align*}90^\circ\end{align*}) angles. The Triangle Sum Theorem states that the sum of the three angles in a triangle is 180\begin{align*}180^\circ\end{align*}. If two of the angles in a triangle are right angles, then the third angle must be 0\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 SU\begin{align*}\angle S \cong \angle U\end{align*}.

#### Example 4

If an isosceles triangle has base angles with measures of 47\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, v\begin{align*}v\end{align*}. Remember that the three angles in a triangle always add up to 180\begin{align*}180^\circ\end{align*}.

47+47+vvv=180=1804747=86\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 116\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, b\begin{align*}b\end{align*}.

116+b+b2bb=180=64=32\begin{align*}116^\circ + b + b & = 180^\circ\\ 2b & = 64^\circ\\ b & = 32^\circ\end{align*}

### Review

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

Determine if the following statements are true or false.

1. Base angles of an isosceles triangle are congruent.
2. Base angles of an isosceles triangle are complementary.
3. Base angles of an isosceles triangle can be equal to the vertex angle.
4. Base angles of an isosceles triangle are acute.

Fill in the proofs below.

1. Given: Isosceles CIS\begin{align*}\triangle CIS\end{align*}, with base angles C\begin{align*}\angle C\end{align*} and S\begin{align*}\angle S\end{align*} IO¯¯¯¯¯¯\begin{align*}\overline{IO}\end{align*} is the angle bisector of CIS\begin{align*}\angle CIS\end{align*} Prove: IO¯¯¯¯¯¯\begin{align*}\overline{IO}\end{align*} is the perpendicular bisector of CS¯¯¯¯¯¯¯\begin{align*}\overline{CS}\end{align*}
Statement Reason
1. 1. Given
2. 2. Base Angles Theorem
3. CIOSIO\begin{align*}\angle CIO \cong \angle SIO\end{align*} 3.
4. 4. Reflexive PoC
5. CIOSIO\begin{align*}\triangle CIO \cong \triangle SIO\end{align*} 5.
6. CO¯¯¯¯¯¯¯¯OS¯¯¯¯¯¯¯\begin{align*}\overline{CO} \cong \overline{OS}\end{align*} 6.
7. 7. CPCTC
8. IOC\begin{align*}\angle IOC\end{align*} and IOS\begin{align*}\angle IOS\end{align*} are supplementary 8.
9. 9. Congruent Supplements Theorem
10. IO¯¯¯¯¯¯\begin{align*}\overline{IO}\end{align*} is the perpendicular bisector of CS¯¯¯¯¯¯¯\begin{align*}\overline{CS}\end{align*} 10.
1. Given: Isosceles ICS\begin{align*}\triangle ICS\end{align*} with C\begin{align*}\angle C\end{align*} and S\begin{align*}\angle S\end{align*} IO¯¯¯¯¯¯\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.

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

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### Vocabulary Language: English

Base

The side of a triangle parallel with the bottom edge of the paper or screen is commonly called the base. The base of an isosceles triangle is the non-congruent side in the triangle.

Base Angles

The base angles of an isosceles triangle are the angles formed by the base and one leg of the triangle.

Base Angles Theorem Converse

The base angles theorem converse states if two angles in a triangle are congruent, then the sides opposite those angles are also congruent.

Isosceles Triangle Theorem

The Isosceles Triangle Theorem states that the perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex angle.

Triangle Sum Theorem

The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees.

Vertical Angles

Vertical angles are a pair of opposite angles created by intersecting lines.