Video: Congruent Polygons

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, let’s 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. Let’s 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, let’s 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 𝐴𝐵𝐶𝐷 is congruent to rectangle 𝐸𝐹𝐺𝐻, 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 𝐴 must be congruent to the angle at 𝐸. Equally, the angle at 𝐵 would be congruent to the angle at 𝐹 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 𝐴𝐵 is congruent to side 𝐸𝐹 and the side 𝐵𝐶 is congruent to the side 𝐹𝐺. So, when we’re writing congruency relationships, we need to pay careful attention to which sides and angles are congruent. And when we’re given a congruency relationship, we can use this to help us work out the corresponding sides and angles that are congruent.

So, now, let’s 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 𝐴𝐵𝐶 is congruent to triangle 𝐶𝐴𝐷. Option B, triangle 𝐴𝐵𝐶 is congruent to triangle 𝐷𝐴𝐶. Option C, triangle 𝐴𝐶𝐵 is congruent to triangle 𝐷𝐴𝐶. Or option D, triangle 𝐵𝐶𝐴 is congruent to triangle 𝐷𝐴𝐶.

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 𝐴𝐵 is congruent to side 𝐶𝐷. We can see from the double mark on the line that side 𝐴𝐷 is congruent to side 𝐵𝐶. We can see that both triangles share the side 𝐴𝐶. So, this means that 𝐴𝐶 is congruent to 𝐴𝐶. 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 𝐴 to 𝐵 and then from 𝐵 to 𝐶, we’d be travelling from the one marking to the two marking. Then, the equivalent journey in our other triangle would be from 𝐶 to 𝐷 along the one marking and then from 𝐷 to 𝐴 along the two marking. So, we could write the relationship as triangle 𝐴𝐵𝐶 is congruent to triangle 𝐶𝐷𝐴.

Notice that we could also keep the order of letters the same and write that triangle 𝐵𝐶𝐴 is congruent to triangle 𝐷𝐴𝐶. Or we could also say that triangle 𝐶𝐴𝐵 is congruent to triangle 𝐴𝐶𝐷. Any of these congruency relationships would be a true statement. But only one of them appears in our answer options. And that’s option D, triangle 𝐵𝐶𝐴 is congruent to triangle 𝐷𝐴𝐶.

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

Given that triangle 𝐴𝐵𝐶 is congruent to triangle 𝑋𝑌𝑍, find the measure of angle 𝐶.

So, here, we have two congruent triangles and we’re asked to work out the missing angle, 𝐶. 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 𝐴. And this will be congruent to angle 𝑋 in triangle 𝑋𝑌𝑍. And as we’re told that this angle 𝑋 is 40 degrees, this means that angle 𝐴 in triangle 𝐴𝐵𝐶 will also be 40 degrees.

We can also see that angle 𝐶 in triangle 𝐴𝐵𝐶 is congruent to angle 𝑍 in triangle 𝑋𝑌𝑍. But we’re not given an angle measure for angle 𝑍. So, we can’t use this directly to help us work out angle 𝐶. Instead, we can use the fact that the angles in a triangle add up to 180 degrees to find the measure of angle 𝐶. Therefore, the measure of angle 𝐶 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 𝐶 is 84 degrees.

Given that 𝑋𝑌𝐾𝑀 is congruent to 𝐴𝐵𝐶𝑀, find the measure of angle 𝐾𝑀𝐶.

In this question, we have two congruent quadrilaterals, 𝐴𝐵𝐶𝑀 on the left side of the diagram and 𝑋𝑌𝐾𝑀 on the right. We’re asked to find the measure of angle 𝐾𝑀𝐶, which is outside of these quadrilaterals. If we knew the measure of this angle, 𝐶𝑀𝐴, 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 𝑋 in the quadrilateral 𝑋𝑌𝐾𝑀 would be congruent with angle 𝐴 in the quadrilateral 𝐴𝐵𝐶𝑀. So, therefore, the angle 𝑀 in quadrilateral 𝐴𝐵𝐶𝑀 is congruent to the angle 𝑀 in quadrilateral 𝑋𝑌𝐾𝑀.

So, therefore, the missing angle 𝐶𝑀𝐴 in quadrilateral 𝐴𝐵𝐶𝑀 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 𝐾𝑀𝐶 is equal to 180 degrees subtract 53 degrees subtract 53 degrees. And, therefore, the measure of angle 𝐾𝑀𝐶 is 74 degrees.

In the next question, we’ll 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 they’re congruent or not.

So, if we start with our sides, with side 𝐶𝐷 on our left quadrilateral, we can see from the one marking that this is congruent with side length 𝑂𝑃 on our quadrilateral 𝑂𝑃𝑀𝑁. We can also see that the side 𝐹𝐸 on the quadrilateral 𝐶𝐷𝐸𝐹 is congruent with side 𝑀𝑁 on the quadrilateral 𝑃𝑀𝑁𝑂. We can see that side 𝐶𝐹 is congruent with side 𝑃𝑀 and side 𝐷𝐸 is congruent to side 𝑂𝑁. So, we’ve demonstrated that we have four corresponding sets of congruent sides. However, this isn’t 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 aren’t the same shape. So, we need to check the angles in our polygons.

So, looking at angle 𝐶 in quadrilateral 𝐶𝐷𝐸𝐹, we could say that this is congruent with angle 𝑀 in quadrilateral 𝑃𝑀𝑁𝑂. Equally, angle 𝐷, which is labelled as 104 degrees, would be congruent with angle 𝑁, which is also 104 degrees. We can see that angle 𝐸 of 76 degrees is congruent with angle 𝑂 of 76 degrees. And our final angle 𝐹 would be congruent with angle 𝑃. So now, we’ve 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, we’ll 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 𝐴𝐵𝐶𝐷.

In this question, we’re 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 𝐴 in our quadrilateral 𝐴𝐵𝐶𝐷 would be congruent with angle 𝐸 in quadrilateral 𝐸𝐹𝐺𝐻. Angle 𝐵 would be congruent with angle 𝐻. Angle 𝐶 is congruent with angle 𝐺. And angle 𝐷 is congruent with angle 𝐹. We can, therefore, say that 𝐴𝐵𝐶𝐷 is congruent to 𝐸𝐻𝐺𝐹.

In order to work out the perimeter of 𝐴𝐵𝐶𝐷, we need to find some of the missing sides on this quadrilateral. We could see that the side 𝐵𝐶 would correspond with the side 𝐺𝐻, meaning that 𝐵𝐶 would also be 4.2. The last unknown side 𝐷𝐶 is corresponding with side 𝐹𝐺. So, it will be of length three. Notice that as these two shapes are congruent, this means that they’ll have the same perimeter. To find the perimeter of 𝐴𝐵𝐶𝐷, 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 weren’t given any units in the question. So, we don’t have any in the answer.

We can now summarize what we’ve 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.