### Video Transcript

In this video, we will learn how to
identify congruent polygons and use their properties to find a missing side length
or angle. So, letβs begin by thinking what
are congruent polygons.

We could say that polygons are
congruent when they have the same number of sides and all corresponding sides and
interior angles are congruent. In other words, we can say that
they are the same shape and size, but they can be rotated or a mirror image of each
other. So here, for example, we have two
rectangles. We can see that corresponding sides
are congruent. And each corresponding angle in the
first rectangle would be equal to that in the second rectangle, meaning that these
two rectangles are congruent. In order to demonstrate that two
polygons are congruent, we have to show that all corresponding sides and interior
angles are congruent.

There are some special congruency
rules for proving that triangles are congruent. Letβs have a look at those
next. The first rule that we can apply to
show that two triangles are congruent is the rule SSS, which stands for
side-side-side. Here, we have two triangles which
we can demonstrate are congruent using the SSS rule. Even though the second triangle has
been flipped, we can still see that corresponding sides are congruent. When the corresponding sides are
congruent, that means that the corresponding angles are also congruent.

The second rule is the SAS rule,
which stands for side-angle-side, where the angle is the included angle between the
two sides. Here, we can see an example of two
triangles which are congruent using the SAS rule. This rule would mean that the third
side in each triangle would also be congruent. So, this rule and the following
rules would show that the three sides are congruent.

The third congruency rule is ASA,
which stands for angle-side-angle, where the side is the included side between the
two angles. For example, our two triangles here
are congruent using this rule.

Our next rule is the
angle-angle-side rule or AAS. Here, we have to show that two
corresponding angles are congruent and any pair of corresponding sides are also
congruent in order to show that the two triangles are congruent.

Our final rule is a special-case
rule, which only applies to right triangles. It can be referred to as RHS,
standing for right angle-hypotenuse-side, or HL, which stands for hypotenuse and leg
in right triangles. In either format of this rule, we
have to show that the hypotenuse is congruent plus another side or leg of the
triangle in each triangle would be congruent and that there is a right angle.

Before we look at some example
questions, letβs go through some notation rules that we use for congruency. The first thing to note is that we
use this symbol, which looks like an equal sign with a wavy line above it, to mean
that two shapes are congruent. So, for example, we could say that
a rectangle π΄π΅πΆπ· is congruent to rectangle πΈπΉπΊπ», which brings us to the
second important point about the order of the letters used. Even without drawing out our
rectangles, we could use the order of the letters to say that the angle at π΄ must
be congruent to the angle at πΈ. Equally, the angle at π΅ would be
congruent to the angle at πΉ and the same for the remaining angles in each
rectangle.

We can also use the notation to
help us work out the congruent sides. For example, the side π΄π΅ is
congruent to side πΈπΉ and the side π΅πΆ is congruent to the side πΉπΊ. So, when weβre writing congruency
relationships, we need to pay careful attention to which sides and angles are
congruent. And when weβre given a congruency
relationship, we can use this to help us work out the corresponding sides and angles
that are congruent.

So, now, letβs have a look at some
questions involving congruent polygons.

The symbol congruent means that
the two objects are congruent. Which statement is true? Option A, triangle π΄π΅πΆ is
congruent to triangle πΆπ΄π·. Option B, triangle π΄π΅πΆ is
congruent to triangle π·π΄πΆ. Option C, triangle π΄πΆπ΅ is
congruent to triangle π·π΄πΆ. Or option D, triangle π΅πΆπ΄ is
congruent to triangle π·π΄πΆ.

We can see in the diagram that
we have a parallelogram which is split into two separate triangles. We can see from the markings
given that side π΄π΅ is congruent to side πΆπ·. We can see from the double mark
on the line that side π΄π· is congruent to side π΅πΆ. We can see that both triangles
share the side π΄πΆ. So, this means that π΄πΆ is
congruent to π΄πΆ. We can, therefore, say that our
two triangles are congruent, using the side-side-side congruency criterion.

In order to write a congruency
relationship between the two triangles, we need to be very careful about the
orders of the letters. In our left triangle, if we
were to travel from π΄ to π΅ and then from π΅ to πΆ, weβd be travelling from the
one marking to the two marking. Then, the equivalent journey in
our other triangle would be from πΆ to π· along the one marking and then from π·
to π΄ along the two marking. So, we could write the
relationship as triangle π΄π΅πΆ is congruent to triangle πΆπ·π΄.

Notice that we could also keep
the order of letters the same and write that triangle π΅πΆπ΄ is congruent to
triangle π·π΄πΆ. Or we could also say that
triangle πΆπ΄π΅ is congruent to triangle π΄πΆπ·. Any of these congruency
relationships would be a true statement. But only one of them appears in
our answer options. And thatβs option D, triangle
π΅πΆπ΄ is congruent to triangle π·π΄πΆ.

In our next question, weβll see an
example, where weβre given a congruency relationship and we need to find a missing
angle.

Given that triangle π΄π΅πΆ is
congruent to triangle πππ, find the measure of angle πΆ.

So, here, we have two congruent
triangles and weβre asked to work out the missing angle, πΆ. Here, we can use the congruency
statement to help us work out which corresponding angles would be congruent. The first angle we can look at
is angle π΄. And this will be congruent to
angle π in triangle πππ. And as weβre told that this
angle π is 40 degrees, this means that angle π΄ in triangle π΄π΅πΆ will also be
40 degrees.

