Chapter 9

Geometry: Transformations, Congruence, and Similarity

By the third century BCE, the Greeks had built up an enormous body of geometric knowledge. This knowledge came from earlier Greek thinkers (like Pythagoras), older civilizations (the Babylonians and Egyptians), and the Greeks' own discoveries. Aristotle and the mathematicians who followed him set out to organize all of this knowledge on a solid logical foundation. The result was the Elements of Geometry, credited to Euclid (a name that may refer to one person or to a group of scholars).

Euclid's system rested on a set of "self-evident" axioms — statements assumed to be true without proof — along with constructions made using only a straightedge and compass. Two figures were called congruent (identical in size and shape) if one could be copied exactly onto the other using these tools. The straightedge and compass were used to copy points, line segments, circles, and angles. Notably, Euclid's system did not rely on numerical measurement. For instance, the "length" of a line segment was simply the distance between the compass's two points when set at the segment's endpoints — not a number read off a ruler.

The goal of the Elements was to derive all known geometric facts from these basic axioms and tools, using strict logical reasoning modeled on Aristotle's methods. It was essential that the logical structure rest on the axioms themselves, not on tradition or common assumptions — even though it was often intuition and hands-on construction that made people believe the axioms were self-evident in the first place. Concepts like "same shape" and "same size" were given precise definitions, all built from a small set of undefined terms (point, line, and so on) that people were expected to understand intuitively. The goal was to keep this list of intuitive concepts as short as possible, with everything else established through definition or rigorous logic. In this system, a statement was proven true only through logical deduction from already-established truths — never through construction, observation, or by physically moving objects from one place to another.

This was a monumental achievement, and it shaped the teaching of geometry for more than 2,000 years. However, almost immediately, philosophers raised an objection: at least one of the "self-evident" axioms wasn't so self-evident after all. This was the axiom about parallel lines. Roughly speaking, two lines are called parallel if they never meet, no matter how far they're extended. Every other axiom could be understood intuitively just by looking at drawings on a finite piece of paper (or papyrus, or sand). But this axiom asks us to imagine extending two lines indefinitely to check whether they ever cross — and worse, there's no way to verify that two lines will never meet, since we can't check forever.

This problem was finally resolved in the 19th century. Mathematicians discovered that there are other types of planar (flat-surface) geometries: in some, all lines eventually intersect; in others, almost all lines never intersect at all. These became known as spherical geometry and hyperbolic geometry, respectively. They were discovered because real-world applications of mathematics required them. Then, in the late 19th century, the mathematician Felix Klein introduced a new, broader concept of geometry — one general enough to include all these different types. Klein's approach treated geometry as something dynamic (based on motion) rather than static (based on fixed pictures). In this system, the central idea is transformation: a specific set of transformations is defined, and geometry becomes the study of which properties of objects stay unchanged under those transformations. This is the approach used in math education today, known as transformational geometry.

Around the same time, other mathematical tools were developing that would eventually connect with this geometric approach. The introduction of coordinates in the 17th century, followed by the development of linear algebra in the 19th century, gave mathematicians a powerful new toolkit for studying geometry. This approach is called coordinate geometry (or, in dimensions higher than two, vector geometry). It offers a fresh perspective, representing geometry entirely through algebra, and through that lens, allows us to rediscover geometric truths. The word "rediscover" is chosen carefully: coordinate geometry gives us precise ways to calculate measurements, but it isn't really a new way to build the subject from scratch. For example, the distance between two points is defined using the points' coordinates — not by laying a ruler between them. In other words, the Pythagorean theorem becomes the very definition of a line segment's length, rather than a fact to be proven.

So why is the Pythagorean theorem true? To the Egyptians, it was simply an observed fact. To Euclid, it followed logically from geometry's self-evident axioms. In transformational and coordinate geometry, it becomes a foundational starting point rather than something proven from other rules.

All of these ideas will be developed systematically throughout secondary school math. In 8th grade, the goal is to let students freely explore the basic concepts of transformational geometry: rigid motions, dilations, congruence, and similarity. In Chapters 9 and 10, students will verify that rigid motions preserve the measurements of line segments and angles, while dilations preserve angle measurements but change line-segment lengths by a constant multiplying factor. From there, students will observe basic geometric facts — some pointed out directly by the teacher, others discovered through their own exploration.

