Def.—A line in any figure, such as AC in the preceding diagram, which is such that, by folding the plane of the figure round it, one part of the diagram will coincide with the other, is called an axis of symmetry of the figure.
Exercises.
1. Prove that the angles at the base are equal without producing the sides. Also by producing the sides through the vertex.
2. Prove that the line joining the point A to the intersection of the lines CF and BG is an axis of symmetry of the figure.
3. If two isosceles triangles be on the same base, and be either at the same or at opposite sides of it, the line joining their vertices is an axis of symmetry of the figure formed by them.
4. Show how to prove this Proposition by assuming as an axiom that every angle has a bisector.
5. Each diagonal of a lozenge is an axis of symmetry of the lozenge.
6. If three points be taken on the sides of an equilateral triangle, namely, one on each side, at equal distances from the angles, the lines joining them form a new equilateral triangle.
PROP. VI.—Theorem.
If two angles (B, C) of a triangle be equal, the sides (AC, AB) opposite to them are also equal.
