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Lesson Explainer: Isosceles Triangle Theorems Mathematics • 11th Grade

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. 𝐴𝐡 and 𝐡𝐢
  2. 𝐴𝐡 and 𝐴𝐢
  3. 𝐡𝐢 and 𝐴𝐢

Answer

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

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

We will now consider the angle properties of isosceles triangles.

Consider the following isosceles triangle, 𝐴𝐡𝐢.

𝐴𝐡 and 𝐡𝐢 are the legs of the isosceles triangle: they are congruent.

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

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

We now have two triangles that have 3 pairs of sides that are congruent since 𝐴𝐡=𝐢𝐡,𝐴𝑀=𝑀𝐢,𝐡𝑀.andiscommontobothtriangles

Hence, the triangles are congruent. We can write that △𝐴𝐡𝑀≅△𝐢𝐡𝑀.

Therefore, we know that there is a pair of equal angles in triangle 𝐴𝐡𝐢: ∠𝐴=∠𝐢.

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 π‘šβˆ πΆπ΅π΄ and π‘šβˆ π·π΄πΆ.

Answer

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

In order to find π‘šβˆ πΆπ΅π΄, 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 △𝐴𝐡𝐢, ∠𝐢𝐴𝐡 and ∠𝐢𝐡𝐴, must be equal in measure.

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

Hence, we have π‘šβˆ π΄πΆπ΅+π‘šβˆ πΆπ΅π΄+π‘šβˆ πΆπ΄π΅=180.∘

From the diagram, we have π‘šβˆ π΄πΆπ΅=38∘, and we know that π‘šβˆ πΆπ΄π΅=π‘šβˆ πΆπ΅π΄. So, we can write 38+π‘šβˆ πΆπ΅π΄+π‘šβˆ πΆπ΅π΄=18038+2(π‘šβˆ πΆπ΅π΄)=1802(π‘šβˆ πΆπ΅π΄)=180βˆ’382(π‘šβˆ πΆπ΅π΄)=142.∘∘∘∘∘∘∘

We can then divide both sides of the equation by 2 to give π‘šβˆ πΆπ΅π΄=71.∘

Next, we need to find π‘šβˆ π·π΄πΆ.

We could calculate this angle if we knew π‘šβˆ π·π΄π΅. To find this angle, we can observe that △𝐷𝐴𝐡 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, π‘šβˆ π·π΄π΅=60∘.

As we previously calculated that π‘šβˆ πΆπ΅π΄=71∘ and since △𝐴𝐡𝐢 is isosceles, we also know that π‘šβˆ πΆπ΄π΅=71∘.

Therefore, we can calculate the required angle as π‘šβˆ π·π΄πΆ=π‘šβˆ πΆπ΄π΅βˆ’π‘šβˆ π·π΄π΅=71βˆ’60=11.∘∘∘

We can give the answers for both of the required angles as follows: π‘šβˆ πΆπ΅π΄=71,π‘šβˆ π·π΄πΆ=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 △𝐴𝐡𝐢.

Answer

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

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

The two angles ∠𝐷𝐴𝐢 and ∠𝐴𝐢𝐡 will be of equal measure, as these are alternate angles. Hence, π‘šβˆ π΄πΆπ΅=π‘šβˆ π·π΄πΆ=39.5.∘

We have established that △𝐴𝐡𝐢 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 △𝐴𝐡𝐢 are ∠𝐡𝐴𝐢 and ∠𝐴𝐡𝐢. Thus, π‘šβˆ π΅π΄πΆ=π‘šβˆ π΄π΅πΆ.

Using the fact that the interior angles of a triangle sum to 180∘, we can write that π‘šβˆ π΅π΄πΆ+π‘šβˆ π΄π΅πΆ+π‘šβˆ π΄πΆπ΅=180.∘

We determined that π‘šβˆ π΄πΆπ΅=39.5∘, and, since π‘šβˆ π΅π΄πΆ=π‘šβˆ π΄π΅πΆ, we have π‘šβˆ π΅π΄πΆ+π‘šβˆ π΅π΄πΆ+39.5=1802(π‘šβˆ π΅π΄πΆ)=180βˆ’39.52(π‘šβˆ π΅π΄πΆ)=140.5.∘∘∘∘∘

Dividing both sides of the equation by 2 gives us π‘šβˆ π΅π΄πΆ=70.25.∘

Since π‘šβˆ π΅π΄πΆ=π‘šβˆ π΄π΅πΆ, we can also write that π‘šβˆ π΄π΅πΆ=70.25.∘

Therefore, the angles of △𝐴𝐡𝐢 can be given as π‘šβˆ π΄π΅πΆ=70.25,π‘šβˆ π΅π΄πΆ=70.25,π‘šβˆ π΄πΆπ΅=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: π‘šβˆ π·πΈπΉ=π‘šβˆ π·πΉπΈ.

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 π‘šβˆ π·πΊπΈ=π‘šβˆ π·πΊπΉ.

