The capital letters A, B, C, D, E, H, I, K, M, O, T, U, V, W, X, and Y are often written as reflection symmetric figures. Some are symmetric about a horizontal line (BCDEHIKOX) whereas others are symmetric about a vertical line (AHIMOTUVWXY). As you can see since some are in both lists (HIOX), there may be more than one line of symmetry. A challenge would be to find words such as DIXIE or COOKBOOK composed entirely of letters with a horizontal line of symmetry or MOM, WAXY, YOUTH (written vertically!) composed entirely of letters with a vertical line of symmetry. After collecting enough of these words you might make them into a crossword puzzle (for extra credit)!
Our textbook states and proves what they call the Flip-Flop Theorem: (reflection is symmetric).
If F and G are points/figures, and rl(F)=G, then rl(G)=F.
From this it can be proved that every segment has two lines of symmetry: itself and its perpendicular bisector. This is the same as the letter I discussed above. Angles only have one line of symmetry: the angle bisector which causes one ray to reflect onto the other ray. A circle has infinitely many lines of symmetry (no matter which way you draw the diameter, the semicircles are reflections of each other). The section concludes with the following important result.
If a figure is symmetric, then any pair of corresponding parts under the symmetry are congruent.
Rorschach inkblots and logos commonly are reflective-symmetric. These symmetries will be useful when applied to various polygons. Symmetry is also important in algebra. The function y=x2 defines a parabola in which the sign of x doesn't matter. This makes it an even function (the exponent of 2 is another clue).
Symmetric Triangles (Isosceles and Equilateral)
Triangles, as mentioned in Numbers lesson 11 and Geometry lesson 2, can be classified either by the number of sides with the same length (0 is scalene, 2 or more is isosceles, all 3 is equilateral) or by the largest angle (acute, right, obtuse). A hierarchy chart combining both situations is given at the left. Due to the overlap, hierarchy charts for either situation are typically given instead. Note: a right/acute/obtuse triangle might be either scalene or isosceles.
Also, our definition of isosceles includes and does not exclude the equilateral triangle. Just as there are special names associated with the sides of a right triangle (hypotenuse and legs), there are special names associated with the angles and sides of an isosceles triangle. The angle determined by the two equal sides is called the vertex angle. The side opposite the vertex angle is called the base. The two angles opposite the equal sides are the base angles (and are equal). These can also be described as the angles at the endpoints of the base.
Three important theorems are as follows. Certain terms will be defined further below.
The line containing the bisector of the vertex angle of an isosceles triangle
is a symmetry line for the triangle.
Step-by-step explanation:
