6.5: Congruent Polygons
Introduction
Geodesic Trouble
After doing all of his research and drawing a design, Dylan began working on the construction of his geodesic dome. He decided to use a combination of rolled newspaper tubes and duct tape. He rolled tubes of newspaper, created triangles with duct tape and then worked on connecting them together.
“It doesn’t look right,” Sarah, Dylan’s sister commented as he was putting the structure together in the living room.
“What do you mean?” Dylan asked tearing off another piece of duct tape.
“It is crooked and I think it will collapse.”
“You don’t know anything,” Dylan snapped turning his back on his sister.
However, when Dylan actually went to connect the triangles together, the structure began to collapse. His sister came back into the room.
“Can I help?” she asked.
“Maybe.”
“Are the triangles congruent?” Sarah asked.
Congruent? Dylan had to think about that one. What would it mean if the triangles weren’t congruent? What does it mean “congruent?” How can one tell if a figure is congruent or not?
In this lesson, you will learn all about the importance of congruence and how to determine congruence.
What You Will Learn
In this lesson you will learn how to do the following skills.
- Recognize congruence of segments and angles, and name congruent polygons by vertices.
- Name all pairs of congruent corresponding parts of given congruent polygons.
- Recognize and apply side–side–side (SSS), side–angle–side (SAS) and angle–side–angle (ASA) rules for triangle congruence.
- Identify congruence of polygons found in real – world objects and architecture.
Teaching Time
I. Recognize Congruence of Segments and Angles, and Name Congruent Polygons
What do we mean when we say that two figures are congruent? To complete all of the work in this lesson, you must first understand what the word “congruent” means. Congruent means exactly the same.
Yes, but it also includes sides of figures and angles too. When we have two figures of any kind that have the same size, shape and measure, we can say that these two figures are congruent.
Let’s look at an example.
Example
Are these two angles congruent?
If you look at these two angles, you will see that they are both \begin{align*}55^\circ\end{align*}. The angles are labeled that way, so we can see that they are equal. This means that they are congruent. We can say that angle \begin{align*}A\end{align*} is congruent to angle \begin{align*}B\end{align*}.
\begin{align*}\angle A \cong \angle B\end{align*}
This is the way we would write a statement about congruence using mathematical notation. Notice the symbol that we used for congruent.
Now write the definition for congruent and its symbol in your notebook.
Just as we said that any two figures can be congruent, we can use this when we look at different types of polygons too. Two polygons can be considered congruent or not congruent. Let’s look at an example and see if we can determine congruence.
Example
Are these two octagons congruent?
Look at these two octagons. They are exactly the same in every way. You can see that if we put one octagon on top of the other octagon that they would match up perfectly. The side lengths are also congruent and the angle measures are congruent. If two polygons are congruent, then it is a given that the side lengths and the angle measures are also congruent.
Example
Are these two hexagons congruent?
These two figures are both hexagons, but they are different hexagons. One is a regular hexagon where all of the sides are congruent, and one is irregular. The irregular hexagon has six sides, but they are different lengths, etc. These two hexagons are not congruent.
II. Name all Pairs of Congruent Corresponding Parts of Given Congruent Polygons
Now that you know how to identify whether or not two figures are congruent, we can look at figuring out congruent parts and angles. First, let’s think again about the four characteristics of congruent polygons.
Congruent Polygons have:
- Same size
- Same shape
- Common angle measures
- Common side lengths
Be sure that you have these notes written down in your notebook.
The last two characteristics can be a bit tricky. Sometimes, you will have two congruent figures, but all of the angles measures won’t be exactly the same. For example, if you had two irregular congruent hexagons, that means that there are different angle measures in the two hexagons - however, they are congruent so there are “matching” angles between the two figures.
Look at this example.
Example
Look at this example. We have two hexagons. They are irregular - which means that all of the side lengths and angles are not the same. However, they are congruent. You can see that one matches the other. Because of this, we have corresponding angles that connect with each angle from the first hexagon to the second hexagon.
