Congruent polygons are exactly the same size and exactly the same shape. All their sides are the same length and all of their angles have the same measure. They are identical. Show Definition: Polygons are congruent when they have the same number of sides, and all corresponding sides and interior angles are congruent. The polygons will have the same shape and size, but one may be a rotated, or be the mirror image of the other.
Polygons are congruent if they are equal in all respects:
One way to think about this is to imagine the polygons are made of cardboard. If you can move them, turn them over and stack them exactly on top of each other, then they are congruent. To see this, click on any polygon below. It will be flipped over, rotated and stacked on another as needed to demonstrate that they are congruent. Try this Click on 'Next' or 'Run'. Each polygon in turn will be flipped over, rotated and stacked on another as needed to show that it is congruent to it. Mathematically speaking, each operation being done on the polygons is one of three types:
Testing for CongruenceThere are four ways to test for congruence of polygons, depending on what you are given to start. See Testing Polygons for congruence.The three types of operation above are called 'transforms'. In effect, they transform a shape to another by changing it in some way - rotation, reflection and translation. What does this mean?If you have shown that two polygons are congruent, then you know that every property of the polygons is also identical. For example they will have the same area, perimeter, exterior angles, apothem etc.Other congruence topics
Congruent TrianglesCongruent Polygons
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In this explainer, we will learn how to identify congruent polygons and use their properties to find a missing side length or angle. Recall that polygons are two-dimensional shapes with straight sides. Each point where two sides of a polygon meet is called a vertex (the plural is βverticesβ). Recalling also that congruent angles are angles that have the same measure and congruent sides are sides that have the same length, we can give a definition of congruent polygons as follows. Two polygons are congruent if there is a correspondence between their vertices such that all corresponding interior angles and all corresponding sides are congruent. Conversely, if two polygons are congruent, then there is a correspondence between their vertices such that all corresponding interior angles and all corresponding sides are congruent. In other words, congruent polygons are polygons whose vertices and sides coincide exactly. Another way to think of this concept is that congruent polygons are polygons with the same shape and size, although they can be rotations, translations, or mirror images of each other. In order to prove that two polygons are congruent, we need to show that
On the other hand, if we are told that two polygons are congruent, this immediately implies that conditions (i) and (ii) must hold. Note that the polygons with the smallest number of sides (three) are triangles. There are special rules for proving the congruency of triangles, and these are covered in another lesson. Here, we start with an example about a type of four-sided polygon: the square. Are two squares congruent if the side length of one square is equal to the side length of the other? AnswerAll squares have four vertices, so we can always form a correspondence between the vertices of one and the vertices of another. To prove that two squares with the same side length are congruent, we need to show that all corresponding interior angles are congruent and all corresponding sides are congruent. We know that all squares have four equal interior angles of 90β (i.e., four right angles). This means that for every interior angle in one square, any corresponding interior angle in the other square must measure the same. Therefore, all corresponding interior angles are congruent. Now, checking the sides, we are told that the two squares have the same side length, which we can label as π. Since both squares have four sides of length π, then for every side in one square, any corresponding side in the other square must have the same length. Therefore, all corresponding sides are congruent. Since we have shown that all corresponding interior angles and all corresponding sides are congruent, this implies that the two squares themselves are congruent. We conclude that the answer to the question is yes, two squares are congruent if the side length of one is equal to the side length of the other. Before moving on to more complicated problems, we introduce a helpful piece of mathematical notation. For objects π and π, we write πβ π to mean that π and π are congruent. For polygons π΄π΅πΆ and πππ, the notation π΄π΅πΆβ πππ implies that the interior angle at vertex π΄ is congruent to the one at vertex π, the interior angle at vertex π΅ is congruent to the one at vertex π, and the interior angle at vertex πΆ is congruent to the one at vertex π. Furthermore, side π΄π΅ is congruent to side ππ, side π΅πΆ is congruent to side ππ, and side πΆπ΄ is congruent to side ππ. The same labeling convention applies to all congruent polygons, irrespective of their number of sides. For example, for two four-sided polygons, we would use the notation π΄π΅πΆπ·β ππππ. By using this notation, we can express detailed information about the properties of congruent polygons in a very concise way. In particular, the order of the vertex letters tells us which interior angle is congruent to which and also which side is congruent to which. Our next example shows how to apply this knowledge. The symbol β means that the two objects are congruent. Which statement is true?
