PROPOSITION VII.

194. Theorem: The opposite angles of a parallelogram are equal, each to each.

Statement: Let A B C H be any parallelogram. Prove that the opposite angles 1 and 3 are equal to each other; and, also, that the opposite angles 2 and 4 are equal to each other.

Construction: Extend the base HC to E and to F, making the exterior angles 5 and 6.

Demonstration: The angle 1 is equal to the angle 5, as they are alternate angles. (BooK JI., PROP. XVI.) The angle 3 is equal to the angle 5, as they are opposite exterior and interior angles on the same side. (BOOK II., PROP. XVII.) The opposite angles 1 and 3, each being equal to the angle 5, are equal to each other. (AXIOM I.)

In similar manner, it may be proved that the opposite. angles 2 and 4 are each equal to the angle 6, and, therefore, they are equal to each other.

Conclusion: As A B C H is any parallelogram, it follows, in general, that: The opposite angles of a parallelogram are equal, each to each.

PROPOSITION VIII.

195. Theorem: If the opposite angles of a quadrilateral are equal, each to each, the figure is a parallelogram.

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Statement: Let 1 2 3 4 be any quadrilateral, with the angle 1 equal to its diagonally opposite angle 3, and the angle 2 equal to the angle 4. Prove that 1 2 3 4 is a parallelogram.

Demonstration: Since the angles 4 and 2 are equal, it is obvious, that the angle 4 is half of their sum. Since the angles 3 and 1 are equal, the angle 3 is half of their sum. The angles 4 and 3 are, then, together equal to half of the sum of the four angles of the quadrilateral, or two right angles. (PROP. VI.) In similar manner, any other two consecutive angles, as 1 and 4, may be proved equal to two right angles.

Since the angles 1 and 2 are together equal to two right angles, the sides 1 4 and 23 are parallel. (BOOK I., PROP. XVIII., COR. II.) Since 1 and 4 are together equal to two right angles, 12 and 43 are parallel. The opposite sides being parallel, the figure is, by definition, a parallelogram.

Conclusion: The figure 1 2 3 4 being any quadrilateral, etc.

PROPOSITION IX.

196. Theorem: If any angle of a parallelogram is a right angle, the figure is a rectangle.

H

Statement: Let ABHC be any parallelogram. angle, as A, be a right angle.

Let any

Prove the figure a rectangle.

Demonstration: The angles A and B are together equal to two right angles. (Book 1., PROP. XVII.) As A is a right angle, B is, also, a right angle. (AXIOM IV.)

As A and B are right angles, their respectively opposite angles, H and C, are right angles. (PROP. VII.) The four angles are, then, all right angles, and the figure is a rectangle.

Conclusion: The figure AB HC being any parallelogram, and the angle ▲ any one of its angles, it is true, in general, that If one angle of a parallelogram, etc.

A

Exercise. 1. If one side of a quadrilateral be extended in both directions, the sum of the exterior angles is equal to the sum of the two opposite interior angles.

PROPOSITION X.

197. Theorem: If any angle of a parallelogram is an oblique angle, the figure is a rhomboid.

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Statement: Let 1234 be any parallelogram in which any angle, as 1, is an oblique angle. The other angles, 2, 3, and 4, are then also oblique, and the parallelogram is a rhomboid.

Demonstration: Since the figure 1 2 3 4 is a parallelogram, the sides 14 and 23 being parallel and intersected by the side 1 2, the angles 1 and 2 are together equal to two right angles. (BOOK I., PROP. XVII.) The angle 1, being oblique, is either greater or less than a right angle; hence, the angle 2 is less or greater than a right angle. (AXIOM XII.)

The angles 1 and 2 being oblique, their respectively opposite angles, 3 and 4, are also oblique, since they are equal to them. (PROP. VII.)

Conclusion: 1 2 3 4 being any parallelogram, it follows, etc.

Exercise. 1. The angles included between the sides of a quadrilateral produced equal four right angles.

MISCELLANEOUS EXERCISES.

198. I. The bisectrices of the interior angles of a parallelogram form a rectangle.

2. The bisectrices of the interior angles of a rectangle form a square.

3. If the points in the sides of a square, at equal distances from the vertices taken in their order, be joined by straight lines, these straight lines will themselves form a square.

4. If the angles adjacent to one base of a trapezoid are equal, those adjacent to the other base are also equal.

5. If from any point in the base of an isosceles triangle parallels to the equal sides are drawn, the perimeter of the parallelogram thus formed is equal to the sum of the equal sides of the triangle.

SECTION III.-DIAGONALS AND DIAMETERS.

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DEFINITIONS.

199. A diameter of a quadrilateral is a straight line joining the middle points of two opposite sides, as A.

200. A trapezoid is a quadrilateral with only two parallel sides, as 1.

201. A trapezium is a quadrilateral with none of its sides parallel, as 2.