We can identify corresponding parts of congruent figures. Corresponding parts can include side lengths and angle measures. When two figures are congruent, then there are corresponding parts.
Let’s look at applying this to our work.
Example
Name each pair of corresponding side lengths for these congruent figures.
Now let’s look at these two congruent pentagons. To name the corresponding sides, we name the sides that match from one pentagon to another pentagon. Here are the corresponding sides and how we can write them using mathematical notation.
\begin{align*}BA & \cong GF\\ BC & \cong GH\\ CD & \cong HK\\ AE & \cong FL\\ ED & \cong LK\end{align*}
We can also look at the corresponding angles for two congruent figures. When two figures are congruent, then the matching angles will also be congruent.
Example
Now when we look at these two parallelograms, we can see that they are congruent. They are the same size and shape, and we can see that all four angles in the first parallelogram have been measured and labeled. We can find the corresponding angle for each of the angles in the second parallelogram by using the angle measures of the first parallelogram.
III. Recognize and Apply Side-Side-Side (SSS), Side – Angle Side (SAS) and Angle – Side – Angle (ASA) Rules for Triangle Congruence
We just worked on identifying the congruent sides and angles of polygons. If the sides and the angles of any two polygons are congruent, then we know that the two polygons are also congruent.
We can also work with triangles, because after all a triangle is also a polygon. Triangles are unique because there are a few rules that we can use to help us to identify whether or not two triangles are congruent. You will notice that if you learn these rules, that you won’t have to compare every angle and every side to determine whether or not two triangles are congruent.
The first rule represents the side-side-side, or SSS, relationship. It says that if each of the three sides of one triangle is congruent to a side in a second triangle, then the two triangles are congruent. We do not need to check the angles. In triangles, angles are always in a fixed relationship with the side opposite them. The wider the angle, the longer the side opposite it must be. If we know the sides are congruent, then we know the angles must be also.
The second rule says that if one angle and the sides adjacent to it in one triangle are congruent to an angle and its adjacent sides in the second triangle, the triangles will be congruent. We call this the side-angle-side (SAS) rule. In other words, this time we only need to make sure one angle and two sides are congruent to know that all parts of the triangles are congruent. However, remember that the angle you use must be located between the two sides.
The third rule tells us that if two angles and the side between them in one triangle are congruent to two angles and the side between them in the second triangle, the two triangles are congruent. This is the angle-side-angle (ASA) rule. We can determine congruence just by knowing two angles and one side.
Alright, here is an example.
Example
Are the triangles below congruent? Explain your reasoning.
Now let’s start by looking at the triangles and looking for the given information. We know that we need three side measurements for SSS, or a side measurement, an angle measurement and a side measurement for SAS, or an angle side angle measurement for ASA.
We know that two pairs of sides match: one pair is 7.5 centimeters and the other is 4 centimeters. This is not enough information to know for sure that the triangles themselves are congruent, because the sides may be in different places. Which rule can we use to check for congruence? We cannot use SSS because we only know the lengths of two sides.
We can use SAS, however. Remember, to use SAS, the angle must be between the two sides. In the first triangle, the angle between the two sides is a right angle, so we know that it is \begin{align*}90^\circ\end{align*}. Does the second triangle also have a right angle? It sure does, and the right angle is between the 7.5 and 4 centimeter sides. Using the SAS rule, we can compare the triangles: 7.5 centimeters (side), \begin{align*}90^\circ\end{align*} (angle), 4 centimeters (side). The triangles must be congruent.
Example
Are these two triangles congruent? Explain your reasoning.
Here we have two triangles with given side lengths. We can see that these two triangles are congruent because their side lengths are congruent. The side lengths of both triangles are labeled and we can prove they are congruent by applying the SSS rule.
You can see how helpful these rules are in thinking about triangles and their congruence.
IV. Identify the Congruence of Polygons Found in Real – World Objects and Architecture
We can see congruent polygons in the world around us all the time. If you think about it, the idea of congruence is very important. If you build steps and each rise of the step is not congruent to the next step, then you will have a very crooked set of stairs. This set of stairs can be dangerous to climb up and down.
