✨ Teacher Examination Regulations (Geometry)
3034
THE NEW ZEALAND GAZETTE.
[No. 98
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 perpen-
dicular 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
to the chord; conversely, the perpendicular to a chord
from the centre bisects the chord.
There is one circle, and one only, which passes through
three given points not in a straight line.
In equal circles (or in the same circle), (i) if two arcs
subtend equal angles at the centres, they are equal;
(ii) conversely, if two arcs are equal, they subtend equal
angles at the centre.
In equal circles (or in the same circle), (i) if two
chords are equal, they cut off equal arcs; (ii) conversely,
if two arcs are equal, the chords of the arcs are equal.
Equal chords of a circle are equidistant from the
centre; and the converse.
The tangent at any point of a circle and the radius
through the point are perpendicular to one another.
If two circles touch, the point of contact lies on the
straight line through the centres.
The angle which an arc of a circle subtends at the
centre is double that which it subtends at any point on
the remaining part of the circumference.
Angles in the same segment of a circle are equal;
and, if the line joining two points subtends equal angles
at two other points on the same side of it, the four points
lie on a circle.
The angle in a semicircle is a right angle; the angle
in a segment greater than a semicircle is less than a right
angle, and the angle in a segment less than a semicircle
is greater than a right angle.
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NZ Gazette 1906, No 98
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Regulations for Examination and Classification of Teachers - Geometry Syllabus
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🎓 Education, Culture & Science26 November 1906
Teacher certification, Examination requirements, Geometry, Triangles, Parallelograms, Areas, Circle theorems, Loci, Geometrical constructions