We can also see that angle πΆ
in triangle π΄π΅πΆ is congruent to angle π in triangle πππ. But weβre not given an angle
measure for angle π. So, we canβt use this directly
to help us work out angle πΆ. Instead, we can use the fact
that the angles in a triangle add up to 180 degrees to find the measure of angle
πΆ. Therefore, the measure of angle
πΆ is equal to 180 degrees subtract 56 degrees and subtract 40 degrees, giving
us 84 degrees. And so, our final answer is
that the measure of angle πΆ is 84 degrees.

Given that πππΎπ is
congruent to π΄π΅πΆπ, find the measure of angle πΎππΆ.

In this question, we have two
congruent quadrilaterals, π΄π΅πΆπ on the left side of the diagram and πππΎπ
on the right. Weβre asked to find the measure
of angle πΎππΆ, which is outside of these quadrilaterals. If we knew the measure of this
angle, πΆππ΄, we could calculate the missing angle. We can use the congruency
statement to help us work out this angle. We could see, for example, that
angle π in the quadrilateral πππΎπ would be congruent with angle π΄ in the
quadrilateral π΄π΅πΆπ. So, therefore, the angle π in
quadrilateral π΄π΅πΆπ is congruent to the angle π in quadrilateral
πππΎπ.

So, therefore, the missing
angle πΆππ΄ in quadrilateral π΄π΅πΆπ would be 53 degrees. We can use the fact that the
angles on a straight line add up to 180 degrees to work out that our angle
πΎππΆ is equal to 180 degrees subtract 53 degrees subtract 53 degrees. And, therefore, the measure of
angle πΎππΆ is 74 degrees.

In the next question, weβll see an
example of how we can prove that two quadrilaterals are congruent.

Are the polygons shown
congruent?

We can remind ourselves that
the word congruent means the same shape and size. A better mathematical
description is that polygons are congruent if all corresponding sides and
interior angles are congruent. If we want to check if these
two quadrilaterals are congruent, we need to check all the corresponding sides
and angles to see if theyβre congruent or not.

So, if we start with our sides,
with side πΆπ· on our left quadrilateral, we can see from the one marking that
this is congruent with side length ππ on our quadrilateral ππππ. We can also see that the side
πΉπΈ on the quadrilateral πΆπ·πΈπΉ is congruent with side ππ on the
quadrilateral ππππ. We can see that side πΆπΉ is
congruent with side ππ and side π·πΈ is congruent to side ππ. So, weβve demonstrated that we
have four corresponding sets of congruent sides. However, this isnβt sufficient
to show that two polygons are congruent. After all, we could, for
example, have a rectangle and a parallelogram which have congruent sides. But these clearly arenβt the
same shape. So, we need to check the angles
in our polygons.

So, looking at angle πΆ in
quadrilateral πΆπ·πΈπΉ, we could say that this is congruent with angle π in
quadrilateral ππππ. Equally, angle π·, which is
labelled as 104 degrees, would be congruent with angle π, which is also 104
degrees. We can see that angle πΈ of 76
degrees is congruent with angle π of 76 degrees. And our final angle πΉ would be
congruent with angle π. So now, weβve shown that we
also have four corresponding sets of congruent angles. This fits with our definition
of congruent polygons. So, yes, these polygons are
congruent.

In our final question, weβll see
how we can use congruency to help us work out missing lengths in polygons.

The two quadrilaterals in the
given figure are congruent. Work out the perimeter of
π΄π΅πΆπ·.

In this question, weβre not
given a congruency statement to help us work out the corresponding congruent
sides. But we can apply a little bit
of logic here. We can begin by noticing that
these shapes are a reflection of each other. We can see that angle π΄ in our
quadrilateral π΄π΅πΆπ· would be congruent with angle πΈ in quadrilateral
πΈπΉπΊπ». Angle π΅ would be congruent
with angle π». Angle πΆ is congruent with
angle πΊ. And angle π· is congruent with
angle πΉ. We can, therefore, say that
π΄π΅πΆπ· is congruent to πΈπ»πΊπΉ.

In order to work out the
perimeter of π΄π΅πΆπ·, we need to find some of the missing sides on this
quadrilateral. We could see that the side π΅πΆ
would correspond with the side πΊπ», meaning that π΅πΆ would also be 4.2. The last unknown side π·πΆ is
corresponding with side πΉπΊ. So, it will be of length
three. Notice that as these two shapes
are congruent, this means that theyβll have the same perimeter. To find the perimeter of
π΄π΅πΆπ·, we add up the lengths around the outside. So, we have 4.2 plus 4.1 plus
1.4 plus three, which is equal to 12.7. And we werenβt given any units
in the question. So, we donβt have any in the
answer.

We can now summarize what weβve
learned in this video. We learned that polygons are
congruent when they have the same number of sides and all corresponding sides and
interior angles are congruent. Congruent polygons are the same
shape and size, but can be rotated or a mirror image of each other. We learned that there are special
congruency criterion for showing that two triangles are congruent. And finally, we learned the very
important fact about the ordering of letters in a congruency statement as this order
indicates the corresponding sides and angles which are congruent.