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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

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.

Discussion

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 $AB$ and any rigid motion. Let $C$ be the image of $A$ and $D$ the image of $B.$

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

The idea is to show that $Q$ lies on $CD$ between $C$ and $D.$ To do this, it must be checked that $CQ+QD$ is equal to $CD.$ Again, since rigid motions preserve distances, $CQ=AP$ and $QD=PB.$${CQ=APQD=PB (I)(II) $

Next, Equations (I) and (II) can be added together and simplified using the Segment Addition Postulate.
${CQ=APQD=PB (I)(II) $

Solve by elimination

SysEqnAdd

$(I):$ Add $II$

${CQ+QD=AP+PBQD=PB $

$(I):$ Segment Addition Postulate

${CQ+QD=ABQD=PB $

Substitute

$(I):$ $AB=CD$

${CQ+QD=CD✓QD=PB $

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 $n-$sided polygons onto $n-$sided polygons.

Example

Consider a rigid motion and a circle $C.$

a If the center and the radius of $C$ are known, how can the image be found?

b If the center and a point on $C$ are given, how can the image be found?

c If three points on $C$ are given, how can the image be found?

a The image of $C$ is the circle centered at the image of the center of $C$ and the same radius as $C.$

b The image of $C$ is the circle centered at the image of the center of $C$ that passes through the image of the given point.

c The image of $C$ is the circle that passes through the images of the three given points.

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.

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

Since rigid motions preserve distances, the image of $C$ is a circle whose radius is equal to the radius of $C.$ Therefore, $C_{′}$ is the circle centered at $P_{′}$ and radius $r.$

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

The image of $C$ is a circle centered at $P_{′}$ with radius $PQ.$ Since $PQ$ and $P_{′}Q_{′}$ are equal, $C_{′}$ is the circle centered at $P_{′}$ passing through point $Q_{′}.$

c Once more, start by finding the images of the three given points.

The point of intersection of the perpendicular bisectors is the center of $C_{′}.$ That way, the image of $C$ can be drawn.

Discussion

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

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.

When stating that two polygons are congruent, 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.

$ABCDE≅JKLMNorCDEAB≅LMNJK $

Example

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.

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?

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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.

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.

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.

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.

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.

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

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.

In the first case described before, $A_{′}B_{′}C_{′}D_{′}$ could be rotated about $B_{′}$ in order to map $B_{′}C_{′}$ onto $QR .$ However, by doing this, $A_{′}B_{′}$ will no longer match with $PQ .$

Although the quadrilaterals have the same side lengths, no matter what rigid motion is applied, $ABCD$ will not match $PQRS.$ The reason is that the quadrilaterals have different shapes.

$AB≅PQ BC≅QR CD≅RSDA≅SP ⇒ ABCD≅PQRS $

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.
Consequently, both size and shape are essential when determining if two figures are congruent.Example

Consider the following pair of quadrilaterals $ABCD$ and $PQRS$ in the coordinate plane.

Furthermore, consider the following sequences of transformations.

- $S_{1}:$ Translation along $v=⟨3,1⟩$ followed by a $90_{∘}$ clockwise rotation about the origin.
- $S_{2}:$ Glide reflection defined by the line $y=-x$ and $v=⟨3,-3⟩.$
- $S_{3}:$ $180_{∘}$ rotation about the origin followed by a reflection across $y=x.$

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

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

Sequence $1$ maps only $AB$ onto $PQ .$ The remaining parts do not match. Consequently, Sequence $1$ does not map $ABCD$ onto $PQRS.$

Next, the translation along $v=⟨3,-3⟩$ will be applied.

As can be seen, the given glide reflection mapped $ABCD$ onto $PQRS.$ Therefore, the given quadrilaterals are congruent.

Next, reflect the resulting image across the line $y=x.$

As the diagram shows, this sequence does not map $ABCD$ onto $PQRS.$ As a result, only the second sequence of transformations maps $ABCD$ onto $PQRS.$

Pop Quiz

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 $Q,$ click on any vertex of the polygon and rotate.

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

Illustration

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.

Closure

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.

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.