### Video Transcript

In this video, we will learn how to use the corollaries of the isosceles triangle theorems to find missing lengths and angles in isosceles triangles. We can recall that an isosceles triangle is simply a triangle that has two congruent sides. And when we are talking about isosceles triangles, we use two important terms. They are the legs and the base. The congruent sides are called the legs of the triangle and the third side is the base.

Now, because isosceles triangles have two congruent sides, this leads us to an important angle property of isosceles triangles. It is the isosceles triangle theorem, and it states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent. So knowing that the two sides are congruent means that in fact the two base angles are congruent. The remaining angle in the isosceles triangle is referred to as the vertex angle. So we know that isosceles triangles by definition have two congruent sides. And by this theorem, it means that they also have two congruent angles. And the converse of this theorem is also true. That is, if two angles of a triangle are congruent, then the sides opposite those angles are also congruent.

We will now consider a number of corollaries to these theorems. These corollaries will allow us to identify additional geometric properties about isosceles triangles. Letβs see the first of these corollaries. This corollary states that the median of an isosceles triangle from the vertex angle bisects it and is perpendicular to the base. We can prove this corollary in the following way. Letβs take this isosceles triangle π΄π΅πΆ, and we draw the median from π΄ to create the point π·. The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side and therefore bisecting that side. We can therefore say that πΆπ· is congruent to π΅π·. We can also write that π΄πΆ is equal to π΄π΅ because we know that the triangle is isosceles and the line segment π΄π· is a shared side in the two triangles π΄π·πΆ and π΄π·π΅.

And as there are now three congruent pairs of sides, then we can say that triangle π΄π·πΆ is congruent to triangle π΄π·π΅ by the SSS, or side-side-side, congurrency criterion. That means that the measure of angle π΄π·πΆ is congruent to the measure of angle π΄π·π΅. Furthermore, since the line segment π΅πΆ is a straight line, then both of these angles π΄π·π΅ and π΄π·πΆ must be 90 degrees. So we have proved this corollary. The median of an isosceles triangle from the vertex angle bisects it and is perpendicular to the base. Because of this corollary, we can also notice that a useful property of the median of an isosceles triangle is that it forms the axis of symmetry of this triangle because it splits the isosceles triangle into two congruent right triangles.

Weβll now see how we can apply this corollary in the following example.

For which values of π₯ and π¦ is line segment π΄π· a perpendicular bisector of line segment π΅πΆ?

In the figure, we can observe that triangle π΄π΅πΆ appears to be an isosceles triangle. Although we canβt prove this, we can use some of the properties of isosceles triangles to help. We can recall that an isosceles triangle is a triangle that has two congruent sides and the median of an isosceles triangle from the vertex angle is a perpendicular bisector of the base. And in this question, we are interested in the perpendicular bisector of the line segment π΅πΆ, which would be the base of this triangle. That means that the line segment π΄π· is only a perpendicular bisector of line segment π΅πΆ in the case of an isosceles triangle. So when this is an isosceles triangle, the two legs π΄π΅ and π΄πΆ will be congruent. And if line segment π΅πΆ is bisected, then π΅π· is equal to πΆπ·.

So letβs take this first equation π΄π΅ is equal to π΄πΆ and fill in the given expressions for these lengths. This would give us three π₯ plus two is equal to five π¦ plus three. We can rearrange this by subtracting three from both sides to give us three π₯ minus one is equal to five π¦. We canβt solve this just yet as there are two unknowns, so letβs label this with equation one. For the second equation, we can fill in the lengths for π΅π· and πΆπ·. This gives five π¦ minus one is equal to 10 minus three π₯. Adding one to both sides gives us a second equation of five π¦ is equal to 11 minus three π₯. We can then find the values of π₯ and π¦ by solving these two equations simultaneously either by substitution or elimination.

We might notice however that both equations have a term of five π¦ on one side of the equation. We can therefore write that three π₯ minus one is equal to 11 minus three π₯. Then, by adding three π₯ and one to both sides of the equation either in one step or two steps, we have that six π₯ is equal to 12, so π₯ is equal to two. We can then substitute π₯ is equal to two into either equation one or two. So using equation one, we have that three times two minus one is equal to five π¦. That gives us that five is equal to five π¦, and so π¦ is equal to one. We can therefore give the answer that the line segment π΄π· is a perpendicular bisector of the line segment π΅πΆ when π₯ is equal to two and π¦ is equal to one.

Weβll now see another corollary. This corollary tells us that the bisector of the vertex angle of an isosceles triangle is a perpendicular bisector of the base. Letβs look at what we mean by this corollary by taking the triangle πΈπΉπΊ, which is isosceles. We can draw in the bisector of the vertex angle, which will meet the line segment πΊπΉ at the point π». And as this is a bisector, we can say that the measure of angle πΊπΈπ» is equal to the measure of angle πΉπΈπ». This corollary would then allow us to state that the base has been bisected, so πΊπ» is equal to πΉπ». And because it does this at right angles, then the measure of angle πΈπ»πΊ and the measure of angle πΈπ»πΉ are both 90 degrees.

