Chapter 9

Geometry: Transformations, Congruence, and Similarity

By the 300s BCE, the Greeks had collected a huge amount of geometric knowledge. This knowledge came from observations made by earlier Greek thinkers (like Pythagoras), older civilizations (the Babylonians and Egyptians), and the Greeks' own discoveries. The philosopher Aristotle and the thinkers who came after him wanted to organize all this knowledge using strict logic. Their work led to a famous book called the Elements of Geometry, written by Euclid (this name might refer to one person or to a whole group of mathematicians).

In this book, geometry was built on a set of "self-evident" axioms—statements assumed to be true without proof—along with constructions made using only a straightedge (an unmarked ruler) and a compass. For example, two shapes were called congruent (identical in size and shape) if you could use a straightedge and compass to copy one shape exactly onto the other. These tools let mathematicians copy points, line segments, circles, and angles. Importantly, they didn't rely on numerical measurements. For instance, the "length" of a line segment was simply the distance between the compass's two points when it was placed on the segment's endpoints.

The whole point of Euclid's Elements was to prove everything using only these basic axioms, these two tools, and logical reasoning (the kind of clear, step-by-step thinking Aristotle developed). It mattered that the structure of geometry rested on the axioms themselves, not on common assumptions or hands-on construction—even though it was often the constructions and familiar patterns that made people believe the axioms in the first place. Ideas like "the same shape" or "the same size" were carefully defined using a small set of basic, undefined terms (like "point" and "line") that people were expected to understand through intuition alone. The goal was to keep this list of intuitive ideas as short as possible, so that everything else could be built from clear definitions and strict logic. In this system, a statement was proven true only by applying logic to already-proven truths—never by simply constructing something, observing it, or moving objects from one place to another.

This achievement was enormous, and it served as the foundation for teaching geometry for more than 2,000 years. However, philosophers noticed a problem almost immediately: at least one of the "self-evident" axioms didn't actually seem so obvious. This was the axiom about parallel lines. Two lines are called parallel if they never meet, no matter how far they extend. Every other axiom could be understood just by looking at drawings on a small piece of paper (or papyrus, or even sand). But this axiom asked people to imagine lines extending infinitely far in every direction to prove they never cross—and worse, there was no way to ever fully confirm that two lines never meet, no matter how far you followed them.

This problem was finally solved in the 1800s. Mathematicians discovered that there are other kinds of flat geometries: in some, all lines eventually cross each other, while in others, almost no lines are ever parallel. These became known as spherical geometry and hyperbolic geometry. They were discovered because real-world math problems required people to understand these alternative geometries. Later in the 1800s, the mathematician Felix Klein created a new, broader way of thinking about geometry—one flexible enough to include all these different types. Instead of focusing on fixed, unmoving shapes, Klein's approach focused on motion and change. In this new system, the central idea is the transformation: you choose a specific set of transformations (ways of moving or changing shapes), and geometry becomes the study of which properties of objects stay the same even after these transformations happen. This approach, called transformational geometry, is the method used in math classrooms today.

Around the same time, other mathematical tools were being developed that would combine with these geometric ideas. Coordinates (like the x and y values used to plot points on a graph) were introduced in the 1600s, and linear algebra was developed in the 1800s. Together, these tools created a powerful new way to study geometry using numbers and equations—this is called coordinate geometry (or vector geometry when working in more than two dimensions). This method gives us a whole new way to represent geometry using algebra, and through it, a new way to "rediscover" geometric ideas. The word rediscover is used on purpose: coordinate geometry gives us a way to calculate exact measurements, but it doesn't create brand-new geometric truths—it just lets us reach the same truths differently. For example, using coordinates, the distance between two points is defined using a formula based on their coordinates, rather than measured with a ruler placed against the two points. In other words, the Pythagorean theorem becomes the very definition of a line segment's length, rather than something we discover afterward.

So why is the Pythagorean theorem true? For the ancient Egyptians, it was simply something they observed to be true in real life. For Euclid, it could be logically proven using his geometric axioms. And in transformational and coordinate geometry, it becomes the starting foundation for those entire systems.

All of these ideas will be explored step-by-step throughout secondary school math. In 8th grade, our goal is to let students freely explore the basic ideas of transformational geometry: rigid motions, dilations, congruence, and similarity. In Chapters 9 and 10, students will confirm that rigid motions keep the lengths of line segments and the sizes of angles exactly the same, while dilations keep angle sizes the same but change line segment lengths by a fixed multiplying factor. From there, students will discover important geometric facts—some explained directly by the teacher, and others found through their own exploration and experimentation.

We believe that success in secondary math depends heavily on how strong a student's geometric intuition is, and that this intuition grows best through hands-on exploration of these core ideas. Because of this, some readers might feel that our approach seems less structured than they expect. If you'd like a more formally organized explanation of transformational geometry, you can find one in the appendix. However, we believe that level of structure is more advanced than what most 8th graders need. Instead, we hope teachers will let the class's own discoveries guide the lessons, even if that means straying a bit from the textbook.

