Lesson Video: Isosceles Triangle Theorems Mathematics • Second Year of Preparatory School

Video Transcript

In this video, we’ll learn how to use the properties of isosceles triangles to find a missing angle or side length. We’ll also look at how we can do this when the given angles or sides are as algebraic expressions. Let’s begin with the definition of an isosceles triangle.

An isosceles triangle is defined as a triangle which has two of its sides equal in length. So in this isosceles triangle 𝐴𝐵𝐶, we can see from the hash marks that 𝐴𝐵 is equal in length to 𝐴𝐶. The unequal side is usually called the base of the isosceles triangle, and these would be the base angles. In this case, it would be angle 𝐵 and angle 𝐶.

This leads us to another important property of an isosceles triangle, and that is that the base angles are always equal. You probably already know that isosceles triangles have two equal angles. But have you ever thought about why? Let’s take our isosceles triangle 𝐴𝐵𝐶 and see if we can prove that there are two equal angles.

We start by drawing a median 𝐴𝑦. This is a line from 𝐴 going to the midpoint of line segment 𝐵𝐶. Because 𝑦 is a midpoint, we can say that the line segment 𝐵𝑦 is equal to the line segment 𝑦𝐶. Next, because we have an isosceles triangle, we know that there’s two equal sides. In this case, we were given that line segment 𝐴𝐵 is equal to line segment 𝐴𝐶.

Finally, as we consider the two smaller triangles, triangle 𝐴𝐵𝑦 and triangle 𝐴𝐶𝑦, we know that they have a common side, which is the line segment 𝐴𝑦. So as we consider our two triangles, 𝐴𝐵𝑦 on the left and triangle 𝐴𝐶𝑦 on the right, we can in fact say that these two triangles are congruent. And that’s because we’ve demonstrated that there are three pairs of sides congruent. So we can use the SSS congruency criterion.

Importantly for us, it means that this angle at 𝐵 is equal to the angle at 𝐶, and so demonstrating the property that these angles are always equal in an isosceles triangle. As we go through our questions on isosceles triangles, we’ll be using the fact that they have two equal sides and two equal angles. So let’s have a look at our first question.

Find the measure of angle 𝐴𝐵𝐶.

In the diagram, we have a triangle. We’re given the angle 𝐶𝐴𝐵 as 58 degrees. And we need to work out this angle of 𝐴𝐵𝐶. We should notice that on this diagram, we have two hash markings on the lines, indicating that there’s two equal sides. So we could say that triangle 𝐴𝐵𝐶 is in fact an isosceles triangle, as an isosceles triangle is defined as a triangle that has two sides equal in length.

An important property of isosceles triangles is that base angles are equal. In an isosceles triangle, the unequal side is referred to as the base, so that would be this line 𝐴𝐵. And the base angles would be the two angles adjacent to this base. But be careful because the base isn’t always at the bottom of the diagram.

We can therefore write that angle 𝐴𝐵𝐶 is equal to angle 𝐵𝐴𝐶. As we’re given that angle 𝐵𝐴𝐶 is 58 degrees, then so is angle 𝐴𝐵𝐶. We can give our answer then that the measure of angle 𝐴𝐵𝐶 is 58 degrees.

Let’s have a look at another question involving the angles in an isosceles triangle.

In the displayed model of a house, what angle does the roof make with the horizontal, given that triangle 𝐴𝐸𝐵 is isosceles?

It’s probably most useful to begin this question by highlighting the triangle 𝐴𝐸𝐵 that we want to consider. This triangle forms the roof of this house, and we’re told that it’s isosceles. We should recall that in an isosceles triangle, we have two sides equal in length and two base angles are equal. So in this diagram, the side 𝐴𝐸 is equal to the side 𝐴𝐵 and the angle 𝐴𝐸𝐵 is equal to the angle 𝐴𝐵𝐸.

Now that we’ve had a look at the diagram, let’s focus on what we’re asked, to find the angle that the roof makes with the horizontal. That means that we’re really looking for the angle created by the slope of the roof and the horizontal axis. Either of the two base angles would give us the answer for this. So let’s see if we can work out one of these angles, angle 𝐴𝐵𝐸.