You will also see congruent polygons in the construction of bridges. Look at this picture of a truss bridge and you can see the importance of the congruent triangles in supporting the structure of the bridge.
Look around you in the world. If you look all around you, you will see many more examples of congruent polygons. Next time you go to the skatepark or hang around on a sports field look around - what would happen if the goal posts of a football stadium weren’t congruent? Or if a soccer field wasn’t designed so that one part is congruent to the other?
Now you will begin to see how important congruent figures are in our world.
Real-Life Example Completed
Geodesic Trouble
Reread this problem. When finished, define congruent and explain why congruent triangles are necessary to have a structure stay balanced. Then give at least one way that Dylan could check and make sure his triangles are congruent. There are three parts to your answer.
After doing all of his research and drawing a design, Dylan began working on the construction of his geodesic dome. He decided to use a combination of rolled newspaper tubes and duct tape. He rolled tubes of newspaper, created triangles with duct tape and then worked on connecting them together.
“It doesn’t look right,” Sarah, Dylan’s sister commented as he was putting the structure together in the living room.
“What do you mean?” Dylan asked tearing off another piece of duct tape.
“It is crooked and I think it will collapse.”
“You don’t know anything,” Dylan snapped turning his back on his sister.
However, when Dylan actually went to connect the triangles together, the structure began to collapse. His sister came back into the room.
“Can I help?” she asked.
“Maybe.”
“Are the triangles congruent?” Sarah asked.
Remember there are three parts to your answer.
Solution to Real – Life Example
First, let’s think about what the word congruent means. Congruent means exactly the same. For an object to be congruent, the side lengths have to be the same. The triangles in the geodesic dome have to be congruent for it to stand up because the triangle is a structure that is well balanced to help with structure and security. Triangles are used in all kinds of construction like roofs and bridges.
Dylan can test the congruence of his triangles in a couple of different ways. First, he can be sure that the side lengths are the same. This is the side-side-side method of testing congruence. He can also test side-angle-side measurements of each triangle to determine congruence.
Vocabulary
Here are the vocabulary words found in this lesson.
- Congruent
- exactly the same, having the same size, shape and measurement.
- Corresponding parts
- When two figures are congruent, there are matching parts for each of the two figures.
- SSS
- determining triangle congruence by comparing the three side lengths of two triangles.
- SAS
- determining triangle congruence by comparing the side, angle, side of two triangles.
- ASA
- determining triangle congruence by comparing the angle, side, angle of two triangles.
Time to Practice
Directions: Answer each question true or false.
- Congruent means that a figure has the same side lengths but not the same angle measures.
- Congruent means exactly the same in every measure.
- Similar means having the same shape, but not the same size.
- Two congruent figures would have the same size and shape.
- Corresponding parts are parts that are in the same figure.
- You need to understand corresponding parts before you can determine if two figures are congruent.
- You can determine if two figures are congruent without knowing any of their measurements.
- Similar figures are also congruent.
- If two triangles are equilateral triangles, then they are automatically congruent.
- If two quadrilaterals have measures of \begin{align*}360^\circ\end{align*}, then they are congruent.
Directions: Use the given information to figure out each congruence statement.
\begin{align*}\Delta ABC \cong \Delta DEF\end{align*}
- \begin{align*}\angle A \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\angle B \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\angle C \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\overline{AB} \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\overline{BC} \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\overline{AC} \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- If line segment \begin{align*}AC\end{align*} has a length of 8, which other segment also has a length of 8?
- If angle \begin{align*}A\end{align*} has a measure of \begin{align*}55^\circ\end{align*}, which other angle has a measure of \begin{align*}55^\circ\end{align*}?
- If angle \begin{align*}B\end{align*} has a measure of \begin{align*}45^\circ\end{align*}, which other angle has a measure congruent to that?
- If these two triangles are congruent, are the side lengths and angle measures the same?
- Would these two triangles look identical?
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