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Page 31 - ... and the converse. If two triangles are equiangular their corresponding sides are proportional ; and the converse. If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar. The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle, and likewise the external bisector externally. The ratio of the areas of similar triangles is equal...

Page 28 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz.

Page 28 - When a straight line cuts two other straight lines, if (i) a pair of alternate angles are equal, or ii) a pair of corresponding angles are equal, or (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.

Page 30 - In a circle, the angle in a semicircle is a right angle ; but 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.

Page 28 - ... into parts in any given proportions. Construction of a triangle equal in area to a given polygon. Construction of tangents to a circle and of common tangents to two circles. Simple cases of the construction of circles from sufficient data. Construction of a fourth proportional to three given straight lines and a mean proportional to two given straight lines. Construction of regular figures of 3, 4, 6 or 8 sides in or about a given circle. Construction of a square equal in area to a given polygon.

Page 29 - If two triangles have the three sides of the one equal to the three sides of SH 19 1 the other, each to each, the triangles are congruent.

Page 27 - Questions may be set in which the use of the set square or of the protractor is forbidden. The questions on Theoretical Geometry shall consist of theorems contained in the annexed Schedule B, together with questions upon these theorems, easy deductions from them, and arithmetical illustrations. Any proof of a proposition shall be accepted which appears to the examiners to form part of a systematic treatment of the subject...

Page 29 - 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...

Page 35 - They will be limited to those branches of Science (such as Physics, Mechanics, and Chemistry) the extension of which is specially important for our national industries.

Page 29 - 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...