Lesson Explainer: Isosceles Triangle Theorems | Nagwa Lesson Explainer: Isosceles Triangle Theorems | Nagwa

Lesson Explainer: Isosceles Triangle Theorems Mathematics

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.

  1. 𝐴𝐵 and 𝐵𝐶
  2. 𝐴𝐵 and 𝐴𝐶
  3. 𝐵𝐶 and 𝐴𝐶

Answer

In this triangle, we can observe that there are two sides of equal length: the lengths of 𝐴𝐵 and 𝐵𝐶 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 𝐴𝐵 and 𝐵𝐶.

That aside, we can note that the third side of an isosceles triangle is referred to as the base. In the figure above, 𝐴𝐶 is the base of 𝐴𝐵𝐶.

We will now consider the angle properties of isosceles triangles.

Consider the following isosceles triangle, 𝐴𝐵𝐶.

𝐴𝐵 and 𝐵𝐶 are the legs of the isosceles triangle: they are congruent.

Let’s take the midpoint 𝑀 of 𝐴𝐶. As 𝑀 is the midpoint, we know that 𝐴𝑀=𝑀𝐶.

We draw a line from 𝐵 to 𝑀. This line segment is the median from vertex 𝐵.

We now have two triangles that have 3 pairs of sides that are congruent since 𝐴𝐵=𝐶𝐵,𝐴𝑀=𝑀𝐶,𝐵𝑀.andiscommontobothtriangles

Hence, the triangles are congruent. We can write that 𝐴𝐵𝑀𝐶𝐵𝑀.

Therefore, we know that there is a pair of equal angles in triangle 𝐴𝐵𝐶: 𝐴=𝐶.

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 𝑚𝐶𝐵𝐴 and 𝑚𝐷𝐴𝐶.

Answer

In the given figure, we can observe that we have an isosceles triangle, 𝐴𝐵𝐶. We know that it is isosceles as it has a pair of congruent sides marked on the figure.

In order to find 𝑚𝐶𝐵𝐴, 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 𝐴𝐵𝐶, 𝐶𝐴𝐵 and 𝐶𝐵𝐴, must be equal in measure.

We recall that the interior angles in a triangle sum to 180.

Hence, we have 𝑚𝐴𝐶𝐵+𝑚𝐶𝐵𝐴+𝑚𝐶𝐴𝐵=180.

From the diagram, we have 𝑚𝐴𝐶𝐵=38, and we know that 𝑚𝐶𝐴𝐵=𝑚𝐶𝐵𝐴. So, we can write 38+𝑚𝐶𝐵𝐴+𝑚𝐶𝐵𝐴=18038+2(𝑚𝐶𝐵𝐴)=1802(𝑚𝐶𝐵𝐴)=180382(𝑚𝐶𝐵𝐴)=142.

We can then divide both sides of the equation by 2 to give 𝑚𝐶𝐵𝐴=71.

Next, we need to find 𝑚𝐷𝐴𝐶.

We could calculate this angle if we knew 𝑚𝐷𝐴𝐵. To find this angle, we can observe that 𝐷𝐴𝐵 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 60. Hence, 𝑚𝐷𝐴𝐵=60.

As we previously calculated that 𝑚𝐶𝐵𝐴=71 and since 𝐴𝐵𝐶 is isosceles, we also know that 𝑚𝐶𝐴𝐵=71.

Therefore, we can calculate the required angle as 𝑚𝐷𝐴𝐶=𝑚𝐶𝐴𝐵𝑚𝐷𝐴𝐵=7160=11.

We can give the answers for both of the required angles as follows: 𝑚𝐶𝐵𝐴=71,𝑚𝐷𝐴𝐶=11.

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 𝐴𝐵𝐶.

Answer

We can observe in the diagram that there is a pair of parallel lines: 𝐴𝐷 and 𝐵𝐶. We also have two line segments marked as congruent: 𝐴𝐶 and 𝐵𝐶. These two congruent line segments form 2 sides of the triangle 𝐴𝐵𝐶. A triangle with two equal sides is defined as an isosceles triangle; hence, 𝐴𝐵𝐶 is an isosceles triangle.

We need to find the measure of all the angles in 𝐴𝐵𝐶. Let’s consider the two parallel lines along with the transversal 𝐴𝐶.

The two angles 𝐷𝐴𝐶 and 𝐴𝐶𝐵 will be of equal measure, as these are alternate angles. Hence, 𝑚𝐴𝐶𝐵=𝑚𝐷𝐴𝐶=39.5.

We have established that 𝐴𝐵𝐶 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 𝐴𝐵𝐶 are 𝐵𝐴𝐶 and 𝐴𝐵𝐶. Thus, 𝑚𝐵𝐴𝐶=𝑚𝐴𝐵𝐶.

Using the fact that the interior angles of a triangle sum to 180, we can write that 𝑚𝐵𝐴𝐶+𝑚𝐴𝐵𝐶+𝑚𝐴𝐶𝐵=180.

We determined that 𝑚𝐴𝐶𝐵=39.5, and, since 𝑚𝐵𝐴𝐶=𝑚𝐴𝐵𝐶, we have 𝑚𝐵𝐴𝐶+𝑚𝐵𝐴𝐶+39.5=1802(𝑚𝐵𝐴𝐶)=18039.52(𝑚𝐵𝐴𝐶)=140.5.

Dividing both sides of the equation by 2 gives us 𝑚𝐵𝐴𝐶=70.25.

Since 𝑚𝐵𝐴𝐶=𝑚𝐴𝐵𝐶, we can also write that 𝑚𝐴𝐵𝐶=70.25.

