Proposition 33. Theorem.

133. If two opposite sides of a quadrilateral are equal and parallel, the figure is a parallelogram.

Hyp. Let ABCD be a quadrilat- D

eral, having AB = and || to DC.

having two sides and the included equal, each to each (104).

1. Prove that the straight lines which bisect two adjacent angles of a parallelogram cut each other at right angles.

2. AB, CD, EF are three equal and parallel straight lines; prove that the triangle ACE is equal to the triangle BDF.

Proposition 34. Theorem.

134. The diagonals of a parallelogram bisect each other.

Hyp. Let ABCD be a, whose D diagonals intersect at O.

To prove AO OC, DO OB.

=

=

having a side and the two adj. Ls equal, each to each (105).

... AO OC, and DO = OB.

B

(109)

EXERCISES.

Q.E.D.

1. If the opposite angles of a quadrilateral are equal, the figure is a parallelogram.

2. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram.

3. If the diagonals of a parallelogram are equal, the figure is a rectangle; if they also intersect at right angles, it is a square.

4. The straight lines joining the middle points of the opposite sides of any quadrilateral bisect each other.

5. The diagonals of a rhombus bisect each other at right angles.

6. The diagonals of a rectangle are equal.

Proposition 35. Theorem.

135. Two parallelograms are equal when two adjacent sides and the included angle of the one are equal respectively to two adjacent sides and the included angle of the other.

Hyp. Let ABCD, A'B'C'D' be two

Os, having

AB = A'B',

.. AD will fall on A'D', and pt. D on pt. D'.

(Hyp.)

Because DC is || to AB, and D'C' is || to A'D',

(124)

.. DC will fall on D'C', and pt. C somewhere on D'C'. Through a given pt. only one st. l. can be drawn || to a given st. l. (Ax. 12)

Also, because BC is || to AD, and B'C' is || to A'D', (124) .. BC will fall on B'C', and pt. C somewhere on B'C'. (Ax. 12)

.. since pt. C falls on D'C' and B'C', it must fall at their pt. of intersection C'.

.. the two s coincide throughout, and are equal. Q. E. D.

136. COR. Two rectangles are equal when they have two adjacent sides equal, each to each.

POLYGONS.

137. A polygon is a plane figure bounded by straight lines; as ABCDE.

The straight lines are called the sides of the polygon; and their sum is called the perimeter of the polygon.

The angles of the polygon are the angles formed by the adjacent sides with each other; and the vertices of these angles are also called the vertices of the polygon.

A

E

B

138. The angles of the polygon measured on the side of the enclosed surface are called interior angles.

An exterior angle of a polygon is an angle between any side and the continuation of an adjacent side.

A diagonal is a line joining any two vertices that are not consecutive; as AD.

139. Polygons are named from the number of their sides, as follows:

A polygon of three sides is a triangle; one of four sides, a quadrilateral; one of five sides, a pentagon; one of six sides, a hexagon; one of seven sides, a heptagon; one of eight sides, an octagon; one of nine sides, a nonagon; one of ten sides, a decagon; one of twelve sides, a dodecagon; one of fifteen sides, a quindecagon.

E

140. An equilateral polygon is one which has all its sides equal. An equiangular polygon is one which has all its angles equal. A regular polygon is one which is both equilateral and equiangular.

141. A convex polygon is one each of whose interior angles is less than a straight angle; as ABCDE.

142. A concave polygon is one in which at least one of the interior angles is reflex (26), as GHKMNO, in which the interior

angle KMN is reflex.*

The polygons considered in this work will be understood to be convex, unless otherwise stated.

N

M

-K

H

It is evident that a polygon has as many angles as sides. 143. Two polygons are mutually equilateral when the sides of the one are equal respectively to the sides of the other, taken in the same order; as

the polygons ABCD, A'B'C'D', in which AB A'B', BC B'C', etc.

=

=

S

144. Two polygons are mutually equiangular when the angles of the A one are equal respectively to the angles of the other, taken in the same order; as the polygons PQRS, P'Q'R'S', in which P = <P', <QQ', etc.

145. In polygons which are mutually equilateral or mutually equiangular, any two corresponding sides or angles are called homologous (95).

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

146. A polygon may be divided into triangles by drawing diagonals from one of its vertices; and the number of triangles into which any polygon can thus be divided is evidently equal to the number of its sides, less two.

When two polygons can be divided by diagonals into the same number of triangles, equal each to each, and similarly placed, the polygons are equal; for they can be applied one to the other, and the corresponding triangles will evidently coincide, and therefore the polygons will coincide throughout.

When two polygons are both mutually equilateral and mutually equiangular, they are equal, for they can be applied one to the other so as to coincide.

*Called also a re-entrant angle,