PROPOSITION XXVIII.-THEOREM.

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

Let AC, A'C', have AB = A'B', ADA'D', and the angle BAD

D

с

D

For they may evidently be applied the one to the other, so as to coincide throughout. (v. Proposition XXIII.)

59. COROLLARY. Two rectangles

are equal when they have equal bases

and equal altitudes.

PROPOSITION XXIX.-THEOREM.

60. The opposite sides of a parallelogram are equal and the opposite angles are equal.

Suggestion. Draw a diagonal AC. ACB

and CAD are equal, by Proposition XXV.
CAB and ACD are equal, by Proposition B

XXV.

Ꭰ

Hence the triangles ABC and ADC are equal, by Proposition VII.

EXERCISES.

1. Theorem.—If one angle of a parallelogram is a right angle, all the angles are right angles, and the figure is a rectangle. 2. Theorem.-If two angles have the sides of one respectively parallel to the sides of the other, they are equal, or supplementary.

-

3. Theorem. Two parallel lines are everywhere equidistant.

B

A

A'

B

B

PROPOSITION XXX.-THEOREM.

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

Suggestion. Let AD be equal and parallel

to BC. Draw a diagonal AC.

The triangles ABC and ADC are equal,

by Proposition VI. Therefore the angles

BAC and ACD are equal, and AB and CD are parallel, by Proposition XXIV.

PROPOSITION XXXI.—THEOREM.

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

Suggestion. Draw a diagonal, and prove the two triangles equal.

PROPOSITION XXXII.-THEOREM.

63. The diagonals of a parallelogram bisect each other.

Suggestion. The triangles AED and BEC are equal, by Proposition VII.

EXERCISES.

A

1. Theorem. The diagonals of a rectangle are equal.

2. Theorem. The diagonals of a rhombus are perpendicular to each other.

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

4. Theorem.-If the diagonals of a parallelogram are equal, the figure is a rectangle.

5. Theorem.-If the diagonals of a parallelogram are perpendicular to each other, the figure is a rhombus.

ARRANGEMENT OF WRITTEN EXERCISES.

64. In writing out a demonstration, brevity of statement and clearness of arrangement should be carefully studied, and symbols and abbreviations may be used with profit. The following list is recommended:

65. In arranging a written demonstration, it is well to begin each statement on a separate line, giving the reason for the statement at the end of the line, if it can be written briefly, or in parenthesis immediately below the line, if it cannot be written briefly. The following examples of demonstrations prepared as written exercises, or for a written examination, will serve as illustrations.

(1)

PROPOSITION XII.-THEOREM.

If two angles of a triangle are unequal, the side opposite the greater angle is greater than the side opposite the less angle.

In

B

ABC, let it be given that ACB > B,

(2)

PROPOSITION XXIV.-THEOREM.

When two straight lines are cut by a third, if the alternate interior angles are equal, the two straight lines are parallel.

Let AB cut CD and EF in the points A and B, making

we are to prove

BACABF,

CD || to EF.

Through G, the middle point of AB, draw HI | to CD.

ABE+ABF = 2 rt.,

[BAD + BAC = 2 rt. /.
ABFBAC,

and the proof given above applies.

Prop. III.

Prop. III.