be understood. It is seldom requisite to consider said to be a straight angle. DEF. 12. When three straight lines are drawn from a point, if one of them be regarded as lying between the other two, the angles which this one (the mean) makes with the other two (the extremes) are said to be adjacent angles : and the angle between the extremes, through which a line would turn in passing from one extreme through the mean to the other extreme, is the sum of the two adjacent angles. DEF. 13. The bisector of an angle is the straight line that divides it into two equal angles. DEF. 14. When one straight line stands upon another straight line and makes the adjacent angles equal, each of the angles is called a right angle. OBS. Hence a straight angle is equal to two right angles ; or, a right angle is half a straight angle. DEF. 15. A perpendicular to a straight line is a straight line that makes a right angle with it. DEF. 16. An acute angle is that which is less than a right angle. DEF. 17. An obtuse angle is that which is greater than one right angle, but less than two right angles. DEF. 18. A reflex angle is a term sometimes used for a major conjugate angle. DEF. 19. When the sum of two angles is a right angle, each is called the complement of the other, or is said to be complementary to the other. a DEF. 20. When the sum of two angles is two right angles, each is called the supplement of the other, or is said to be supplementary to the other. Def. 21. The opposite angles made by two straight lines that intersect are called vertically opposite angles. DEF. 22. A plane rectilineal figure is a portion of a plane surface inclosed by straight lines. When there are more than three inclosing straight lines the figure is called a polygon. DEF. 23. A polygon is said to be convex when no one of its angles is reflex. DEF. 24. A polygon is said to be regular when it is equilateral and equiangular ; that is, when all its sides and angles are equal. Dec. 25. A diagonal is the straight line joining the vertices of any angles of a polygon. which have not a common arm. DEF. 26. The perimeter of a rectilineal figure is the sum of its sides. DEF. 27. The area of a figure is the space inclosed by its boundary Def. 28. A triangle is a figure contained by three straight lines. DEF. 29. A quadrilateral is a polygon of four sides, a pentagon one of five sides, a hexagon one of six sides, and so on. GEOMETRICAL AXIOMS. 2. 1. Magnitudes that can be made to coincide are equal. Two straight lines that have two points in common lie wholly in the same straight line. I. POSTULATES. Let it be granted that A straight line may be drawn from any one point to any other point. A terminated straight line may be produced to any length in a straight line. 3. A circle may be described from any centre, with a radius equal to any finite straight line. 2. SECTION 1. ANGLES AT A POINT. THEOR. I. All right angles are equal to one another. perpendicular can be drawn to that line. COR. 2. The complements of equal angles are equal. Cor. 3. The supplements of equal angles are equal. THEOR. 2. If a straight line stands upon another straight line, it makes the adjacent angles together equal to two right angles. COR. All the angles made by any number of straight lines meeting at a point are together equal to four right angles. THEOR. 3. If the adjacent angles made by one straight line with two others are together equal to two right angles, these two straight lines are in one straight line. THEOR. 4. If two straight lines cut one another, the vertically opposite angles are equal to one another. SECTION 2. TRIANGLES. Def. 30. An isosceles triangle is that which has two sides equal. DEF. 31. A right-angled triangle is that which has one of its angles a right angle. An obtuse-angled triangle is that which has one of its angles an obtuse angle. All other triangles are called acute-angled triangles. DEF. 32. Any side of a triangle may be called the base, and the opposite angular point is then called the vertex. Def. 33. The side of a right-angled triangle which is opposite to the right angle is called the hypotenuse. DEF. 34. Figures that may be made by superposition to coin cide with one another are said to be identically equal; or they are said to be equal in all respects. THEOR. 5. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles included by these sides equal, then the triangles are identically equal, and of the angles those are equal which are opposite to the equal sides. [By Superposition.]* THEOR. 6. If two triangles have two angles of the one equal to two angles of the other, each to each, and have likewise the sides between the vertices of these angles equal, then the triangles are identically equal, and of the sides those are equal which are opposite to the equal angles. [By Superposition.] THEOR. 7. If two sides of a triangle are equal, the angles opposite to those sides are equal. [By a single application of Theor. 5, or directly by Superposition.] COR. If a triangle is equilateral, it is also equiangular. THEOR. 8. If two angles of a triangle are equal, the sides * Throughout this Syllabus a method of proof has been indicated wherever it was felt that this would make the principles upon which the Syllabus is drawn up more readily understood. opposite to those angles are equal. [By Theor. 6 or directly by Superposition.] COR. If a triangle is equiangular, it is also equilateral. THEOR. 9. If any side of a triangle is produced, the exterior angle is greater than either of the interior opposite angles. THEOR. 10. Any two angles of a triangle are together less than two right angles. COR. 1. If a triangle has one right angle or obtuse angle, its remaining angles are acute. Cor. 2. From a given point outside a given straight line, only one perpendicular can be drawn to that line. THEOR, II. If two sides of a triangle are unequal, the greater side has the greater angle opposite to it. THEOR. 12. If two angles of a triangle are unequal, the greater angle has the greater side opposite to it. THEOR. 13. Any two sides of a triangle are together greater than the third side. Cor. The difference of any two sides of a triangle is less than the third side. THEOR. 14. If from the ends of a side of a triangle two straight lines are drawn to a point within the triangle, these are together less than the two other sides of the triangle, but contain a greater angle. THEOR. 15. 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; and of the others, those which make equal angles with the perpendicular are equal; and that which makes a greater angle with the perpendicular is greater than that which makes a less angle. |