### Video Transcript

In this video, weโll learn how to
use the properties of isosceles triangles to find a missing angle or side
length. Weโll also look at how we can do
this when the given angles or sides are as algebraic expressions. Letโs 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
๐ด๐ต๐ถ, we can see from the hash marks that ๐ด๐ต is equal in length to ๐ด๐ถ. 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 ๐ต
and angle ๐ถ.

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? Letโs take our isosceles triangle
๐ด๐ต๐ถ and see if we can prove that there are two equal angles.

We start by drawing a median
๐ด๐ฆ. This is a line from ๐ด going to the
midpoint of line segment ๐ต๐ถ. Because ๐ฆ is a midpoint, we can
say that the line segment ๐ต๐ฆ is equal to the line segment ๐ฆ๐ถ. Next, because we have an isosceles
triangle, we know that thereโs two equal sides. In this case, we were given that
line segment ๐ด๐ต is equal to line segment ๐ด๐ถ.

Finally, as we consider the two
smaller triangles, triangle ๐ด๐ต๐ฆ and triangle ๐ด๐ถ๐ฆ, we know that they have a
common side, which is the line segment ๐ด๐ฆ. So as we consider our two
triangles, ๐ด๐ต๐ฆ on the left and triangle ๐ด๐ถ๐ฆ on the right, we can in fact say
that these two triangles are congruent. And thatโs because weโve
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 ๐ต is equal to the angle at ๐ถ, and so demonstrating the property that
these angles are always equal in an isosceles triangle. As we go through our questions on
isosceles triangles, weโll be using the fact that they have two equal sides and two
equal angles. So letโs have a look at our first
question.

Find the measure of angle
๐ด๐ต๐ถ.

In the diagram, we have a
triangle. Weโre given the angle ๐ถ๐ด๐ต as 58
degrees. And we need to work out this angle
of ๐ด๐ต๐ถ. We should notice that on this
diagram, we have two hash markings on the lines, indicating that thereโs two equal
sides. So we could say that triangle
๐ด๐ต๐ถ 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 ๐ด๐ต. And the base angles would be the
two angles adjacent to this base. But be careful because the base
isnโt always at the bottom of the diagram.

We can therefore write that angle
๐ด๐ต๐ถ is equal to angle ๐ต๐ด๐ถ. As weโre given that angle ๐ต๐ด๐ถ is
58 degrees, then so is angle ๐ด๐ต๐ถ. We can give our answer then that
the measure of angle ๐ด๐ต๐ถ is 58 degrees.

Letโs 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 ๐ด๐ธ๐ต is
isosceles?

Itโs probably most useful to begin
this question by highlighting the triangle ๐ด๐ธ๐ต that we want to consider. This triangle forms the roof of
this house, and weโre told that itโs 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 ๐ด๐ธ
is equal to the side ๐ด๐ต and the angle ๐ด๐ธ๐ต is equal to the angle ๐ด๐ต๐ธ.

Now that weโve had a look at the
diagram, letโs focus on what weโre asked, to find the angle that the roof makes with
the horizontal. That means that weโre 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 letโs see if we can work out one
of these angles, angle ๐ด๐ต๐ธ.

In order to do this, weโll 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 ๐ต๐ด๐ธ plus angle ๐ด๐ธ๐ต plus angle ๐ด๐ต๐ธ is equal to 180 degrees. Weโre given that angle ๐ต๐ด๐ธ is
107 degrees. So if we subtract 107 degrees from
both sides of this equation, we get that angle ๐ด๐ธ๐ต plus angle ๐ด๐ต๐ธ is equal to
73 degrees. As we have an isosceles triangle,
we know that our two base angles are equal. So angle ๐ด๐ธ๐ต is equal to angle
๐ด๐ต๐ธ.

We could think of this then that
two times angle ๐ด๐ต๐ธ is 73 degrees. And so to find angle ๐ด๐ต๐ธ, we
must divide both sides of this equation by two, which means that the measure of
angle ๐ด๐ต๐ธ 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 donโt worry. Weโll just need to apply some of
our algebra skills along with the properties that we know of isosceles
triangles.

Find the values of ๐ฅ and ๐ฆ.

In this triangle question, the
important thing to notice is these markings on the line, which indicate that itโs 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, itโs these two base
angles of nine ๐ฆ minus three degrees and ๐ฅ plus one degrees.

We might begin trying to solve this
by writing the equation that nine ๐ฆ minus three degrees equals ๐ฅ plus one
degrees. However, in order to solve an
equation like this that has two unknowns, the ๐ฅ and ๐ฆ, weโd need another equation
linking ๐ฅ and ๐ฆ, which we donโt have. So letโs try another approach. This time, weโll 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 ๐ฆ minus three and ๐ฅ plus one. If we write that 96 degrees plus
nine ๐ฆ minus three degrees plus ๐ฅ 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 ๐ฆ minus three degrees plus ๐ฅ 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 ๐ฆ minus three degrees must
be 42 degrees, and ๐ฅ plus one degrees must be 42 degrees.