Therefore, the angles of 𝐴𝐵𝐶 can be given as 𝑚𝐴𝐵𝐶=70.25,𝑚𝐵𝐴𝐶=70.25,𝑚𝐴𝐶𝐵=39.5.

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 𝐷𝐸𝐹, with two congruent angles: 𝑚𝐷𝐸𝐹=𝑚𝐷𝐹𝐸.

We construct the angle bisector of 𝐸𝐷𝐹 down to the base 𝐸𝐹 and label the point of intersection of the angle bisector and the base as 𝐺.

Now, we have two triangles, 𝐷𝐸𝐺 and 𝐷𝐹𝐺, that have two pairs of congruent angles (since 𝐸𝐷𝐹 was split into two congruent angles, 𝐸𝐷𝐺 and 𝐹𝐷𝐺, 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 180. Hence, we have that 𝑚𝐷𝐺𝐸=𝑚𝐷𝐺𝐹.

Using this angle property, alongside the fact that 𝑚𝐷𝐸𝐹=𝑚𝐷𝐹𝐸 and that 𝐷𝐺 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 𝐷𝐸𝐺𝐷𝐹𝐺.

Hence, the corresponding sides 𝐷𝐸 and 𝐷𝐹 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 𝑚𝐹𝐶𝐴=134 in the figure below, two sides are equal. Which sides are these?

Answer

The given figure has 3 parallel lines: 𝐷𝐸, 𝐶𝐹, and 𝐵𝐴. 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, 𝐴𝐵𝐶 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, 𝑚𝐹𝐶𝐴=134.

Let’s consider the parallel lines 𝐷𝐸 and 𝐹𝐶 along with the transversal 𝐷𝐶. The angles 𝐶𝐷𝐸 and 𝐹𝐶𝐷 are alternate angles and, as such, their measures are equal. Thus, we have 𝑚𝐹𝐶𝐷=𝑚𝐶𝐷𝐸=134.

Next, we can consider the parallel lines 𝐷𝐸 and 𝐴𝐵 along with the transversal 𝐵𝐷. We recall that, in parallel lines, the interior angles on the same side of the transversal are supplementary. 𝐶𝐷𝐸 and 𝐶𝐵𝐴 are supplementary. Hence, given that 𝑚𝐶𝐷𝐸=134, we have 𝑚𝐶𝐷𝐸+𝑚𝐶𝐵𝐴=180134+𝑚𝐶𝐵𝐴=180𝑚𝐶𝐵𝐴=180134=46.

We can calculate 𝑚𝐹𝐶𝐵 by using the fact that the angle measures on a straight line sum to 180. Given that 𝑚𝐹𝐶𝐷=134, we have 𝑚𝐹𝐶𝐵+𝑚𝐹𝐶𝐷=180𝑚𝐹𝐶𝐵+134=180𝑚𝐹𝐶𝐵=180134=46.

Alternatively, we could also calculate 𝑚𝐹𝐶𝐵 by observing that this angle is alternate to 𝐶𝐵𝐴; thus, we have 𝑚𝐹𝐶𝐵=𝑚𝐶𝐵𝐴=46.

We can calculate the angle at the top of 𝐴𝐵𝐶, 𝑚𝐴𝐶𝐵, by subtracting 𝑚𝐹𝐶𝐵 (46) from 𝑚𝐹𝐶𝐴 (134). This gives us 𝑚𝐴𝐶𝐵=𝑚𝐹𝐶𝐴𝑚𝐹𝐶𝐵=13446=88.

We can now calculate the third angle in 𝐴𝐵𝐶; that is 𝑚𝐶𝐴𝐵. We recall that the interior angle measures in a triangle sum to 180. Since 𝑚𝐶𝐵𝐴=46 and 𝑚𝐴𝐶𝐵=88, we have 𝑚𝐶𝐵𝐴+𝑚𝐴𝐶𝐵+𝑚𝐶𝐴𝐵=18046+88+𝑚𝐶𝐴𝐵=180134+𝑚𝐶𝐴𝐵=180𝑚𝐶𝐴𝐵=180134=46.

We can observe that, within 𝐴𝐵𝐶, we have two congruent angles: 𝑚𝐶𝐴𝐵=𝑚𝐶𝐵𝐴. 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 𝐴𝐶𝐵𝐶.and

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 𝐴𝐵𝐶 isosceles?

Answer

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 𝐴𝐵𝐶. Therefore, we will need to calculate the angle measures in the triangle to see if two angles have equal measure or not.

Given that 𝑚𝐴𝐵𝐷=142, we can use the property that the angle measures on a straight line sum to 180 to calculate 𝑚𝐴𝐵𝐶. Thus, we have 𝑚𝐴𝐵𝐷+𝑚𝐴𝐵𝐶=180142+𝑚𝐴𝐵𝐶=180𝑚𝐴𝐵𝐶=180142=38.

Next, we know that the angle measures in a triangle sum to 180. Given that 𝑚𝐶𝐴𝐵=83 and 𝑚𝐴𝐵𝐶=38, we have 𝑚𝐶𝐴𝐵+𝑚𝐴𝐵𝐶+𝑚𝐴𝐶𝐵=18083+38+𝑚𝐴𝐶𝐵=180121+𝑚𝐴𝐶𝐵=180𝑚𝐴𝐶𝐵=180121=59.

We have 3 different angle measures in the triangle. Since no 2 angles are congruent, the answer is no, the triangle 𝐴𝐵𝐶 is not isosceles.

We now summarize the key points.

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.

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