Using this angle property, alongside the fact that π‘šβˆ π·πΈπΉ=π‘šβˆ π·πΉπΈ 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 △𝐷𝐸𝐺≅△𝐷𝐹𝐺.

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 π‘šβˆ πΉπΆπ΄=134∘ in the figure below, two sides are equal. Which sides are these?

Answer

The given figure has 3 parallel lines: 𝐷𝐸, 𝐢𝐹, and 𝐡𝐴. 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, △𝐴𝐡𝐢 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, π‘šβˆ πΉπΆπ΄=134∘.

Let’s consider the parallel lines 𝐷𝐸 and 𝐹𝐢 along with the transversal 𝐷𝐢. The angles 𝐢𝐷𝐸 and 𝐹𝐢𝐷 are alternate angles and, as such, their measures are equal. Thus, we have π‘šβˆ πΉπΆπ·=π‘šβˆ πΆπ·πΈ=134.∘

Next, we can consider the parallel lines 𝐷𝐸 and 𝐴𝐡 along with the transversal 𝐡𝐷. We recall that, in parallel lines, the interior angles on the same side of the transversal are supplementary. ∠𝐢𝐷𝐸 and ∠𝐢𝐡𝐴 are supplementary. Hence, given that π‘šβˆ πΆπ·πΈ=134∘, we have π‘šβˆ πΆπ·πΈ+π‘šβˆ πΆπ΅π΄=180134+π‘šβˆ πΆπ΅π΄=180π‘šβˆ πΆπ΅π΄=180βˆ’134=46.∘∘∘∘∘∘

We can calculate π‘šβˆ πΉπΆπ΅ by using the fact that the angle measures on a straight line sum to 180∘. Given that π‘šβˆ πΉπΆπ·=134∘, we have π‘šβˆ πΉπΆπ΅+π‘šβˆ πΉπΆπ·=180π‘šβˆ πΉπΆπ΅+134=180π‘šβˆ πΉπΆπ΅=180βˆ’134=46.∘∘∘∘∘∘

Alternatively, we could also calculate π‘šβˆ πΉπΆπ΅ by observing that this angle is alternate to ∠𝐢𝐡𝐴; thus, we have π‘šβˆ πΉπΆπ΅=π‘šβˆ πΆπ΅π΄=46.∘

We can calculate the angle at the top of △𝐴𝐡𝐢, π‘šβˆ π΄πΆπ΅, by subtracting π‘šβˆ πΉπΆπ΅Β (46)∘ from π‘šβˆ πΉπΆπ΄Β (134)∘. This gives us π‘šβˆ π΄πΆπ΅=π‘šβˆ πΉπΆπ΄βˆ’π‘šβˆ πΉπΆπ΅=134βˆ’46=88.∘∘∘

We can now calculate the third angle in △𝐴𝐡𝐢; that is π‘šβˆ πΆπ΄π΅. We recall that the interior angle measures in a triangle sum to 180∘. Since π‘šβˆ πΆπ΅π΄=46∘ and π‘šβˆ π΄πΆπ΅=88∘, we have π‘šβˆ πΆπ΅π΄+π‘šβˆ π΄πΆπ΅+π‘šβˆ πΆπ΄π΅=18046+88+π‘šβˆ πΆπ΄π΅=180134+π‘šβˆ πΆπ΄π΅=180π‘šβˆ πΆπ΄π΅=180βˆ’134=46.∘∘∘∘∘∘∘∘∘

We can observe that, within △𝐴𝐡𝐢, we have two congruent angles: π‘šβˆ πΆπ΄π΅=π‘šβˆ πΆπ΅π΄. 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 𝐴𝐢𝐡𝐢.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 𝐴𝐡𝐢 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 △𝐴𝐡𝐢. Therefore, we will need to calculate the angle measures in the triangle to see if two angles have equal measure or not.

Given that π‘šβˆ π΄π΅π·=142∘, we can use the property that the angle measures on a straight line sum to 180∘ to calculate π‘šβˆ π΄π΅πΆ. Thus, we have π‘šβˆ π΄π΅π·+π‘šβˆ π΄π΅πΆ=180142+π‘šβˆ π΄π΅πΆ=180π‘šβˆ π΄π΅πΆ=180βˆ’142=38.∘∘∘∘∘∘

Next, we know that the angle measures in a triangle sum to 180∘. Given that π‘šβˆ πΆπ΄π΅=83∘ and π‘šβˆ π΄π΅πΆ=38∘, we have π‘šβˆ πΆπ΄π΅+π‘šβˆ π΄π΅πΆ+π‘šβˆ π΄πΆπ΅=18083+38+π‘šβˆ π΄πΆπ΅=180121+π‘šβˆ π΄πΆπ΅=180π‘šβˆ π΄πΆπ΅=180βˆ’121=59.∘∘∘∘∘∘∘∘∘

We have 3 different angle measures in the triangle. Since no 2 angles are congruent, the answer is no, the triangle 𝐴𝐡𝐢 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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