In order to do this, we’ll need to remember an important fact about the angles in a triangle. And that is that the angles in a triangle add up to 180 degrees. This means that we can write that angle 𝐵𝐴𝐸 plus angle 𝐴𝐸𝐵 plus angle 𝐴𝐵𝐸 is equal to 180 degrees. We’re given that angle 𝐵𝐴𝐸 is 107 degrees. So if we subtract 107 degrees from both sides of this equation, we get that angle 𝐴𝐸𝐵 plus angle 𝐴𝐵𝐸 is equal to 73 degrees. As we have an isosceles triangle, we know that our two base angles are equal. So angle 𝐴𝐸𝐵 is equal to angle 𝐴𝐵𝐸.

We could think of this then that two times angle 𝐴𝐵𝐸 is 73 degrees. And so to find angle 𝐴𝐵𝐸, we must divide both sides of this equation by two, which means that the measure of angle 𝐴𝐵𝐸 is 36.5 degrees as a decimal. We can therefore give our answer that the angle that the roof makes with the horizontal is 36.5 degrees.

This next question looks a little more complicated. But don’t worry. We’ll just need to apply some of our algebra skills along with the properties that we know of isosceles triangles.

Find the values of 𝑥 and 𝑦.

In this triangle question, the important thing to notice is these markings on the line, which indicate that it’s an isosceles triangle, as we have two sides of equal length. We should also remember that isosceles triangles have two equal angles. In this case, it’s these two base angles of nine 𝑦 minus three degrees and 𝑥 plus one degrees.

We might begin trying to solve this by writing the equation that nine 𝑦 minus three degrees equals 𝑥 plus one degrees. However, in order to solve an equation like this that has two unknowns, the 𝑥 and 𝑦, we’d need another equation linking 𝑥 and 𝑦, which we don’t have. So let’s try another approach. This time, we’ll try thinking about the sum of the angles in a triangle.

Using the fact that the angles in a triangle add up to 180 degrees might give us a way to find out an actual numerical value for nine 𝑦 minus three and 𝑥 plus one. If we write that 96 degrees plus nine 𝑦 minus three degrees plus 𝑥 plus one degrees equals 180 degrees, then we could subtract 96 degrees from both sides of this equation to give us that these two values of nine 𝑦 minus three degrees plus 𝑥 plus one degrees must equal 84 degrees. And since we know that these two angles are equal to each other, that means that each of these must be equal to half of 84 degrees. So nine 𝑦 minus three degrees must be 42 degrees, and 𝑥 plus one degrees must be 42 degrees.

Writing these two equations and solving them would give us the values of 𝑥 and 𝑦. Beginning with nine 𝑦 minus three degrees equals 42 degrees, we could add three to both sides. So nine 𝑦 is equal to 45. Dividing both sides by nine would give us that 𝑦 equals five. In order to solve the second equation, 𝑥 plus one degrees equals 42 degrees, we just need to subtract one from both sides. So 𝑥 is equal to 41.

Before we finish, it’s always worth checking our values. If 𝑥 is 41, then this angle is 42 degrees. If 𝑦 is five, then this angle is also 42 degrees. The first check could be that these two angles are indeed equal. And the final check would be to see if 42 plus 42 plus 96 does indeed give 180 degrees, and it does. So we can give our answers that 𝑥 is equal to 41 and 𝑦 is equal to five.

Let’s have a look at another question.

Which of the following is true? Option (A) 𝐴𝐵 equals 𝐴𝐶, option (B) 𝐶𝐴 equals 𝐶𝐵, option (C) 𝐵𝐶 equals 𝐵𝐴.

In the diagram, we have a triangle 𝐴𝐵𝐶, we’ve got a line 𝐵𝐶, and we’ve got this ray 𝐵𝐴. In the options that we’re given, we’re really looking to see if there’s a pair of lines which are equal in length. We’re not given any hash marks on any of the lines, which would indicate that two are equal. So let’s have a look at the angles instead.

If we use the fact that the angles on a straight line add up to 180 degrees, then we should be able to work out this angle 𝐴𝐶𝐵 and this angle 𝐴𝐵𝐶. Starting with angle 𝐴𝐶𝐵, we can write that that’s equal to 180 degrees subtract 98 degrees. We can work this out by calculating 180 degrees subtract 100 degrees and then adding on two, which would give us a value of 82 degrees. We can add this value to the diagram. The next angle, angle 𝐴𝐵𝐶, must be equal to 180 degrees subtract 131 degrees. Subtracting 130 degrees and then another one degree would give us 49 degrees.

