160. Definitions.

POLYGONS.

A polygon is a figure formed by a series of segments, AB, BC, CD, etc., leading back to the starting point 4.

The segments are the sides of the polygon and the points A, B, C, D, etc., are its vertices. The angles A, B, C, D, etc., are the angles of the polygon.

A polygon is convex if no side when produced enters it. Otherwise it is concave.

Only convex polygons are here considered.

A polygon is equiangular if all its angles are equal and equilateral if all its sides are equal.

A polygon is regular if it is both equiangular and equilateral.

A segment connecting two non-adjacent vertices is a diagonal of the polygon.

The perimeter of a polygon is the sum of its sides.

161. THEOREM. The sum of the angles of a polygon having n sides is (2n-4) right angles.

Proof: Connect one vertex with each of the other non-adjacent vertices, thus forming a set of triangles. Evidently the sum of the angles of these triangles equals the sum of the angles of the polygon.

Now show that if the polygon has n sides there are (n-2) triangles. The sum of the angles of one triangle is 2 rt. 4. Hence, the sum of the angles of all the triangles, that is, the sum of the angles of the polygon, is

162. THEOREM. The sum of the exterior angles of a polygon, formed by producing the sides in succession, is four right angles.

Outline of Proof:

angles is 2 n rt. 4.

The sum of both exterior and interior (Why?)

The sum of the interior angles is (2 n − 4) rt. 4. (Why?) Hence, the sum of the exterior angles is 4 rt. 4. (Why?) Write out the proof in detail, using the figure.

1. What is the sum of the angles of a polygon of 3 sides? of 4 sides? of 5 sides? of 6 sides? of 10 sides? of 18 sides?

2. Find each angle of a regular polygon of 3 sides, 4 sides, 5 sides, 6 sides, 8 sides, 14 sides, n sides.

3. Construct a regular triangle, thus obtaining an angle of 60°. 4. Construct a regular quadrilateral. What is its common name? 5. Prove that a regular hexagon ABCDEF may be constructed as follows: Let A be any point on a circle with center O. With A as center and OA as radius describe arcs meeting the circle in B and in F. With B as center and the same radius describe an arc meeting the circle in C, and so for points to D and E. See § 159, Ex. 18.

SYMMETRY.

164. Two points A and A' are said to be symmetric points with respect to a line 7 if 7 is the perpendicular bisector of the segment AA'.

A

A figure is symmetric with respect to an axis if for every point P in the figure there is also a point p' in the figure such that P and P' are symmetric points with P respect to l. This is called axial symmetry. Two separate figures may have an axis of symmetry between them.

-A'

165. THEOREM. Two figures which are symmetric with respect to a line are congruent.

Given two figures F and F symmetric

with respect to a line l.

P

P

F'

F

To Prove that FF'.

Proof: This is evident, since, by

folding figure F over on the line 7

as an axis, every point in F will fall upon a corresponding point in F (Why?).

166. COROLLARY. If points A and A' and also B and B' are symmetric with respect to a line 1, then the segments AB and A'B' are symmetric with respect to l.

B

P

P'

B'

1. How many axes of symmetry has a square? A rectangle? A rhombus? An isosceles trapezoid?

2. If a diagonal of a rectangle is an axis of symmetry, what kind of a rectangle is it?

3. If a triangle has an axis of symmetry, what kind of a triangle is it? Assume that the axis passes through one vertex.

4. If a triangle has two axes of symmetry, what kind of a triangle is it?

5. How many axes of symmetry has an equilateral triangle?

6. How many axes of symmetry has a

regular pentagon (five-sided figure)?

7. How many axes of symmetry has a regular hexagon?

8. Show that one figure of § 40 has an axis of symmetry. State this as a theorem.

168. PROBLEM. Given a polygon P and a line l not meeting it, to construct a polygon P' such that P and P' shall be symmetric with respect to l.

SOLUTION. Let A, B, C, D, E be the vertices of the given polygon P, and I the given line.

Construct A', B', C', D', E' symmetric respectively to A, B, C, D, E with respect to the line 7.

Then the polygon P' formed by joining the points A', B', C', D', E', A' in succession is symmetric to P.

Proof : Give the proof in full.

Figures having an axis of symmetry are very common in all kinds of decoration and architectural construction.

169. Two points A and A' are symmetric with respect to a point o if the segment AA' is bisected by 0.

A figure is symmetric with respect to a A point o if for every point P of the figure

Α'

there is a point P' also in the figure such that P and P' are symmetric with respect to 0.

Such a figure is said to have central symmetry with respect to the point. The point is called the center of symmetry. A circle has central symmetry.

Two separate figures may have a center of symmetry between them.

1. Prove that if A and A' and also B and B' are symmetric with respect to a point O, then the segments AB and A'B' are symmetric with respect to 0.

2. Prove that if the triangles ABC and A'B'C' are symmetric with respect to a point O, then they are congruent.

171. PROBLEM. Given a polygon P and a point O outside of it, to construct a polygon P' symmetric to P with respect to 0.

B

E

B'

SOLUTION. Construct points symmetric to the vertices of P. Connect these points, forming the polygon P', and prove this is the polygon sought.