We believe that success in much of secondary math depends on how strong a student's geometric intuition is, and that this intuition develops best through open-ended exploration of fundamental ideas. Because of this, our approach here may look less structured than some readers expect. If you'd like a more formal, step-by-step development of transformational geometry, see the appendix. We suspect that appendix will feel too structured for 8th graders — we hope teachers will instead let the class shape its own path of discovery, even if that means straying from the textbook.

In Chapter 2, Section 2, students learned that a line's slope can be calculated as rise/run using any two points on the line. To explain why this works, we introduced translations (shifts) and dilations (resizing transformations) and examined their properties. We relied on two key facts about dilations: a dilation has exactly one fixed point (called the center of the dilation), and every other point moves either away from or toward that center. There's also a positive number, r, by which the dilation multiplies every length. Note that if r = 1, nothing moves at all — this special case is called the identity transformation.

In this chapter, we'll examine transformations of the plane more closely, aiming to understand the shape and size of a geometric figure regardless of where it sits on the plane. Students have already encountered shifts, flips, and rotations informally; here, we reintroduce them formally as motions of the plane that preserve two basic measurements: angles and lengths. When discussing these motions (and dilations), it helps to think dynamically rather than statically — we're not picking up a shape and dropping it somewhere else, but actually moving it to a new location.

A rigid motion of the plane is a transformation that turns lines into lines and preserves both the lengths of line segments and the measures of angles. In other words, under a rigid motion, a line segment and its new position (its "image") have the same length, and an angle and its image have the same measure. One example of a rigid motion is a translation (what we've been calling a shift). There are two other basic types: reflections (flips) and rotations (turns).

We now say two figures are congruent — meaning they have the same shape and size — if some sequence of rigid motions can transform one into the other. This is a new way of thinking about equivalence between figures, but it doesn't change the underlying meaning: if two figures are congruent according to a classical Euclidean construction, then some sequence of rigid motions will also take one to the other, and vice versa. The advantage of thinking in terms of motions rather than constructions is that it connects more directly to how geometry is used in science and engineering. For example, when installing a beam in a house, you don't use a straightedge-and-compass construction — you physically move the beam into place. If you wanted to build a robot to do that job, you'd need to describe the process using rigid motions, not classical constructions.

In the second section, we turn to dilations and scale factors. A dilation preserves lines and angles but changes the length of line segments by a fixed scale. We say two figures are similar (meaning they have the same shape, though not necessarily the same size) if some combination of rigid motions and dilations can transform one into the other.

The focus of 8th-grade geometry is to explore transformations, congruence, and similarity through hands-on experimentation, building familiarity with how constructing a new image of an object relates to physically moving that object to a new location. We concentrate on the "what" and "how" of geometry here, while high school geometry will build on this foundation to explain the "why." In real-world science and industry, people constantly create visual representations (called graphics) of their work — whether they're in medicine, finance, architecture, or construction. In 8th grade, we lay the groundwork for these essential skills.

To start, students should have a chance to discuss what "same shape" and "same size and shape" actually mean. That's the purpose of the example below, along with many of the introductory exercises in the workbook.

Figure 1 shows several groups of shapes. In Figure A, every image is the same size and shape — we can move any one onto any other using a rigid motion. In the other figures, no rigid motion can transform one shape into another. Try to figure out how to move the first shape in Figure A onto the others. Why doesn't this work for the other groups of shapes? Notice that in Figure B, the shapes share the same form but differ in size, while in Figures C and D, the shapes differ in both size and shape.

[Figure 1: Figure A, Figure B, Figure C, Figure D]

Let's review some basic geometric facts from earlier grades:

1. A line is fully determined by any two distinct points on it — just place a straightedge against the points and draw.
2. Two lines either coincide (are actually the same line), intersect at exactly one point, or never intersect at all. You might wonder: if two lines don't cross anywhere on my paper, how can I know for certain they never cross? This exact question sparked a controversy in Euclid's time that lasted nearly 2,000 years, as discussed earlier in this chapter.
3. Two circles either don't intersect, intersect at exactly one point, or intersect at exactly two points. If they seem to intersect at more than two points, they're actually the same circle.
4. Two lines that never intersect are called parallel. If two lines intersect and every angle formed at that intersection point has the same 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

Goal: Understand congruence in terms of translations, rotations, and reflections (collectively, rigid motions) using rulers, compasses, physical models, transparencies, and geometry software.

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

a) Lines map to lines, and line segments map to line segments...