AnswerThe diagram shows a four-sided polygon (or quadrilateral) split into two triangles that share the side π΄πΆ. From the wording of the question, we know that one of the four answer options is correct, so we may assume the two triangles are congruent. Recall that if two polygons are congruent, then there is a correspondence between their vertices such that all the corresponding interior angles and sides are congruent. By comparing the two triangles, we need to work out which interior angle is congruent to which. This will then enable us to use mathematical notation to describe the congruence relationship between the triangles, so that we can pick the correct answer option. The three different side lengths in each triangle are marked with either a single dash, a double dash, or with no dashes (the shared side). In β³π΄π΅πΆ, the interior angle at vertex π΄ (written β π΄) is between the shared side and the side with a single dash. Similarly, β π΅ is between the sides with single and double dashes and β πΆ is between the side with a double dash and the shared side. In β³π΄πΆπ·, tracking the interior angles in the same order by sides, we see that β πΆ is between the shared side and the side with a single dash, β π· is between the sides with single and double dashes, and β π΄ is between the side with a double dash and the shared side. These correspondences tell us that β π΄β³π΄π΅πΆβ β πΆβ³π΄πΆπ·,β π΅β³π΄π΅πΆβ β π·β³π΄πΆπ·,β πΆβ³π΄π΅πΆβ β π΄β³π΄πΆπ·,ofofofofofof as shown in the diagram below. We can express this congruence relationship by the notation β³π΄π΅πΆβ β³πΆπ·π΄, but this is not one of the four available answer options. Consequently, it is important to remember that there is more than one way to describe the same congruence relationship, depending on the vertex we start at and the direction of travel around the polygon. For instance, instead of starting at vertex π΄ in β³π΄π΅πΆ, we could have started at π΅ or πΆ, so the following three statements are equivalent: β³π΄π΅πΆβ β³πΆπ·π΄,β³π΅πΆπ΄β β³π·π΄πΆ,β³πΆπ΄π΅β β³π΄πΆπ·. Additionally, if we travel around the polygon in the opposite direction, we get three more equivalent statements: β³πΆπ΅π΄β β³π΄π·πΆ,β³π΅π΄πΆβ β³π·πΆπ΄,β³π΄πΆπ΅β β³πΆπ΄π·. We have now listed all possible congruence relationships between the two triangles. The only one from this list that appears as an answer option is β³π΅πΆπ΄β β³π·π΄πΆ, so statement B is correct. In the above example, we knew that the given triangles had a congruence relationship, but in many questions, we will be asked to check whether or not two polygons are congruent. Are the polygons shown congruent? AnswerRecall that two polygons are congruent if there is a correspondence between their vertices such that all the corresponding interior angles and sides are congruent. Therefore, if we can show that these conditions are satisfied, then the polygons must be congruent. From the diagram, the polygons πΆπ·πΈπΉ and ππππ are both parallelograms, so in theory, we can form a correspondence between their vertices. Starting with vertex πΆ of parallelogram πΆπ·πΈπΉ, it has an interior angle of 76β. Comparing with parallelogram ππππ, we see that the only possible corresponding vertices are π or π, so β πΆβ β πβ π.or Repeating this step for vertices π·, πΈ, and πΉ of πΆπ·πΈπΉ, we deduce that β π·β β πβ π,β πΈβ β πβ π,β πΉβ β πβ π.ororor Next, we compare side lengths. Starting with side πΆπ· of parallelogram πΆπ·πΈπΉ, we see that its length is marked with a single dash. Comparing with parallelogram ππππ, we see that the only possible corresponding sides are ππ or ππ, so πΆπ·β ππππ.or Repeating this process for the other sides of πΆπ·πΈπΉ, we get π·πΈβ ππππ,πΈπΉβ ππππ,πΉπΆβ ππππ.ororor This means we have a choice of