✨ Mathematics Syllabus
Feb. 16.] THE NEW ZEALAND GAZETTE. 773.
(iii.) A pair of interior angles on the same side of the cutting line are together equal to two right angles—
then the two straight lines are parallel; and the converse.
Straight lines which are parallel to the same straight line are parallel to one another.
If there are three or more parallel straight lines, and the intercepts made by them on any straight line that cuts them are equal, then the corresponding intercepts on any other straight line that cuts them are also equal.
Triangles and other Rectilinear Figures.
The sum of the angles of a triangle is equal to two right angles.
If the sides of a convex polygon are produced in order, the sum of the angles so formed is equal to four right angles.
If two triangles have two sides of the one equal to two sides of the other, each to each, and also the angles contained by those sides equal, the triangles are congruent.
If two triangles have two angles of the one equal to two angles of the other, each to each, and also one side of the one equal to the corresponding side of the other, the triangles are congruent.
If two sides of a triangle are equal, the angles opposite to these are equal; and the converse.
If two triangles have the three sides of the one equal to the three sides of the other, each to each, the triangles are congruent.
If two right-angled triangles have their hypotenuses equal, and one side of the one equal to one side of the other, the triangles are congruent.
If two sides of a triangle are unequal, the greater side has the greater angle opposite to it; and the converse.
Of all the straight lines that can be drawn to a given straight line from a given point outside it, the perpendicular is the shortest.
The opposite sides and angles of a parallelogram are equal, each diagonal bisects the parallelogram, and the diagonals bisect one another.
Areas.
Parallelograms on the same or equal bases and of the same altitude are equal in area.
Triangles on the same or equal bases and of the same altitude are equal in area.
Equal triangles on the same or equal bases are of the same altitude.
Illustrations and explanations of the geometrical theorems corresponding to the following algebraical identities:—
k (a + b + c + ..) = ka + kb + kc + ..
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b) (a - b).The square on a side of a triangle is greater than, equal to, or less than the sum of the squares on the other two sides, according as the angle contained by those sides is obtuse, right, or acute. The difference in the cases of inequality is twice the rectangle contained by one of the two sides and the projection on it of the other.
Loci.
The locus of a point which is equidistant from two fixed points is the perpendicular bisector of the straight line joining the two fixed points.
The locus of a point which is equidistant from two intersecting straight lines consists of the pair of straight lines which bisect the angles between the two given lines.
The Circle.
A straight line, drawn from the centre of the circle to bisect a chord which is not a diameter, is at right angles
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NZ Gazette 1912, No 15
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Class D Teacher Examination - Elementary Mathematics II. Syllabus
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🎓 Education, Culture & ScienceTeacher examination, Class D, Mathematics, Geometry, Theorems, Loci, Circle