# Isosceles triangle

An isosceles triangle is a triangle that has at least two sides of equal length. Since the sides of a triangle correspond to its angles, this means that isosceles triangles also have two angles of equal measure. The figure below shows an isosceles triangle example.

The tally marks on the sides of the triangle indicate the congruence (or lack thereof) of the sides while the arcs indicate the congruence of the angles. This is typical triangle notation.

## What is an isosceles triangle

The isosceles triangle definition is a triangle that has two congruent sides and angles. To identify if a triangle is isosceles, check whether it has two congruent sides and angles; if it does, it is isosceles; if it doesn't, it is not isosceles.

### What does an isosceles triangle look like

An isosceles triangle can look like various different things depending on the type of isosceles triangle. A real life example of an isosceles triangle is a slice of pizza given that we exclude the curved part of the pizza crust, as shown in the figure below.

The yellow part of the pizza forms an isosceles triangle, as shown by the side and angle markings.

## Parts of an isosceles triangle

The parts of an isosceles triangle are its legs, base, vertex angle, base angle, and altitudes.

- Legs - the congruent sides of the triangle.
- Base - the third side of the triangle that is not congruent to the other two.
- Vertex angle - the angle opposite the base of the isosceles triangle.
- Base angle - the angles adjacent to the base of the isosceles triangle; these are the two congruent angles.
- Altitude - the perpendicular distance from the vertex of a triangle to the opposite side. Any triangle has three altitudes. The altitude from the base of an isosceles triangle to its opposite vertex divides the triangle into two congruent right triangles.

The figure below shows these parts of an isosceles triangle.

### Base angles of an isosceles triangle

The base angles of an isosceles triangle are the same in measure. Refer to triangle ABC below.

AB ≅ AC so triangle ABC is isosceles. ABC can be divided into two congruent triangles by drawing line segment AD, which is also the height of triangle ABC.

Using the Pythagorean Theorem where l is the length of the legs, . Based on this, △ADB ≅ △ADC by the Side-Side-Side theorem for congruent triangles since BD ≅ CD, AB ≅ AC, and AD ≅ AD. So, ∠B ≅ ∠C, since corresponding parts of congruent triangles are also congruent.

## Properties of an isosceles triangle

- Two congruent sides.
- Two congruent angles.
- Side opposite the vertex angle is the base.
- The altitude drawn from the vertex angle to the base divides an isosceles triangle into two congruent right triangles.

## Types of isosceles triangles

There are a few different types of isosceles triangles. Generally, triangles are categorized as acute, obtuse, right, isosceles, scalene, and equilateral. Isosceles triangles can further be categorized as acute, obtuse, and right. Also, all equilateral triangles are also classified as isosceles since they have 3 congruent sides and angles.

### Isosceles acute triangle

An isosceles acute triangle is a triangle with 2 congruent sides and angles in which all the angles are acute.

### Isosceles obtuse triangle

An isosceles obtuse triangle is a triangle with 2 congruent sides and angles in which the non-congruent angle is obtuse.

### Right isosceles triangle

An isosceles right triangle is a triangle with 2 congruent sides and angles in which the non-congruent angle measures 90°.

Because the sum of a triangle's interior angles is equal to 180°, the remaining two angles in an isosceles right triangle measure 45° (90 + 45 + 45 = 180°).

#### 45-45-90 triangle

A 45-45-90 triangle is a special type of right triangle. The sides of a 45-45-90 triangle have the following relationship:

This relationship allows us to find the missing lengths using simple algebra given the length of either the hypotenuse or one of the congruent sides.

Example

Find the lengths of the missing sides of the 45-45-90 triangle below.

Side b = 12 because it has the same measure as the known side. To find side c, we use the relationship . Since x in this case is 12, we find c by multiplying by :

### Golden triangle

The golden triangle is a special isosceles triangle that is also referred to as the sublime triangle. It has a vertex angle measuring 36° and base angles of 72°. The legs of the Golden triangle are in the golden ratio relative to the base. Furthermore, when a base angle is bisected, two smaller isosceles triangles are formed, and the angle bisector divides the side into two lengths also related by the golden ratio.