In Chapter 2, Section 2, students learned that the slope of a line can be found using "rise over run," calculated from any two points on the line. To explain why this works, we introduced translations (shifts) and dilations (resizing), and looked at their properties as ways of transforming the plane. We used two key facts about dilations: first, a dilation has exactly one point that doesn't move, called the center of dilation, while every other point moves either closer to or farther from that center. Second, there's always a positive number, r, that the dilation uses to multiply every length in the shape. Note that if r equals 1, nothing moves at all—this special case is called the identity transformation, since no point changes position.

In this chapter, we'll take a deeper look at transformations of the plane, aiming to understand the shape and size of a geometric figure no matter where it's located. Students have already learned about shifts, flips, and turns; here, we'll revisit them as motions of the plane that keep two key measurements exactly the same: angle sizes and lengths. When discussing these motions (and dilations), it helps to think of them as dynamic actions—not as static, unmoving pictures. We aren't picking an object up and setting it back down somewhere else; we're actually moving it smoothly to its new position.

A rigid motion of the plane is a transformation that turns lines into lines and keeps both line segment lengths and angle measurements exactly the same. In other words, a line segment and its "image" (its new position after the transformation) will always have the same length, and an angle and its image will always have the same measure. One example of a rigid motion is a translation (previously called a "shift"). There are two other basic types: reflections (flips) and rotations (turns). We now say two figures are congruent (the same shape and size) if there's some sequence of rigid motions that moves one figure exactly onto the other.

This is simply a new way of describing something we already understood using Euclid's constructions—the meaning hasn't changed. If two shapes are congruent according to a classical construction, there will always be some sequence of rigid motions that moves one onto the other. And if we can move one shape onto another using rigid motions, there will always be a construction that does the same thing. The advantage of thinking in terms of motion, rather than construction, is that it connects more directly to how geometry is used in real science and engineering. For example, when builders install a support beam in a house, they don't use a compass-and-straightedge construction—they physically move the beam into place. If we wanted to build a robot to do that job, we'd need to describe the task using rigid motions, not classical constructions.

In the second section of this chapter, we'll explore dilations and scale factors. A dilation keeps lines and angles the same, but it changes the length of line segments by a set scale. We say two shapes are similar (meaning they have the same shape, though possibly different sizes) if some combination of rigid motions and dilations can transform one into the other.

The main focus of 8th grade geometry is to explore transformations, congruence, and similarity through hands-on experimentation, and to become comfortable connecting two related ideas: constructing a new copy of a shape, and physically moving that shape to a new location. Our focus here is on the "what" and "how" of geometry, while high school geometry will build on this to explain the "why." In real careers—in science, medicine, finance, architecture, and construction—people constantly create diagrams and drawings (called graphics) to represent their work. In 8th grade, we begin building the foundational skills needed for this kind of work.

To start, students should have a chance to discuss what it really means for two shapes to be "the same shape" versus "the same size and shape." That's the purpose of the example below, along with many of the early exercises in the workbook.

Figure 1 shows several groups of shapes. In Figure A, every shape is exactly the same size and shape, and any one of them can be moved onto any other using a rigid motion. In the other figure groups, no rigid motion can turn one shape into another. Try figuring out how the first shape in Figure A could be moved to match the others in that group. Then think about why this isn't possible for the other groups of shapes. Notice that in Figure B, the shapes share the same form but are different sizes, while in Figures C and D, the shapes differ in both size and shape.

Figure A
Figure B
Figure C
Figure D

Figure 1

Let's review some basic geometric facts that you've learned in earlier grades:

1. A line is defined by any two different points on it—you can draw the line by placing a straightedge against both points.
2. Two lines are either exactly the same line, cross at exactly one point, or never cross at all. You might wonder: if two lines don't cross anywhere on my paper, how can I know for sure they'll never cross at all? This exact question puzzled thinkers back in Euclid's time, and it sparked debate that lasted almost 2,000 years.
3. Two circles either don't intersect at all, touch at exactly one point, or cross at exactly two points. If they seem to meet at more than two points, they must actually be the same circle.
4. Two lines that never intersect are called parallel. If two lines do intersect and every angle formed where they cross is equal in measure, the lines are called perpendicular.
5. In any triangle, the sum of the lengths of any two sides is always greater than the length of the third side.

Section 9.1: Rigid Motions and Congruence

Understand congruence in terms of translations, rotations, and reflections (rigid motions), using a ruler and compass, physical models, transparencies, or geometry software.

Verify experimentally the properties of rotations, reflections, and translations:

a) Lines are transformed into lines, and line segments into line segments...