The CCSSM calls for students to learn to recognize line-symmetric figures as early as Grade 4:

Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

In high school, students can be expected to use logical reasoning to demonstrate that a given figure is line-symmetric. In order to do this, we need to take the intuitive idea about “folding the figure along a line into matching parts” and turn it into a statement that is mathematically precise.

Definition: let line m be given. A figure S is line-symmetric with respect to m if the reflection over line m maps S onto itself. That is, r_m(S)=S.

An angle is an extremely simple example of a line-symmetric figure. Students can explore this phenomenon by tracing an angle on a piece of patty paper, then folding the paper so that the two sides coincide.

In this post, we’ll explore the logic underlying what some texts refer to as “The Side-Switching Theorem.” The aim of the proof is to show that there is a line in which the sides of the angle are mirror images of one another. That is, we’ll find a suitable reflection r with r(\overrightarrow{OA}) = \overrightarrow{OB} and r(\overrightarrow{OB}) = \overrightarrow{OA}. It seems intuitively correct to say that the mirror should be the line that bisects \angle AOB. How can we prove that this is so?

angles are reflection-symmetric

Line OC bisects \angle AOB.

Click here to interact with the figure in GeoGebra.

The plan for this proof is to show that \angle BOC is the mirror image of \angle AOC in line m. To establish this, we’ll show that these two angles 1) have a common initial side, 2) have the same degree measure, and 3) have terminal sides that lie in a common half-plane. As long as these conditions are met, we can be sure that these two angles coincide. (The technical machinery that makes this argument work is the Protractor Postulate.)

  1. Let m be the line through O and C, and let r be the reflection over line m. We’ll show that the sides of \angle AOB are symmetric with respect to line m.
  2. Do r(\overrightarrow{OA}) and \overrightarrow{OB} lie in the same half-plane relative to m?
    • Since \overrightarrow{OA} lies to the left of line m, its mirror image must lie to the right of m, as does \overrightarrow{OB}.
  3. Do r(\angle AOC) and \angle BOC have a common initial side?
    • Since points O and C are on line m, r(\overrightarrow{OC}) = \overrightarrow{OC}.
  4. Do r(\angle AOC) and \angle BOC have the same degree measure?
    • Since line OC bisects \angle AOB, we have \angle AOC = \angle BOC. And since reflections preserve angles, we can be sure that r(\angle AOC) = \angle BOC as well.
  5. It now follows that r(\angle AOC) coincides with \angle BOC. In particular, r(\overrightarrow{OA}) = \overrightarrow{OB}, which implies that r(\overrightarrow{OB}) = \overrightarrow{OA} also. This completes the proof that \angle AOB is line-symmetric with respect to its angle bisector OC.

The proof above makes use of three important ideas: 1) the definition of reflection, 2) the fundamental assumption that reflections preserve angles, and 3) the Protractor Postulate.