In this explainer, we will learn how to use the isosceles triangle theorems to find missing lengths and angles in isosceles triangles.
We know that there are four different types of triangles: equilateral, isosceles, scalene, and right triangles. The properties of the angles and sides determine what type a particular triangle is.
Here, we will focus on isosceles triangles. The word congruent is useful when discussing angles and sides: congruent angles are of equal measure (in degrees, for example), and congruent sides have equal lengths.
Let’s recap the exact definition of an isosceles triangle.
Definition: Isosceles Triangle
An isosceles triangle is a triangle that has two congruent sides.
The congruent sides are called the legs of the triangle, and the third side is called the base.
In this definition, we have been given some useful terminology for the sides of an isosceles triangle. Knowing which of the sides in an isosceles triangle is equal to another (a leg), or whether it is the third side (the base), gives us a way to reference these sides, and, as we will see later, to consider important properties about the angles.
We typically think of isosceles triangles drawn with the base as the horizontal side, but, of course, the orientation of the triangle does not matter. Congruent sides in the isosceles triangle would always be referred to as the legs, regardless of the position they take.
In the first example, we will identify the legs of an isosceles triangle.
Example 1: Identifying the Legs of an Isosceles Triangle
Consider the following triangle.
Identify the legs of the triangle.
In this triangle, we can observe that there are two sides of equal length: the lengths of and are both given as 2 cm.
By definition, a triangle that has two congruent sides is an isosceles triangle. These two congruent sides are called the legs of the triangle.
Therefore, we can give the answer that the legs of the triangle are and .
That aside, we can note that the third side of an isosceles triangle is referred to as the base. In the figure above, is the base of .
We will now consider the angle properties of isosceles triangles.
Consider the following isosceles triangle, .
and are the legs of the isosceles triangle: they are congruent.
Let’s take the midpoint of . As is the midpoint, we know that .
We draw a line from to . This line segment is the median from vertex .
We now have two triangles that have 3 pairs of sides that are congruent since
Hence, the triangles are congruent. We can write that .
Therefore, we know that there is a pair of equal angles in triangle :
Thus, we have proved that an isosceles triangle has two congruent angles. The congruent angles are identified as the angles created by a leg and a base, or the angles opposite the legs.
This angle property is often referred to as the isosceles triangle theorem.
Theorem: Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
The congruent angles are called the base angles. The third angle is called the vertex angle.
We will now see how we can use this angle property of isosceles triangles to help us find the measures of missing angles.
Example 2: Finding the Measure of Base Angles in Isosceles Triangles
Find and .
In the given figure, we can observe that we have an isosceles triangle, . We know that it is isosceles as it has a pair of congruent sides marked on the figure.
In order to find , we use the angle property of isosceles triangles: an isosceles triangle has two congruent angles, which are the angles opposite the two congruent sides. Therefore, the base angles of , and , must be equal in measure.
We recall that the interior angles in a triangle sum to .
Hence, we have
From the diagram, we have , and we know that . So, we can write
We can then divide both sides of the equation by 2 to give
Next, we need to find .
We could calculate this angle if we knew . To find this angle, we can observe that has 3 equal sides. This means that it is an equilateral triangle. Equilateral triangles have the angle property that all 3 angles are equal; they are all . Hence, .
As we previously calculated that and since is isosceles, we also know that .
Therefore, we can calculate the required angle as
We can give the answers for both of the required angles as follows:
We will now see another example where we need to use the angle properties of isosceles triangles to help us find the measure of unknown angles.
Example 3: Finding the Angles in a Triangle Using Isosceles Triangle Theorems
Find the angles of .
We can observe in the diagram that there is a pair of parallel lines: and . We also have two line segments marked as congruent: and . These two congruent line segments form 2 sides of the triangle . A triangle with two equal sides is defined as an isosceles triangle; hence, is an isosceles triangle.
We need to find the measure of all the angles in . Let’s consider the two parallel lines along with the transversal .