Writing these two equations and
solving them would give us the values of ๐ฅ and ๐ฆ. Beginning with nine ๐ฆ minus three
degrees equals 42 degrees, we could add three to both sides. So nine ๐ฆ is equal to 45. Dividing both sides by nine would
give us that ๐ฆ equals five. In order to solve the second
equation, ๐ฅ plus one degrees equals 42 degrees, we just need to subtract one from
both sides. So ๐ฅ is equal to 41.

Before we finish, itโs always worth
checking our values. If ๐ฅ is 41, then this angle is 42
degrees. If ๐ฆ 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 ๐ฅ
is equal to 41 and ๐ฆ is equal to five.

Letโs have a look at another
question.

Which of the following is true? Option (A) ๐ด๐ต equals ๐ด๐ถ, option
(B) ๐ถ๐ด equals ๐ถ๐ต, option (C) ๐ต๐ถ equals ๐ต๐ด.

In the diagram, we have a triangle
๐ด๐ต๐ถ, weโve got a line ๐ต๐ถ, and weโve got this ray ๐ต๐ด. In the options that weโre given,
weโre really looking to see if thereโs a pair of lines which are equal in
length. Weโre not given any hash marks on
any of the lines, which would indicate that two are equal. So letโs 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 ๐ด๐ถ๐ต and this angle ๐ด๐ต๐ถ. Starting with angle ๐ด๐ถ๐ต, we can
write that thatโs 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 ๐ด๐ต๐ถ, must
be equal to 180 degrees subtract 131 degrees. Subtracting 130 degrees and then
another one degree would give us 49 degrees.

Now that weโve found these two
angles in this diagram, we might think that itโs not very helpful. But letโs see if we can have a look
at calculating the other angle in the triangle ๐ด๐ต๐ถ. We should remember that the angles
in a triangle add up to 180 degrees. So weโll have our unknown angle
๐ถ๐ด๐ต plus angle ๐ด๐ถ๐ต, which we worked out as 82 degrees, plus the angle ๐ด๐ต๐ถ,
which we worked out as 49 degrees, all must add to give 180 degrees. This means angle ๐ถ๐ด๐ต plus 131
degrees equals 180 degrees. Subtracting 131 degrees from both
sides of this equation gives us that angle ๐ถ๐ด๐ต is equal to 49 degrees. We can then add that onto our
diagram and see if thereโs 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 weโve got an
isosceles triangle here in triangle ๐ด๐ต๐ถ. Any triangle that has two equal
angles must have two equal sides. And therefore, itโs an isosceles
triangle. The two sides that are equal will
be this side ๐ถ๐ด and this side ๐ถ๐ต. We can then give our answer that
๐ถ๐ด equals ๐ถ๐ต, which was the option given in option (B).

In the final question, weโll find
the area of an isosceles triangle.

Calculate the area of the triangle
๐ด๐ต๐ถ.

The first thing we might note when
weโre looking at this triangle is that itโs 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, weโll 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, weโd 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 โ if we wish. As weโve 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 letโs consider this triangle on
the left. We can say that this line from ๐ด
meets the line ๐ต๐ถ at point ๐ฆ. In order to use the Pythagorean
theorem, we need to know the length of this line segment ๐ต๐ฆ. Now, you might think that itโs very
clear that itโs six centimeters. But how can we be absolutely sure
that it is six centimeters?

Letโs consider the two
triangles. Weโve got triangle ๐ด๐ต๐ฆ on the
left and triangle ๐ด๐ถ๐ฆ on the right. If we consider a pair of sides, we
know that side ๐ด๐ต is equal to side ๐ด๐ถ. Side ๐ด๐ฆ is common to both
triangles. And finally, angle ๐ด๐ฆ๐ต is equal
to angle ๐ด๐ฆ๐ถ. Theyโre both 90 degrees. We can say then that triangle
๐ด๐ต๐ฆ is congruent with triangle ๐ด๐ถ๐ฆ by using the right angle hypotenuse side
congruency criterion.

You might not need to show that
level of working in every question. But itโs good to demonstrate that
it means that this length of ๐ต๐ฆ is the same as the length of ๐ฆ๐ถ. Theyโll 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 ๐ด
to ๐ต๐ถ connecting at the midpoint ๐ฆ will be perpendicular to the base ๐ต๐ถ.

Letโs continue with this question
and apply the Pythagorean theorem. Using the triangle ๐ด๐ต๐ฆ, we can
see that thereโs a hypotenuse of 10 and the other two sides will be six and โ. So we write 10 squared is equal to
six squared plus โ squared. Evaluating the squares, 100 is
equal to 36 plus โ squared. Subtracting 36 from both sides
gives us 64 is equal to โ squared. We should recognize that 64 is a
perfect square. So when we take the square root,
weโll have โ is equal to eight. And the units will be the length
units of centimeters.

In this question, remember that
weโre finding the area, not just the perpendicular height. So weโll use the area formula
now. When weโre filling in our values
for the base and the height, remember that weโre using this base of 12 centimeters,
not six centimeters, and weโre 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 weโre working with an area,
weโll need square units. We can give our answer then that
the area of triangle ๐ด๐ต๐ถ is 48 square centimeters.

We can now summarize what weโve
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.