Weβll now see how we can apply this in the following example.

Fill in the blank. In this figure, if π΄π΅ equals π΄πΆ, the intersection of line segments π΄π· and π΅πΆ is equal to the set containing π·, where π·πΆ equals eight centimeters and the measure of angle πΆπ΄π· equals the measure of angle π΅π΄π· equals 35 degrees, the length of line segment π΅π· is what centimeters. The second part of this question asks us to find the measure of angle π΄πΆπ΅.

The first very important piece of information we are given is that π΄π΅ is equal to π΄πΆ. As this triangle π΄π΅πΆ has two congruent sides, then it means that triangle π΄π΅πΆ is an isosceles triangle. We are also told that this line segment π΄π· intersects the base π΅πΆ at the point π·. We can also fill in the angle measures that the measure of angle πΆπ΄π΅ and π΅π΄π· are both 35 degrees. We can observe that since these two angles are equal in measure, it means that the vertex angle π΄ of this isosceles triangle has been bisected.

We can recall that one of the corollaries to the isosceles triangle theorems states that the bisector of the vertex angle of an isosceles triangle is a perpendicular bisector of the base. So in triangle π΄π΅πΆ, the base π΅πΆ must be bisected. And that means that the line segment π΅π· is congruent to the line segment π·πΆ. And therefore, π΅π· must also be eight centimeters. Therefore, the value of eight would complete the blank in the first part of this question.

We can now look at the second part of the question where we need to determine the measure of angle π΄πΆπ΅. To do this, letβs go back to the corollary where we are told that this bisector of the vertex angle is a perpendicular bisector of the base. This means that the measure of angle π΄π·πΆ is 90 degrees. By considering triangle π΄πΆπ· and using the fact that the interior angles in a triangle sum to 180 degrees, we can work out that the measure of angle π΄πΆπ·, which is the same as the measure of angle π΄πΆπ΅, must be equal to 180 degrees subtract 35 degrees plus 90 degrees. This simplifies to 55 degrees. We have therefore answered the second part of this question, but of course itβs always worthwhile checking our answers.

For the second part of this question, we recall that we have already said that triangle π΄π΅πΆ is an isosceles triangle. That means that if angle π΄πΆπ΅ is 55 degrees, then the other base angle of this isosceles triangle π΄π΅πΆ must also be 55 degrees. Adding the three interior angle measures would give us 55 degrees plus 55 degrees plus 70 degrees. And that would give us 180 degrees, which we know is the sum of the interior angles in the triangle. And therefore, we have confirmed that the measure of angle π΄πΆπ΅ must be 55 degrees.

We will now see the final corollary to the isosceles triangle theorems. This corollary 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. Letβs look at this diagrammatically with the isosceles triangle πππ
. If we have a straight line through the vertex angle perpendicular to the base, then the base is bisected and the vertex angle is bisected. And weβll now see how we can apply this corollary in the next example.

Find the measure of angle π·π΄π΅.

In this triangle π΄π΅πΆ, we can observe that we have two congruent sides marked. π΄π΅ is equal to π΄πΆ. We can therefore recognize that triangle π΄π΅πΆ must be an isosceles triangle. We can see on the diagram that the measure of angle π΄π·π΅ is 90 degrees. In other words, we can say that the line segment π΄π· is perpendicular to the base πΆπ΅. And this is a very important piece of information because it means that we can apply one of the corollaries to the isosceles triangle theorems. This corollary 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.

Since we know that π΄π· is perpendicular to the base, then the important thing is that this line segment π΄π· bisects the vertex angle. The angle π·π΄π΅ that we need to work out the measure of is part of the vertex angle of the isosceles triangle. Because we know that this angle πΆπ΄π΅ has been bisected, then the measure of angle πΆπ΄π· is congruent to the measure of angle π·π΄π΅. Both of these angles will be 25 degrees. We can therefore give the answer that the measure of angle π·π΄π΅ is 25 degrees.

We will now summarize the key points of this video. We began by recalling the definition of an isosceles triangle, which is that it is a triangle that has two congruent sides. The isosceles triangle theorem tells us that if two sides of a triangle are congruent, then the angles opposite those sides are also congruent. The converse of this theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are also congruent.

We then saw three separate corollaries, the first of which states that the median of an isosceles triangle from the vertex angle is a perpendicular bisector of the base and bisects the vertex angle. The second corollary states that the bisector of the vertex angle of an isosceles triangle is a perpendicular bisector of the base. The third corollary 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. Finally, we also noted that the axis of symmetry of an isosceles triangle is the median that bisects the base.