126. A Diagonal of a parallelogram is a line joining any two opposite vertices.

NOTE.-Let the pupil illustrate.

127. The Altitude of a parallelogram or trapezoid is the perpendicular distance between its bases.

PARALLELOGRAMS.

THEOREM XL.

128. In any parallelogram, the opposite sides and angles are

and AB

=

=

AB CD, AC BD, and a = b.

From the first two equations,

or

Lo+ ≤ n = ≤ d + 2 m,

LACD

= L ABD.

Q.E.D.

129. COR. 1.-A diagonal of a parallelogram divides it into two equal triangles.

130. COR. 2.-Parallels intercepted between parallels are equal.

THEOREM XLI.

131. Two parallelograms are equal if two sides and the included angle of the one are respectively equal to two sides and the included angle of the other.

Let ABCD and EFGH be two □s, having AD = EH, a = ▲ b.

AB =

EF, and

THEOREM XLII.

132. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

= =

In the quadrilateral ABCD, let AB CD, and AC=BD.

THEOREM XLIII.

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

Let ABCD be a quadrilateral, having AB and CD equal and II.