The two angles and will be of equal measure, as these are alternate angles. Hence,
We have established that is an isosceles triangle, and we can recall that isosceles triangles have a pair of congruent angles that are the base angles opposite the two congruent sides. The base angles of are and . Thus,
Using the fact that the interior angles of a triangle sum to , we can write that
We determined that , and, since , we have
Dividing both sides of the equation by 2 gives us
Since , we can also write that
Therefore, the angles of can be given as
We will now consider if the converse of the isosceles triangle theorem holds; that is, if a triangle has two congruent angles, does it have two congruent sides?
Let’s take a triangle , with two congruent angles: .
We construct the angle bisector of down to the base and label the point of intersection of the angle bisector and the base as .
Now, we have two triangles, and , that have two pairs of congruent angles (since was split into two congruent angles, and , by the angle bisector). In fact, if two triangles have two pairs of congruent angles, then the third pair of angles in each triangle must be the same, as the angles in both triangles must sum to . Hence, we have that
Using this angle property, alongside the fact that and that is a common side to both triangles, we can then apply the ASA triangle congruence criterion. This criterion states that, if two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, then the triangles are congruent.
Thus, we can write that .
Hence, the corresponding sides and must be congruent. Therefore, we have proved that the converse of the isosceles triangle theorem is true.
Definition: Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, the sides opposite those angles are also congruent.
We will now see how we can apply the converse of the isosceles triangle theorem in the following example.
Example 4: Identifying the Pair of Segments in a Triangle with Equal Length Using Isosceles Triangle Theorems
Given that in the figure below, two sides are equal. Which sides are these?
The given figure has 3 parallel lines: , , and . We can use the angle properties of parallel lines and transversals to help us work out the measures of some of the unknown angles in this figure. We may wonder if, in fact, is isosceles, but, in order to prove this, we will need to determine the angle measures in this triangle.
We can begin by adding the information given in the question, .
Let’s consider the parallel lines and along with the transversal . The angles and are alternate angles and, as such, their measures are equal. Thus, we have
Next, we can consider the parallel lines and along with the transversal . We recall that, in parallel lines, the interior angles on the same side of the transversal are supplementary. and are supplementary. Hence, given that , we have
We can calculate by using the fact that the angle measures on a straight line sum to . Given that , we have
Alternatively, we could also calculate by observing that this angle is alternate to ; thus, we have
We can calculate the angle at the top of , , by subtracting from . This gives us
We can now calculate the third angle in ; that is . We recall that the interior angle measures in a triangle sum to . Since and , we have
We can observe that, within , we have two congruent angles: . By the converse of the isosceles triangle theorem, we know that a triangle that has two congruent angles must have two congruent sides. This type of triangle is by definition an isosceles triangle.
We can give the answer that the two sides that are equal are
We have seen that an isosceles triangle is defined as a triangle that has two congruent sides. It also has two congruent angles. The converse of the isosceles triangle theorem informs us that, if a triangle has two equal angles, it will have two equal sides.
Therefore, we have two different ways in which we can identify an isosceles triangle: if it has two congruent sides or if it has two congruent angles. Knowing either of these properties will demonstrate that the triangle is isosceles.
In the final example, we will determine if a given triangle is isosceles.
Example 5: Determining Whether a Given Triangle Is Isosceles
Is the triangle isosceles?
We recall that a triangle is isosceles if it has two congruent sides. Isosceles triangles also have two equal angles, and a triangle can be described as isosceles if we can demonstrate that it has either two congruent sides or two congruent angles.
In the given figure, we are not given any information about the lengths of the sides in . Therefore, we will need to calculate the angle measures in the triangle to see if two angles have equal measure or not.
Given that , we can use the property that the angle measures on a straight line sum to to calculate . Thus, we have
Next, we know that the angle measures in a triangle sum to . Given that and , we have
We have 3 different angle measures in the triangle. Since no 2 angles are congruent, the answer is no, the triangle is not isosceles.
We now summarize the key points.
- An isosceles triangle is a triangle that has two congruent sides. The congruent sides are called the legs of the triangle, and the third side is called the base.
- The isosceles triangle theorem states that, if two sides of a triangle are congruent, then the angles opposite those sides are congruent. The congruent angles are called the base angles. The third angle is called the vertex angle.
- The converse of the isosceles triangle theorem states that, if two angles of a triangle are congruent, then the sides opposite those angles are congruent.
- We can prove that a triangle is isosceles by either demonstrating that the triangle has two congruent sides or two congruent angles.