Now that we’ve found these two angles in this diagram, we might think that it’s not very helpful. But let’s see if we can have a look at calculating the other angle in the triangle 𝐴𝐵𝐶. We should remember that the angles in a triangle add up to 180 degrees. So we’ll have our unknown angle 𝐶𝐴𝐵 plus angle 𝐴𝐶𝐵, which we worked out as 82 degrees, plus the angle 𝐴𝐵𝐶, which we worked out as 49 degrees, all must add to give 180 degrees. This means angle 𝐶𝐴𝐵 plus 131 degrees equals 180 degrees. Subtracting 131 degrees from both sides of this equation gives us that angle 𝐶𝐴𝐵 is equal to 49 degrees. We can then add that onto our diagram and see if there’s anything to notice.

Well, we should hopefully see that we have in fact got two angles that are the same size. Both of these are 49 degrees. This means that we’ve got an isosceles triangle here in triangle 𝐴𝐵𝐶. Any triangle that has two equal angles must have two equal sides. And therefore, it’s an isosceles triangle. The two sides that are equal will be this side 𝐶𝐴 and this side 𝐶𝐵. We can then give our answer that 𝐶𝐴 equals 𝐶𝐵, which was the option given in option (B).

In the final question, we’ll find the area of an isosceles triangle.

Calculate the area of the triangle 𝐴𝐵𝐶.

The first thing we might note when we’re looking at this triangle is that it’s an isosceles triangle. We can tell this from the hash notation indicating that we have two sides of equal length, which therefore fits the definition of an isosceles triangle. In order to find the area of a triangle, we’ll need the formula that the area of a triangle is equal to half multiplied by the base multiplied by the perpendicular height.

If we were to try and immediately calculate this however, we’d have a problem. This value of 10 centimeters represents the slant height of the triangle but not the perpendicular height. The perpendicular height would look like this. We can even define it with the letter ℎ if we wish. As we’ve created two right triangles here, we might consider the Pythagorean theorem, which tells us that the square on the hypotenuse is equal to the sum of the squares on the other two sides.

So let’s consider this triangle on the left. We can say that this line from 𝐴 meets the line 𝐵𝐶 at point 𝑦. In order to use the Pythagorean theorem, we need to know the length of this line segment 𝐵𝑦. Now, you might think that it’s very clear that it’s six centimeters. But how can we be absolutely sure that it is six centimeters?

Let’s consider the two triangles. We’ve got triangle 𝐴𝐵𝑦 on the left and triangle 𝐴𝐶𝑦 on the right. If we consider a pair of sides, we know that side 𝐴𝐵 is equal to side 𝐴𝐶. Side 𝐴𝑦 is common to both triangles. And finally, angle 𝐴𝑦𝐵 is equal to angle 𝐴𝑦𝐶. They’re both 90 degrees. We can say then that triangle 𝐴𝐵𝑦 is congruent with triangle 𝐴𝐶𝑦 by using the right angle hypotenuse side congruency criterion.

You might not need to show that level of working in every question. But it’s good to demonstrate that it means that this length of 𝐵𝑦 is the same as the length of 𝑦𝐶. They’ll both be six centimeters. This working also proves an important property of isosceles triangles that the median to the base of an isosceles triangle is perpendicular to the base. In other words, this line from 𝐴 to 𝐵𝐶 connecting at the midpoint 𝑦 will be perpendicular to the base 𝐵𝐶.

Let’s continue with this question and apply the Pythagorean theorem. Using the triangle 𝐴𝐵𝑦, we can see that there’s a hypotenuse of 10 and the other two sides will be six and ℎ. So we write 10 squared is equal to six squared plus ℎ squared. Evaluating the squares, 100 is equal to 36 plus ℎ squared. Subtracting 36 from both sides gives us 64 is equal to ℎ squared. We should recognize that 64 is a perfect square. So when we take the square root, we’ll have ℎ is equal to eight. And the units will be the length units of centimeters.

In this question, remember that we’re finding the area, not just the perpendicular height. So we’ll use the area formula now. When we’re filling in our values for the base and the height, remember that we’re using this base of 12 centimeters, not six centimeters, and we’re multiplying that by a half and then by eight, the perpendicular height. We can simplify before we multiply to give us a value of 48. And as we’re working with an area, we’ll need square units. We can give our answer then that the area of triangle 𝐴𝐵𝐶 is 48 square centimeters.

We can now summarize what we’ve learnt in this video. Firstly, we saw that isosceles triangles have two sides equal in length. Secondly, we saw some terminology. The unequal side in an isosceles triangle is called the base. Next, we saw that because isosceles triangles have two sides equal in length, this means that they have two equal angles. The base angles will be equal in size. And finally, as we saw in our last question, the median to the base of an isosceles triangle is perpendicular to the base.