All Lines are Congruent

A standard high school geometry textbook talks about congruence in terms of three types of objects: Line segments, angles, and polygons. Congruence is then defined in terms of measurable parameters: “Two figures are congruent if they have the same size and the same shape” (Carnegie’s Bridge to Algebra Student Text, 2008, p. G-9). Math Open Reference has a similar definition: “Two objects are congruent if they have the same dimensions and shape.” Oxford’s Concise Dictionary of Mathematics says: “Two geometrical figures are congruent if they are identical in shape and size.”

Most of the time, the question of congruence of lines and rays is ignored. However, when the subject does come up, there is some confusion expressed. The curriculum I’m working with for the summer school program I’m teaching (which shall remain nameless) explicitly implies, in fact, that rays are not congruent to each other, and that lines are not congruent to each other; this belief is echoed on this site. When researching the topic, I also found this Q&A, where two people agree that rays are congruent if and only if they share a common endpoint (I’ve added my own voice of reason to the conversation).

Here is a different definition of geometric congruence: “Two geometric figures are said to be congruent if one can be transformed into the other by an isometry” (Weisstein, Eric W. “Congruent” MathWorld). A similar definition comes from John Sullivan’s Math 302 class notes: “two geometric figures are called congruent if there is an isometry taking one to the other.”

An isometric transformation involves moving one object around until it fits on top of the other. In two dimensions, it might even be better to think of moving the plane itself around. Imagine an object etched onto a piece of glass, and then another one etched on wood. If you can move the glass around (including flipping it over) in such a way that the two objects line up perfectly, then one of the objects is an isometric transformation of the other.

Sullivan’s class notes specifically argue that all rays are isometric transformations of each other, while William Smith states theorems both that all rays are congruent (p. 23) and that all straight angles are congruent (p. 29). However, Smith’s argument about straight angles is specifically that straight angles consist of pairs of opposing rays, and since all rays are congruent, all pairs of opposing rays are congruent, so he’s equating an angle with its rays, a topic I want to tackle in a separate item.

Since isometric transformations tend to come after line segment congruence in standard secondary-level geometry syllabi, it’s not surprising that textbooks would want to avoid that definition. And using a definition of length with regards to congruence of lines and rays creates a special problem that smart students might well pick up on, and unwary (or unknowledgeable) teachers might well stumble on responding to.

Specifically, with regards to line segment, students are told that the shape is irrelevant because all segments are the same shape. If that’s true, then shape is likewise irrelevant with regards to rays and lines. However, rays and lines both have infinite length. So, if congruence of line segments is based solely on length, then that would tend to imply that rays are congruent to lines, which they are not. An infinitely long portion of a line with an endpoint cannot be transformed onto an infinitely long portion of a line without an endpoint.

This feels a bit like us wanting to have our cake and eat it too. We want students to marvel at the abstraction of the infiniteness of lines and rays, but we want to avoid one of the more blatant ramifications of that infiniteness. And regardless, if students can understand that lines are infinite, they can understand that all lines are congruent.

If, however, sources want to avoid the topic altogether, the “size and shape” definition of congruence is serviceable for finite shapes, and the congruence of all lines and of all rays is generally trivial anyway. I’m fine with sources avoiding that portion of the topic altogether. What bothers me is when sources are blatantly wrong about it.

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