26
THE NEW ZEALAND GAZETTE.
[No. 1

(b) Construct and cut out two triangles which have the two sides
and the included angle equal. By measurement and
calculation, and also by superimposing, prove equality in
all respects—viz., area, other side, and other angles.
(c) Any two sides of a triangle are together greater than the
third side.
(d) (i) If one side of a triangle is produced, the exterior angle is
greater than either of the interior opposite angles.
(ii) The exterior angle is equal to the sum of the two interior
opposite angles.
(iii) The sum of the three angles of any triangle is equal to two
right angles.
(iv) Any two angles of a triangle are together less than two
right angles.
(v) A triangle cannot have more than one right angle.
(e) Construction of pentagon, hexagon, and octagon, and measure-
ment of the interior angles of each. The geometrical figures
that lend themselves to the formation of a pavement.
(f) The diagonals of a parallelogram bisect each other.
(g) The straight line joining the middle points of two sides of a
triangle is parallel to the base and equal to half the base.
(h) If in a triangle a straight line is drawn parallel to the base
through the middle point of one side it bisects the other
side, hence lead to the method of dividing a given straight
line into any given number of equal parts.
(i) The area of a parallelogram is—
(i) Equal to the area of a rectangle on the same base and
between the same parallels.
(ii) Double the area of a triangle on the same base and
between the same parallels.

ADDITIONAL MATHEMATICS (OPTIONAL).

1. Algebra.—The following course constitutes about half the programme
   outlined in the Department’s syllabus for the Intermediate and P.S.E.
   Examinations :—
   (a) Fundamental operations.
   (b) Easy fractions.
   (c) Factors of expressions that are the product of two binomial fac-
   tors. Factors of such expressions as a³ + b³.
   (d) Common multiples and divisors to correspond.
   (e) Simple equations involving one or two unknown quantities.
   (f) Easy problems.

2. Geometry.—The following has been selected from the programme in
   Geometry as prescribed for the Intermediate and Public Service Examina-
   tions. No theoretical proofs required in Part I but only in Part II. The
   truth of the theorems in Part I should be established by intuition and
   experiment, and the teaching of formal geometry should be based on the
   acceptance of these theorems.
   Part I—
   (a) If a straight line stands on another straight line, the sum of the
   adjacent angles so formed is equal to two right angles, and
   the converse.
   (b) When a straight line cuts two other straight lines, if a pair of
   corresponding angles are equal, the two straight lines are
   parallel; and the converse.
   (c) When a straight line cuts two other straight lines, if (i) a pair
   of alternate angles are equal, or (ii) a pair of interior angles
   on the same side of the cutting lines are supplementary, then
   the two straight lines are parallel; and the converse.
   (d) Straight lines which are parallel to the same straight line are
   parallel to one another.
   (e) If two triangles have two sides of the one equal to two sides of
   the other, each to each, and also the angles included by those
   sides equal, the triangles are congruent.
   (f) If two triangles have three sides of the one equal to three sides
   of the other, each to each, the triangles are congruent.

Part II, section 1—
(a) The sum of the angles of a triangle is equal to two right angles.
(b) If two sides of a triangle are equal, the angles opposite those sides
are equal; and the converse.
(c) If two sides of a triangle are unequal, the greater side has the
greater angle opposite to it; and the converse.