correspondences, but to prove that the two parallelograms are congruent, it is sufficient to find one set of correspondences that works. Choosing β πΆβ β π implies that β π·β β π, β πΈβ β π, and β πΉβ β π. Therefore, our answer is yes, the two polygons are congruent, with πΆπ·πΈπΉβ ππππ. We can see this more clearly if we rotate the polygon πΆπ·πΈπΉ, as shown in the diagram below. Note that if we had chosen β πΆβ β π instead, it would follow that β π·β β π, β πΈβ β π, and β πΉβ β π. Again, the two polygons would be congruent, but this time with πΆπ·πΈπΉβ ππππ. This can be seen by rotating the polygon πΆπ·πΈπΉ as below. Once we have identified two polygons as being congruent, we can sometimes use their properties to find a missing side length or angle in geometric problems. Letβs look at an example of this type. Given that πππΎπβ π΄π΅πΆπ, find the measure of β πΎππΆ. AnswerRecall that congruent polygons are the same shape and size, but they can be rotations, translations, or mirror images of each other. We are told that πππΎπβ π΄π΅πΆπ, and from the diagram, we see that polygon πππΎπ is actually a reflection of polygon π΄π΅πΆπ in the straight line that passes through π, perpendicular to the line segment π΄π. As β πΎππ=53β with β πΆππ΄β β πΎππ, we deduce that β πΆππ΄=53β. Therefore, we know two of the three angles at π above the line segment π΄π. The missing angle is β πΎππΆ, which we have been asked to work out. Recalling the fact that angles on a straight line sum to 180β, we have the equation β πΆππ΄+β πΎππ+β πΎππΆ=180.β Subtracting β πΆππ΄ and β πΎππ from both sides gives β πΎππΆ=180ββ πΆππ΄ββ πΎππ,β and substituting the values β πΆππ΄=53β and β πΎππ=53β, we get β πΎππΆ=180β53β53=74.ββββ Thus, we have calculated that β πΎππΆ=74β. In our final example, we apply the properties of congruent polygons in a geometric context. The perimeter of the polygon π΄π΅πΆπ·πΈ is 176 cm and π΄π΅πΆπ·πΈβ πΉππΏπ·πΈ. Given that πΈββο©ο©ο©ο©βπ΄πΉ and π·πΈ=48cm, find the perimeter of the figure π΄π΅πΆπ·πΏππΉ. AnswerRecall that congruent polygons are the same shape and size, but they can be rotations, translations, or mirror images of each other. The question states that π΄π΅πΆπ·πΈβ πΉππΏπ·πΈ, and from the diagram, we see that polygon π΄π΅πΆπ·πΈ is a reflection of polygon πΉππΏπ·πΈ in the straight line containing the line segment π·πΈ. We have been asked to find the perimeter of the figure π΄π΅πΆπ·πΏππΉ, which is the new shape formed from the two original polygons by excluding the shared side π·πΈ, as shown below. In the question, the notation βο©ο©ο©ο©βπ΄πΉ tells us that π΄πΉ is a line segment, so the horizontal sides π΄πΈ of polygon π΄π΅πΆπ·πΈ and πΈπΉ of polygon πΉππΏπ·πΈ join to make the single horizontal side π΄πΉ of figure π΄π΅πΆπ·πΏππΉ. To calculate the perimeter of π΄π΅πΆπ·πΏππΉ, we need to add together the perimeters of π΄π΅πΆπ·πΈ and πΉππΏπ·πΈ, but in both cases, we must exclude the length of the side π·πΈ (we write this length as just π·πΈ). Then, writing π(π΄π΅πΆπ·πΏππΉ) for the perimeter of π΄π΅πΆπ·πΏππΉ and so on, we have π(π΄π΅πΆπ·πΏππΉ)=(π(π΄π΅πΆπ·πΈ)βπ·πΈ)+(π(πΉππΏπ·πΈ)βπ·πΈ)=π(π΄π΅πΆπ·πΈ)+π(πΉππΏπ·πΈ)β2Γπ·πΈ. From the question, the perimeter of the polygon π΄π΅πΆπ·πΈ is 176 cm, with π΄π΅πΆπ·πΈβ πΉππΏπ·πΈ. This implies that the perimeter of the polygon πΉππΏπ·πΈ is also 176 cm, so substituting these values into the above equation, we get π(π΄π΅πΆπ·πΏππΉ)=176+176β2Γπ·πΈ=352β2Γπ·πΈ. Finally, we know that π·πΈ=48cm, so substituting this value gives π(π΄π΅πΆπ·πΏππΉ)=352β2Γ48=352β96=256. All the lengths were given in centimeters, so the perimeter of the figure π΄π΅πΆπ·πΏππΉ is 256 cm. Let us finish by recapping some key concepts from this explainer.
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