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Congruence, Proof, and Constructions

Rigid Motion and Congruence

A method used to determine whether two segments or angles are congruent is to find and compare their measures. Additionally, their congruence can be determined through the use of rigid motions. Engage with this lesson to learn how this is done.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Identifying Congruent Figures

In the following applet, four figures are shown. One of the three figures at the bottom is the result of applying a rigid motion or a sequence of rigid motions to the upper figure. Match the preimage with its image.

Applet to pick the correct image

Discussion

Congruence and Rigid Motions

When working with the different types of rigid motions, it was used that they map a segment into another segment. To verify that this is true, consider a segment and any rigid motion. Let be the image of and the image of

Segments AB and CD

Because rigid motions preserve distances, is equal to Now, to check that every point of was actually mapped onto consider a point on different from the endpoints. Let be the image of under the rigid motion.

Point P on AB and point Q not on CD
The idea is to show that lies on between and To do this, it must be checked that is equal to Again, since rigid motions preserve distances, and Next, Equations (I) and (II) can be added together and simplified using the Segment Addition Postulate.
Solve by elimination

Segment Addition Postulate

Consequently, lies on which proves that the image of every point of lies on Therefore, rigid motions map segments into segments.
Interactive graph where P can be moved along AB and Q is moved accordingly on CD
Not only can rigid motions map segments into segments, but rigid motions also map angles into congruent angles. Because of these two properties, to find only the image of the vertices provides enough information to map the image of a polygon under a rigid motion.

Rigid motions map sided polygons onto sided polygons.

Now, consider a circle. The image of a circle is also a circle. However, circles do not have vertices, so how can the image be found? The following example investigates how to handle this situation.

Example

Congruence in Circles

Consider a rigid motion and a circle

Circle C
a If the center and the radius of are known, how can the image be found?
b If the center and a point on are given, how can the image be found?
c If three points on are given, how can the image be found?

Answer

a The image of is the circle centered at the image of the center of and the same radius as
b The image of is the circle centered at the image of the center of that passes through the image of the given point.
c The image of is the circle that passes through the images of the three given points.

Hint

a Use the fact that rigid motions preserve distances.
b With the image of the center and the image of the given point, the image circle can be drawn.
c Consider the segments connecting the images of the given points. Then, draw their perpendicular bisectors. These lines intersect at the center of the circle.

Solution

a The first step is to find the image of the center of under the rigid motion.
Image of the center of the circle C

Since rigid motions preserve distances, the image of is a circle whose radius is equal to the radius of Therefore, is the circle centered at and radius

Image of the circle C
b As in Part A, start by finding the image of the center of and the image of the given point.
Image of the center and image of the given point

The image of is a circle centered at with radius Since and are equal, is the circle centered at passing through point

Image of the circle C
c Once more, start by finding the images of the three given points.
Image of the three points of the circle C
The image of is a circle passing through and Next, find the center of by drawing the perpendicular bisectors of and
Perpendicular bisectors of A'B' and B'C'
The point of intersection of the perpendicular bisectors is the center of That way, the image of can be drawn.
Image of the three points of the circle C

Discussion

Using Rigid Motions to Determine Congruence

Since rigid motions map segments into congruent segments and angles into congruent angles, rigid motions can be used to define when a pair of geometric figures are congruent.

Concept

Congruent Figures

Two figures are congruent figures if there is a rigid motion or sequence of rigid motions that maps one of the figures onto the other. As a result, congruent figures have the same size and shape. To denote algebraically that two figures are congruent, the symbol is used.
Showing that to figures are congruent by mapping one onto the other
When writing a polygon congruence, the corresponding vertices must be listed in the same order. For the polygons above, two of the possible congruence statements can be written as follows.

Example

Determining if Figures are Congruent

Emily and her family are spending the weekend at a lake house. On Saturday afternoon, Emily begins to work on her geometry homework that asks her to determine if the following pair of polygons are congruent.

Pair of possibly congruent figures

Unfortunately, Emily left her ruler and protractor at her house and only brought a pencil and a piece of tracing paper.

a How can Emily do the homework?
b Are the polygons congruent?

Answer

a Emily can use the definition of congruent figures in terms of rigid motions. She can draw one of the polygons on the tracing paper and use it to perform different rigid motions on it and try to match both polygons.
b Yes, by applying a sequence of rigid motions, one polygon can be mapped onto the other.

Hint

a Use the definition of congruent figures in terms of rigid motions, and remember how to perform rigid motions using paper. Start by translating one polygon in order to match a pair of corresponding vertices.
b After applying some rigid motions, if the two figures match perfectly, then they are congruent.

