In this explainer, we will learn how to use the corollaries of the isosceles triangle theorems to find missing lengths and angles in isosceles triangles.
Let’s begin by recapping the exact definition of an isosceles triangle. Recall that congruent means having the same measure; for example, congruent sides have the same length and congruent angles have the same measure.
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.
Because isosceles triangles have two congruent sides, this leads us to an important angle property of isosceles triangles.
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.
In , , so .
We know that isosceles triangles, by definition, have two congruent sides, and by the previous theorem, they have two congruent angles. The converse of this is true; that is, if a triangle has two angles that are congruent, then it is an isosceles triangle. This is defined in the theorem below.
Theorem: Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, the sides opposite those angles are also congruent.
In this explainer, we will consider a number of corollaries to these theorems. These corollaries allow us to identify additional geometric properties about isosceles triangles. Let’s see the first of these corollaries.
Corollary of the Isosceles Triangle Theorems: The Median of an Isosceles Triangle
The median of an isosceles triangle from the vertex angle bisects it and is perpendicular to the base.
We can prove this corollary as follows.
A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side and thus bisecting that side. In isosceles triangle below, the median from the vertex angle, , is a line segment joining to the midpoint () of the base, .
Since is the midpoint of , we know that .
We can also write that and
As there are three congruent sides, then is congruent to by the SSS congruency criterion.
Hence, we can write that , and since is a straight line, these angles must both be . We can also note that , so the median from the vertex of an isosceles triangle also bisects the vertex.
Given that is a median, we can say that the median from the vertex angle bisects the base, but here we have also demonstrated that it does this at an angle of .
Therefore, the median of an isosceles triangle from the vertex angle is a perpendicular bisector of the triangle.
Because of this corollary, we can observe that a useful property of the median of an isosceles triangle is that it also forms the axis of symmetry of an isosceles triangle and splits the isosceles triangle into two congruent right triangles.
We will now see how we can apply this corollary in the following example.
Example 1: Finding Values in a Triangle given a Perpendicular Bisector
For which values of and is a perpendicular bisector of ?
Answer
In the figure, we can observe that appears to be an isosceles triangle. Although we cannot prove this, we can use some of the properties of isosceles triangles to help. An isosceles triangle is a triangle that has two congruent sides. We recall that the median of an isosceles triangle from the vertex angle is a perpendicular bisector of the base.
Hence, is only a perpendicular bisector of in the case of an isosceles triangle. Therefore, we need to determine the values of and such that (two congruent legs) and (the base is bisected).
We can begin by substituting in the given expressions for and . This gives
In the same way, for and , we can form a second equation as follows:
To find the values of and , we solve these two equations simultaneously, either by substitution or elimination. They are:
As both equations have a term of on the right-hand side of each equation, we can set the left-hand side of each equation equal to one another and solve for . This gives us
Next, we substitute into either equation (1) or (2). Substituting into equation (1) gives
Thus, we can answer that is a perpendicular bisector of when and .
Let’s consider another corollary.
Corollary of the Isosceles Triangle Theorems: Bisector of the Vertex Angle
The bisector of the vertex angle of an isosceles triangle is a perpendicular bisector of the base.
We can prove this in the following way. Consider the isosceles triangle . The bisector of the vertex angle, , has been drawn.
We can label the point where the bisector crosses as . Since is an angle bisector, we know that .
There are now 2 pairs of congruent sides:
With the congruent pair of included angles, we can say that is congruent to by the SAS condition (two equal sides and the included angle). You may be familiar with this criterion although we will not directly prove it in this explainer.
This means that corresponding angles are congruent and . Since these two angles lie on the straight line , then they must both equal . Furthermore, the sides and are corresponding, and so they are also congruent.
Thus, we have proved the corollary that the bisector of the vertex angle is also the perpendicular bisector of the isosceles triangle .
Let’s see how we can apply this in the following example.
Example 2: Finding a Missing Length in an Isosceles Triangle Using the Bisector of the Vertex Angle
- Fill in the blank: In this figure, if , , where , and , the length of is cm.
- Find .
Answer
For the figure, we are given the information that has two congruent sides (). This means that is an isosceles triangle. We are also given that the intersection point of and is the point .
Part 1
Let’s begin by adding the angle measurements, , to the figure along with marking the congruent sides.
We observe that, since these two angles are equal in measure, this means that the vertex angle, , of the isosceles triangle has been bisected. We recall that one of the corollaries of the isosceles triangle theorems tells us that the bisector of the vertex angle of an isosceles triangle is a perpendicular bisector of the base. This allows us to calculate the length of . Note that, if we did not have this information about the angles, we would not have been able to prove that is a perpendicular bisector.
