201. A polygon which is both equilateral and equiangular is called Regular.

202. Two polygons are Mutually Equilateral if the sides of the one are equal respectively to the sides of the other taken in the same order.

203. Two polygons are Mutually Equiangular if the angles of the one are equal respectively to the angles of the other taken in the same order.

204. Two polygons may be mutually equiangular without being mutually equilateral.

205. Except in the case of triangles, two polygons may be mutually equilateral without being mutually equiangular.

206. A polygon of three sides is a Trigon or Triangle; one of four sides is a Tetragon or Quadrilateral; one of five sides is a Pentagon; one of six sides is a Hexagon; one of seven sides is a Heptagon; one of eight, an Octagon; of nine, a Nonagon; of ten, a Decagon; of twelve, a Dodecagon; of fifteen, a Quindecagon.

207. The Surface of a polygon is that part of the plane enclosed by its perimeter.

208. A Parallelogram is a quadrilateral whose opposite sides are parallel.

209. A Trapezoid is a quadrilateral with two sides parallel.

II. General Properties.

THEOREM XXVIII.

210. If two polygons be mutually equilateral and mutually equiangular, they are congruent.

PROOF. Superposition: they may be applied, the one to the other, so as to coincide.

EXERCISES. 29. Is a parallelogram a trapezoid? How could a triangle be considered a trapezoid?

THEOREM XXIX.

211. The sum of the interior angles of a polygon is two less straight angles than it has sides.

HYPOTHESIS. A polygon of n sides.

CONCLUSION. Sum of 's = (n-2) st. 's.

PROOF. If we can draw all the diagonals from any one vertex with out cutting the perimeter, then we have a triangle for every side of the polygon, except the two which make our chosen vertex. Thus, we have (n − 2) triangles, whose angles make the interior angles of the polygon. But, by 174, the sum of the angles in each triangle is a straight angle,

.. Sum of 's in polygon = (n-2) st. 's.

212. COROLLARY I. From each vertex of a polygon of n sides are (n 3) diagonals.

213. COROLLARY II. The sum of the angles in a quadrilateral is a perigon.

214. COROLLARY III. Each angle of an equiangular poly(n-2) st. 's gon of n sides is

n

THEOREM XXX.

215. In a convex polygon, the sum of the exterior angles, one at each vertex, made by producing each side in order, is a perigon.

HYPOTHESIS. A convex polygon of n sides, each in order produced at one end.

CONCLUSION. Sum of exterior 's= perigon.

PROOF. Every interior angle, as A, and its adjacent exterior angle, as x, together = st. X.

.. all interior's + all exterior 4's = n st. ¥'s.

EXERCISES. 30. How many diagonals can be drawn in a polygon of n sides?

31. The exterior angle of a regular polygon is one-third of a right angle: find the number of sides in the polygon.

32. The four bisectors of the angles of a quadrilateral form a second quadrilateral whose opposite angles are supplemental. 33. Divide a right-angled triangle into two isosceles triangles.

34. In a right-angled triangle, the sect from the mid-point of the hypothenuse to the right angle is half the hypothenuse.

III. Parallelograms.

THEOREM XXXI.

216. If two opposite sides of a quadrilateral are equal and parallel, it is a parallelogram.

B

D

HYPOTHESIS. ABCD a quadrilateral, with AB = and || CD. CONCLUSION. AD || BC.

PROOF. Join AC. Then X BAC = 4 ACD.

(168. If a transversal cuts two parallels, the alternate angles are equal.)

By hypothesis, AB = CD, and AC is common,

:. 4 =
BCA CAD,

(124. Triangles are congruent if two sides and the included angle are equal in each.)

:. BC || AD.

(165. Lines making alternate angles equal are parallel.)

EXERCISES. 35. Find the number of elements required to determine a parallelogram.

36. The four sects which connect the mid-points of the consecutive sides of any quadrilateral, form a parallelogram.

37. The perpendiculars let fall from the extremities of the base of an isosceles triangle on the opposite sides will include an angle supplemental to the vertical angle of the triangle.

38. If BE bisects the angle B of a triangle ABC, and CE bisects the exterior angle ACD, the angle E is equal to one-half the angle A.