# University of New Zealand. — Examination for Matriculation, December, 1885. — Euclid

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## Euclid.

 1 Define a right angle, a rectangle, and a triangle. Classify triangles with respect to their angles, and with respect to their sides. Draw a right-angled isosceles triangle. 2 If two triangles have two sides of the one equal to two sides of the other each to each, and have likewise the angles contained by those sides equal, the triangles shall be equal in every respect. What name is given to the method of proof adopted in this proposition? Quote the axiom on which it depends. 3 The angles which one straight line makes with another upon the one side of it are together equal to two right angles. The sides of a triangle are produced beyond the base; shew that if the angles at the base are equal, the angles on the other side of the base shall also be equal. 4 Define parallel straight lines, a parallelogram, and a rhombus. Shew that if the diagonal of a parallelogram bisects one of the angles through which it passes the parallelogram is a rhombus. 5 To describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle. 6 If the square described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it, the angle contained by those two sides is a right angle. If the longest sides of a right-angled triangle are 12 and 13, find the shortest side. 7 If a straight line be divided into any two parts, the squares of the whole line and of one of the parts are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. page break 8 To describe a square that shall be equal to a given rectilineal figure. 9 Through E, the bisection of the diagonal BD of a quadrilateral ABCD, draw FEG parallel to AC, and shew that AG will bisect the figure. Candidates coming up for the Preliminary Medical Examination are requested to omit questions 3, 6, and 9, and to substitute for them the three following questions:— 10 The diameter is the greatest chord in a circle, and of all others that which is nearer to the centre is greater than one more remote. A point P is given within a circle; shew how to draw-through that point the longest and the shortest possible chords. 11 If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles made by this line with the tangent are equal to the angles which are in the alternate segments of the circle. 12 Upon a given straight line to describe a segment of a circle containing an angle equal to a given angle. Having given the base, the vertical angle, and the altitude of a triangle; construct the triangle.