Solution

a Since Emily has no ruler and protractor, she cannot compare the figures' side lengths or angle measures. However, she can check if the polygons are congruent using rigid motions. By using paper and pencil she can perform rigid motions on the figures. The first step is to identify possible corresponding parts.
Pointing Corresponding Vertices
Then, draw one of the figures on tracing paper and translate it so that the corresponding vertices match each other.
Translating one figure to match the corresponding vertices
Since the rest of the vertices do not match, a single translation is not enough to map one figure onto the other. The next step is to try to match a pair of corresponding sides. To do so, rotate the paper about the matching vertices.
Rotating one figure to match corresponding sides
As seen, the rest of the sides still do not match. Therefore, another transformation is needed. Notice that the two given polygons have opposite orientations, which suggests that performing a reflection is helpful. In this case, the line of reflection is the line containing the matching sides.
Reflecting the figure along the matching sides
This time, all sides of both figures match.
b Since the two polygons matched perfectly and only rigid motions were applied, it can be concluded that the polygons are congruent.

Discussion

Figures That Are Not Congruent

When two figures are not congruent, no sequence of rigid motions maps one figure onto the other. Even so, it is possible that a rigid motion maps certain parts of the preimage onto their corresponding parts, but not all. For example, consider the following pair of quadrilaterals.

A rectangle and and parallelogram with corresponding congruent sides
A translation can map onto However, the remaining parts of do not match the parts of Equivalently, can be mapped onto by a translation. Still, the remaining parts do not match as before.
Translating one rectangle so that a pair of side match
In the first case described before, could be rotated about in order to map onto However, by doing this, will no longer match with
Rotating A'B'C'D' about B' to map B'C' onto QR
Although the quadrilaterals have the same side lengths, no matter what rigid motion is applied, will not match The reason is that the quadrilaterals have different shapes. Like the shapes of figures, their sizes matter for the figures to be congruent. The next pair of squares have the same shape, but they are not congruent.
A rectangle and and parallelogram with corresponding congruent sides
Consequently, both size and shape are essential when determining if two figures are congruent.

Example

Determining Congruence Using a Sequence of Transformations

Consider the following pair of quadrilaterals and in the coordinate plane.

Two quadrilaterals on the coordinate plane

Furthermore, consider the following sequences of transformations.

Are the given quadrilaterals congruent?
In the affirmative case, which of the listed sequences maps onto

Hint

Use the orientation of both quadrilaterals to discard one of the sequences. Then, apply the remaining sequences to one at a time, to find the image. Compare the image to

Solution

To determine if any of the given sequences maps onto apply them to one at a time, and compare each resulting image to

Sequence

At first glance, it seems like the two quadrilaterals have opposite orientations. If this is true, the first sequence cannot map onto because both translations and rotations maintain the orientation. To verify it, apply the sequence to
Applying Sequence 1 to ABCD
Sequence maps only onto The remaining parts do not match. Consequently, Sequence does not map onto

Sequence

A glide reflection is the composition of a reflection and a translation performed in any order. Since a reflection is involved, this sequence is a good candidate. To apply it to start with the reflection.
Reflecting ABCD across y=-x
Next, the translation along will be applied.
Translating the reflected image along vector v
As can be seen, the given glide reflection mapped onto Therefore, the given quadrilaterals are congruent.

Sequence

Like the previous sequence, this third sequence also involves a reflection and thus it is a good candidate. Here, the rotation is applied first. Since it is a rotation, the direction of the rotation will not affect the resulting image.
Rotating ABCD about the origin
Next, reflect the resulting image across the line
Reflecting the rotated image across y=x
As the diagram shows, this sequence does not map onto As a result, only the second sequence of transformations maps onto

Pop Quiz

Practice Identifying Congruent Figures

In the following applet, the left-hand side polygon can be translated and rotated.

  • To translate the polygon, click on it and slide.
  • To rotate the polygon about click on any vertex of the polygon and rotate.

By applying these rigid motions, determine whether the given pair of polygons are congruent.

Random polygons

Illustration

Similar but Not Congruent

Because of copyright, in marketing, two different companies cannot have the same name or logo. Here, knowing how to determine if two figures are congruent is useful in dealing with intellectual property. Sometimes lesser-known brands use logos almost congruent to those of well-known brands. The intention is to receive instant recognition. For instance, consider the well-known logo of MathLeaks and the logo of a startup company called MathLovers.

Showing two almost equal logos
Even though the two logos appear to be the same, there is no rigid motion that maps one logo onto the other. Therefore, they are not congruent. Consequently, there is no legal issue.

Closure

Congruence in Real Life

Congruent figures are everywhere in daily life. For example, the figures formed by lines on one half of a basketball court are congruent to the figures formed on the other half. However, verifying this congruence through rigid motions is a difficult thing to do, given the circumstance of not being able to see the rigid motions occur.

Basketball Court
Therefore, it would be convenient to find a way of determining whether two figures are congruent without using rigid motions. The good news is that there is a simpler way for polygons, which will be studied in the next lesson.
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