14 A SYLLABUS OF [Alternative proofs, (i) by Theors. 14 and 5. (ii) By Theors. 6 and 5.] THEOR. 16. If two triangles have two sides of the one equal to two sides of the other, each to each, but the bases unequal, then the included angles are unequal, the angle of that which has the greater base being greater than the angle of the other. [By Rule of Conversion.] THEOR. 17. If two triangles have two angles of the one equal to two angles of the other, each to each, and have likewise the sides opposite to one pair of equal angles equal, then the triangles are identically equal, and of the sides those are equal which are opposite to equal angles. [By Superposition and Theor. 9.] THEOR. 18. Any two angles of a triangle are together less than two right angles. COR. I. 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. 19. 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. COR. Not more than two equal straight lines can be drawn from a given point to a given straight line. THEOR. 20. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles opposite to one of the equal sides in each equal, then the angles opposite to the other two equal sides are either equal or supplementary, and in the former case the triangles are identically equal. [By Superposition.] COR. Two such triangles are identically equal (1) If the two angles given equal are right angles or obtuse angles. (2) If the angles opposite to the other two equal sides are both acute, or both obtuse, or if one of them is a right angle. (3) If the side opposite the given angle in each triangle is not less than the other given side. SECTION 3. PARALLELS AND PARALLELOGRAMS. DEF. 35. Parallel straight lines are such as are in the same plane and being produced to any length both ways do not meet. AXIOM 5. Two straight lines that intersect one another cannot both be parallel to the same straight line. DEF. 36. A trapezium is a quadrilateral that has only one pair of opposite sides parallel. This figure is sometimes called a trapezoid. DEF. 37. A parallelogram is a quadrilateral whose opposite sides are parallel. DEF. 38. When a straight line intersects two other straight lines it makes with them eight angles, which have received special names in relation to one another. 16 A SYLLABUS OF 2/1 3/4 6/5 18 Thus in the figure 1, 2, 7, 8 are called exterior angles, and 3, 4, 5, 6, interior angles ; again, 4 and 6, 3 and 5 are called alternate angles; lastly, 1 and 5, 2 and 6, 3 and 7, 4 and 8, are called corresponding angles. DEF. 39. The orthogonal projection of one straight line on another straight line is the portion of the latter intercepted between perpendiculars let fall on it from the extremities of the former. THEOR. 21. If a straight line intersects two other straight lines and makes the alternate angles equal, the straight lines are parallel. [Contrapositive of Theor. 9.] THEOR. 22. If two straight lines are parallel, and are intersected by a third straight line, the alternate angles are equal. [By Rule of. Identity, using Ax. 5.] THEOR. 23. If a straight line intersects two other straight lines and makes either a pair of alternate angles equal, or a pair of corresponding angles equal, or a pair of interior angles on the same side supplementary ; then, in each case, the two pairs of alternate angles are equal, and the four pairs of corresponding angles are equal, and the two pairs of interior angles on the same side are supplementary. THEOR. 24. Straight lines that are parallel to the same straight line are parallel to one another. [Contrapositive of Ax. 5.) THEOR. 25. If a side of a triangle is produced, the exterior angle is equal to the two interior opposite angles; and the three interior angles of a triangle are together equal to two right angles. Cor. In a right-angled triangle the two acute angles are complementary THEOR. 26. All the interior angles of any polygon together with four right angles are equal to twice as many right angles as the figure has sides. COR. All the exterior angles of any convex polygon are together equal to four right angles. THEOR. 27. The adjoining angles of a parallelogram are supple mentary, and the opposite angles are equal. COR. If one of the angles of a parallelogram is a right angle, all its angles are right angles. DEF. 40. The figure is then called a rectangle. OR. 28. The opposite sides of a parallelogram are equal to one another, and a diagonal divides it into two identically equal triangles. COR. If the adjoining sides of a parallelogram are equal, all its sides are equal. DEF. 41. The figure is then called a rhombus. DEF. 42. A square is a rectangle that has all its sides equal. THEOR. 29. If two parallelograms have two adjoining sides of the one respectively equal to two adjoining sides of the other, and likewise an angle of the one equal to an angle of the other; the parallelograms are identically equal. [By Superposition.] COR. Two rectangles are equal, if two adjoining sides of the one are respectively equal to two adjoining sides of the other; and two squares are equal, if a side of the one is equal to a side of the other. THEOR. 30. If a quadrilateral has two opposite sides equal and parallel, it is a parallelogram. THEOR. 31. Straight lines that are equal and parallel have equal 18 A SYLLABUS OF projections on any other straight line; conversely, parallel straight lines that have equal projections on another straight line are equal, and equal straight lines that have equal projections on another straight line make equal angles with that line, or are parallel to it. THEOR. 32. If there are three parallel straight lines, and the intercepts made by them on any straight line that cuts them are equal, then the intercepts on any other straight line that cuts them are equal. COR. I. The straight line drawn through the middle point of one of the sides of a triangle parallel to the base passes through the middle point of the other side. COR, 2. The straight line joining the middle points of two sides of a triangle is parallel to the base. [Cor. I. and Rule of Identity.] SECTION 4. PROBLEMS. PROB. 1. To bisect a given angle. a given point in it. PROB. 3. To draw a perpendicular to a given straight line from a given point outside it. PROB. 4. To bisect a given straight line. PROB. 5. At a given point in a given straight line to make an angle equal to a given angle. FROB. 6. To draw a straight line through a given point parallel to a given straight line. |