Since has been bisected, then we know that
From the diagram, we have that . Hence,
Therefore, we can fill in the blank: the length of is 8 cm.
Part 2
To find , we can use the fact that is a perpendicular bisector of . This means that .
Using and the fact that the interior angle measures in a triangle sum to , we have
Since is the same angle as , we can give the answer that is .
We will now see another corollary of the isosceles triangle theorem.
Corollary of the Isosceles Triangle Theorems: Perpendicular to the Base
The straight line that passes through the vertex angle of an isosceles triangle and is perpendicular to the base bisects the base and the vertex angle.
We can prove this by considering an example of a straight line passing through the vertex angle of an isosceles triangle, below, at to the base. We can define the point at which the line intersects as .
At this point, we need to establish how this line affects the vertex angle and the base .
Given that the line is perpendicular to the base, we recognize that
As the triangle is isosceles, we know that
We can observe that and are the hypotenuses of their respective right triangles, and .
Furthermore, since is a common side to both and , we know that this a congruent side in these triangles. Therefore, by applying the RHS (right angle-hypotenuse-side) congruence criterion, we can prove that
The corollary statement can then be verified: the base is bisected, and since and are the corresponding sides of two congruent triangles, the sides themselves are also congruent. This means that the base is bisected. The vertex angle has also been bisected, as we know that its two component angles, and , are the corresponding angles of two congruent triangles and, hence, have equal measures.
We will now see how we can apply this corollary in the following example.
Example 3: Finding the Measure of an Angle in an Isosceles Triangle Using Its Properties
Find .
Answer
In the figure, we observe that the largest triangle, , has two congruent sides marked to show us that . Therefore, must be an isosceles triangle.
We are given that . This means that the line segment, , from the vertex angle in the isosceles triangle is perpendicular to the base. We recall that one of the corollaries to the isosceles triangle theorems states that the straight line that passes through the vertex angle of an isosceles triangle and is perpendicular to the base bisects the base and the vertex angle. This is useful in allowing us to determine .
Since has been bisected, then
From the diagram, we have ; hence, we can determine that the answer is
We will now see two examples where we need to apply several of these corollaries to the isosceles triangle theorems to find the measures of unknown angles and sides.
Example 4: Proving a Geometric Statement Using the Corollaries of the Isosceles Triangle Theorems
In the following figure, if , , , , and is the midpoint of , find and .
Answer
We are given the information that and . This means that, in each of and , there is a pair of congruent sides. A triangle with two congruent sides is, by definition, an isosceles triangle. Therefore, and are both isosceles. Knowing the properties of isosceles triangles given in the corollaries of the isosceles triangle theorems will be useful in calculating the required angles.
Let’s begin by constructing the segment , which creates the first required angle, .
As is the midpoint of , we know that must be a median of the isosceles triangle . We recall that the median of an isosceles triangle from the vertex angle is a perpendicular bisector of the base. As is a perpendicular bisector to , then
Within , we can use the property that the internal angle measures in a triangle sum to to help us find . Given that and , we have
Next, we must determine . We construct the segment to create this angle.
In the same process as before, we know that is isosceles. Since is a midpoint of , then the median from the vertex angle is a perpendicular bisector of the base. This means that
We can give the answers for and as
We will now see one final example.
Example 5: Proving a Geometric Statement Using the Corollaries of the Isosceles Triangle Theorems
In the figure below, , , , and . Draw and cut at , then draw and cut at . Find and .
Answer
We can begin by constructing the two segments and , which are perpendicular to and respectively. We can also fill in the given angle information, which is , and mark on the diagram that .
In the lower triangle, , we have two congruent line segments. This means that must be an isosceles triangle. There are a number of ways in which we can find . One method is to use and recall that the internal angle measures in a triangle sum to . Given that and , we have
Next, we need to calculate the required angle, . Given that is an isosceles triangle, we can use the fact that the straight line that passes through the vertex angle of an isosceles triangle and is perpendicular to the base bisects the base and the vertex angle. This means that
Given that , , and with and meeting at point , then is a straight line. We can then observe that and are vertically opposite and, hence, are congruent.
Therefore, we can give the answer that the unknown angle measures are
We now summarize the key points.
Key Points
- An isosceles triangle is a triangle that has two congruent sides.
- If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
- If two angles of a triangle are congruent, then the sides opposite those angles are also congruent.
- The median of an isosceles triangle from the vertex angle is a perpendicular bisector of the base and bisects the vertex angle.
- The bisector of the vertex angle of an isosceles triangle is a perpendicular bisector of the base.
- The straight line that passes through the vertex angle of an isosceles triangle and is perpendicular to the base bisects the base and the vertex angle.
- The axis of symmetry of an isosceles triangle is the median that bisects the base.
