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 be given. A figure is line-symmetric with respect to if the reflection over line maps onto itself. That is, .
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 with and . It seems intuitively correct to say that the mirror should be the line that bisects . How can we prove that this is so?
Line bisects .
Click here to interact with the figure in GeoGebra.
The plan for this proof is to show that is the mirror image of in line . 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.)
- Let be the line through and , and let be the reflection over line . We’ll show that the sides of are symmetric with respect to line .
- Do and lie in the same half-plane relative to ?
- Since lies to the left of line , its mirror image must lie to the right of , as does .
- Do and have a common initial side?
- Since points and are on line , .
- Do and have the same degree measure?
- Since line bisects , we have . And since reflections preserve angles, we can be sure that as well.
- It now follows that coincides with . In particular, , which implies that also. This completes the proof that is line-symmetric with respect to its angle bisector .