Lesson Video: Congruent Polygons Mathematics • 8th Grade

In this video, we will learn how to identify congruent polygons and use their properties to find a missing side length or angle.

13:33

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, letβ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. Letβ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, letβ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 π΄π΅πΆπ· is congruent to rectangle πΈπΉπΊπ», 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 πΈ. Equally, the angle at π΅ would be congruent to the angle at πΉ 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 πΈπΉ and the side π΅πΆ is congruent to the side πΉπΊ. So, when weβre writing congruency relationships, we need to pay careful attention to which sides and angles are congruent. And when weβre given a congruency relationship, we can use this to help us work out the corresponding sides and angles that are congruent.

So, now, letβ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 π΄π΅πΆ is congruent to triangle πΆπ΄π·. Option B, triangle π΄π΅πΆ is congruent to triangle π·π΄πΆ. Option C, triangle π΄πΆπ΅ is congruent to triangle π·π΄πΆ. Or option D, triangle π΅πΆπ΄ is congruent to triangle π·π΄πΆ.

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 πΆπ·. We can see from the double mark on the line that side π΄π· is congruent to side π΅πΆ. We can see that both triangles share the side π΄πΆ. So, this means that π΄πΆ is congruent to π΄πΆ. 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 πΆ, weβd be travelling from the one marking to the two marking. Then, the equivalent journey in our other triangle would be from πΆ 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΅πΆ is congruent to triangle πΆπ·π΄.

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

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

Given that triangle π΄π΅πΆ is congruent to triangle πππ, find the measure of angle πΆ.

So, here, we have two congruent triangles and weβre asked to work out the missing angle, πΆ. 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 πππ. And as weβre told that this angle π is 40 degrees, this means that angle π΄ in triangle π΄π΅πΆ will also be 40 degrees.

We can also see that angle πΆ in triangle π΄π΅πΆ is congruent to angle π in triangle πππ. But weβre not given an angle measure for angle π. So, we canβt use this directly to help us work out angle πΆ. Instead, we can use the fact that the angles in a triangle add up to 180 degrees to find the measure of angle πΆ. Therefore, the measure of angle πΆ 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 πΆ is 84 degrees.

Given that πππΎπ is congruent to π΄π΅πΆπ, find the measure of angle πΎππΆ.

In this question, we have two congruent quadrilaterals, π΄π΅πΆπ on the left side of the diagram and πππΎπ on the right. Weβre asked to find the measure of angle πΎππΆ, which is outside of these quadrilaterals. If we knew the measure of this angle, πΆππ΄, 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 πππΎπ would be congruent with angle π΄ in the quadrilateral π΄π΅πΆπ. So, therefore, the angle π in quadrilateral π΄π΅πΆπ is congruent to the angle π in quadrilateral πππΎπ.

So, therefore, the missing angle πΆππ΄ in quadrilateral π΄π΅πΆπ 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 πΎππΆ is equal to 180 degrees subtract 53 degrees subtract 53 degrees. And, therefore, the measure of angle πΎππΆ is 74 degrees.

In the next question, weβ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 theyβre congruent or not.

So, if we start with our sides, with side πΆπ· on our left quadrilateral, we can see from the one marking that this is congruent with side length ππ on our quadrilateral ππππ. We can also see that the side πΉπΈ on the quadrilateral πΆπ·πΈπΉ is congruent with side ππ on the quadrilateral ππππ. We can see that side πΆπΉ is congruent with side ππ and side π·πΈ is congruent to side ππ. So, weβve demonstrated that we have four corresponding sets of congruent sides. However, this isnβ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 arenβt the same shape. So, we need to check the angles in our polygons.

So, looking at angle πΆ in quadrilateral πΆπ·πΈπΉ, we could say that this is congruent with angle π in quadrilateral ππππ. 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 πΈ of 76 degrees is congruent with angle π of 76 degrees. And our final angle πΉ would be congruent with angle π. So now, weβ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, weβ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 π΄π΅πΆπ·.

In this question, weβ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 π΄ in our quadrilateral π΄π΅πΆπ· would be congruent with angle πΈ in quadrilateral πΈπΉπΊπ». Angle π΅ would be congruent with angle π». Angle πΆ is congruent with angle πΊ. And angle π· is congruent with angle πΉ. We can, therefore, say that π΄π΅πΆπ· is congruent to πΈπ»πΊπΉ.

In order to work out the perimeter of π΄π΅πΆπ·, we need to find some of the missing sides on this quadrilateral. We could see that the side π΅πΆ would correspond with the side πΊπ», meaning that π΅πΆ would also be 4.2. The last unknown side π·πΆ is corresponding with side πΉπΊ. So, it will be of length three. Notice that as these two shapes are congruent, this means that theyβll have the same perimeter. To find the perimeter of π΄π΅πΆπ·, 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 werenβt given any units in the question. So, we donβt have any in the answer.

We can now summarize what weβ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.