6.10: Recognizing and Understanding Congruent Polygons
Have you ever built a geodesic dome? Take a look at this dilemma.
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 Concept, you will learn all about the importance of congruence and how to determine congruence.
Guidance
What do we mean when we say that two figures are congruent? To complete all of the work in this Concept, 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.
Take a look at this situation.
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 and see if we can determine congruence.
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.
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.
Take a look at these figures.
Here 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.
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.
Use the following figures to answer each question.
Example A
What is the angle measure of angle F?
Solution: \begin{align*}100^\circ\end{align*}
Example B
Angle D is congruent to which other two angles?
Solution: F and H
Example C
What is the measure of angle G?
Solution: \begin{align*}80^\circ\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
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 because he can see that the side lengths are the same.
Vocabulary
- 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.
Guided Practice
Here is one for you to try on your own.
Are these two hexagons congruent?
Solution
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.
Video Review
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:The two figures shown are congruent. Use the illustration to answer each question.
11. If angle B has a measure of \begin{align*}75^\circ\end{align*}, which other angle has the same measure?
12. If angle F is \begin{align*}120^\circ\end{align*}, which other angle has the same measure?
13. True or false. Angle E and angle K have the same measure.
14. True or false. Angle C and angle H have the same measure.
15. Name this figure.
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Image Attributions
Here you'll recognize and understand congruent polygons.