Triangle ABC is similar to triangle ADC. Triangle ABD is an isosceles triangle, but not in the golden ratio. Line segments BD = AD = AC.

### Is an equilateral triangle isosceles

An equilateral triangle is an isosceles triangle. An isosceles triangle is defined as a triangle with 2 congruent sides and angles. An equilateral triangle is defined as a triangle with 3 congruent sides and angles. Thus, by their definitions, an equilateral triangle is always also an isosceles triangle. However, whenever we have an equilateral triangle, by convention it will be referred to as an equilateral triangle not an isosceles triangle.

## Isosceles triangle formulas

Below are some formulas for calculating the area, perimeter, and altitudes of an isosceles triangle.

### Area of an isosceles triangle

The formula for the area of an isosceles triangle is:

For an isosceles triangle, the base is the non-congruent side and the height is the altitude of the base.

#### Heron's formula

Heron's formula is a formula for the area of a triangle given that all sides of the triangle are known. If we know or can find all the side lengths of an isosceles triangle, we can find the area of the triangle using Heron's formula,

where a, b, and c are the sides of the triangle and s is its semiperimeter. For an isosceles triangle, since two of the sides are congruent, we can simplify Heron's formula to:

where s is:

### Isosceles triangle perimeter

The perimeter of an isosceles triangle formula is:

where the congruent sides have length of "a" and the non-congruent side has length of "b."

### Altitude of an isosceles triangle

The equations for calculating the three altitudes of an isosceles triangle are as follows:

### How to find the sides of an isosceles triangle

The height of an isosceles triangle is the perpendicular line segment drawn from base of the triangle to the opposing vertex.

Using the Pythagorean Theorem, we can find that the base, legs, and height of an isosceles triangle have the following relationships:

#### Law of cosines

Depending on what information is provided, we can use the law of cosines to find the sides and angles of an isosceles triangle. The law of cosines is:

If we know two sides of a triangle and the angle between them, we can use the law of cosines to find the third side. If we know all three sides, we can use the law of cosines to find all angles of a triangle.

For an isosceles triangle, knowing just one of the legs and the vertex angle allows us to use the law of cosines, since the legs both have the same measure.

Example

Use the law of cosines to find the base of the isosceles triangle below.

The two legs measure 12. In this case we will call both legs a, and substitute their values for "a" and "b" in the law of cosines. The angle between them is 35°. Plugging this into the law of cosines,

Thus, the base of the isosceles triangle has a length of 7.217.

### Finding the sides of an isosceles triangle with trigonometry

The altitude from the vertex angle of an isosceles triangle divides the triangle into two congruent right triangles. This enables us to use basic trigonometric relationships to find the sides and angles of the triangle.

Referencing the triangle above, depending on what parts of the triangle are known, we can use trigonometric relationships to solve for the other parts. Recall the relationships:

Example

Find the length of the legs of the isosceles triangle.

The altitude of the isosceles triangle divides it into two congruent right triangles, so we know that the base of 12 forms a leg of each of the right triangles and has a value of 6. With this information as well as the base angle of 78°, we can use tangent to find the altitude:

Now that we know two legs of the right triangle, we can use the Pythagorean theorem, a^{2} + b^{2} = c^{2}, to find the length of the legs:

Thus, the length of the legs is 28.85841.

## Isosceles, scalene, and equilateral

The table below shows the differences between isosceles, scalene, and equilateral triangles.

Isosceles triangle | Scalene triangle | Equilateral triangle |
---|---|---|

2 congruent sides | 0 congruent sides | 3 congruent sides |

2 congruent angles | 0 congruent angles | 3 congruent angles (all 60°) |

## Symmetry in an isosceles triangle

The altitude of an isosceles triangle is also a line of symmetry.

Leg AB reflects across altitude AD to leg AC. Similarly, leg AC reflects to leg AB. Base BC reflects onto itself when reflecting across the altitude.

## Apothem of a regular polygon

The apothem of a regular polygon is also the height of an isosceles triangle formed by the center and a side of the polygon, as shown in the figure below.

For the regular pentagon ABCDE above, the height of isosceles triangle BCG is an apothem of the polygon (FG).