4. How many angles altogether are about a triangle? How many at most of these angles are different in magni tude? What is the least number of angles of different magnitudes about a triangle? 5. In Fig. of 53°, if Q be any point on PC, ^\PAQ=^PBQ. 6. In Fig. of 53°, if the ▲PCB be revolved about PC as an axis, it will become coincident with PCA. 7. The medians to the sides of an isosceles triangle are equal to one another. 8. Prove 58° from the axiom "a straight line is the shortest distance between two given points." 9. Show from 60° that a triangle cannot have two of its angles right angles. 10. If a triangle has a right angle, the side opposite that angle is greater than either of the other sides. 11. What is the locus of a point equidistant from two sides of a triangle? 12. Find the locus of a point which is twice as far from one of two given lines as from the other. 13. Find the locus of a point equidistant from a given line and a given point. SECTION IV. PARALLELS, ETC. 70°. Def.-Two lines, in the same plane, which do not intersect at any finite point are parallel. Next to perpendicularity, parallelism is the most important directional relation. It is denoted by the symbol ||, which is to be read "parallel to" or "is parallel to as occasion may require. The idea of parallelism is identical with that of sameness of direction. Two line-segments may differ in length or in direction or in both. If, irrespective of direction, they have the same length, they are equal; if, irrespective of length, they have the same direction, they are parallel; and if both length and direction are the same they are equal and parallel. Now when two segments are equal one may be made to coincide with the other by superposition without change of length, whether change of direction is required or not. So when they are parallel one may be made to coincide with the other without change of direction, whether change of length is required or not. Axiom.-Through a given point only one line can be drawn parallel to a given line. This axiom may be derived directly from 24°. 71°. Theorem.—Two lines which are perpendicular to the same line are parallel. N P KIM L and M are both to N, then L is to M. Proof.-If L and M meet at any point, two perpendiculars are drawn from that point to the line N. But this is impossible (61°). Therefore L and M do not meet, or they are parallel. Cor. All lines perpendicular to the same line are parallel to one another. 72°. Theorem.-Two lines which are parallel are perpendicular to the same line, or they have a common perpendicular. (Converse of 71°.) L is || to M, and L is 1 to N; then M is to N. Proof.-If M is not to N, through any point P in M, let K be to N. Then But Therefore K is to L. M is to L. K and M are both || to L, which is impossible unless K and M coincide. Therefore L and M are both 1 to N, or N is a common perpendicular. (71°) (hyp.) (70°, Ax.) 73°. Def.-A line which crosses two or more lines of any system of lines is a transversal. Thus EF is a transversal to the lines A In general, the angles formed by a transversal to any two lines are distinguished as follows C e g/h F E a/b B c/d D ά and e, c and g, b and ƒ, d and h are pairs of corresponding angles. c and f, e and d are pairs of alternate angles. c and e, d and fare pairs of interadjacent angles. 74°. When a transversal crosses parallel lines— I. The alternate angles are equal in pairs. 2. The corresponding angles are equal in pairs. 3. The sum of a pair of interadjacent angles is a straight Similarly the other corresponding angles are equal in pairs. Cor. It is seen from the theorem that the equality of a pair of alternate angles determines the equality in pairs of corresponding angles, and also determines that the sum of a pair of interadjacent angles shall be a straight angle. So that the truth of any one of the statements 1, 2, 3 determines the truth of the other two, and hence if any one of the statements be proved the others are indirectly proved also. 75°. Theorem.--If a transversal to two lines makes a pair of alternate angles equal, the two lines are parallel. (Converse of 74° in part.) If LAEF=LEFD, AB and CD are parallel. Proof. Draw PQ as in 74°, I to AB, G/ A P B E (59°) Cor. It follows from 74° Cor. that if a pair of corresponding angles are equal to one another, or if the sum of a pair of interadjacent angles is a straight angle, the two lines are parallel. 76°. Theorem.-The sum of the internal angles of a triangle is a straight angle. B Then BC is a transversal to the parallels AB and CE ; LABC=4BCE. Also, AC is a transversal to the same parallels ; ¿BAC=✩ECD. (74°, 1) (74°, 2) LABC+LBAC=4BCD = supplement of LBCA. LA+LB+LC = 1. q.e.d. Cor. An external angle of any triangle is equal to the sum of the opposite internal angles. (49°, 3) 77°. From the property that the sum of the three angles of any triangle is a straight angle, and therefore constant, we deduce the following 1. When two angles of a triangle are given the third is given also; so that the giving of the third furnishes no new information. 2. As two parts of a triangle are not sufficient to deter mine it, a triangle is not determined by its three angles, and hence one side, at least, must be given (66°, 1). 3. The magnitude of any particular angle of a triangle does not depend upon the size of the triangle, but upon the form only, i.e., upon the relations amongst the sides. 4. Two triangles may have their angles respectively equal and not be congruent. But such triangles have the same form and are said to be similar. 5. A triangle can have but one obtuse angle; it is then called an obtuse-angled triangle. A triangle can have but one right angle, when it is called a right-angled triangle. All other triangles are called acute-angled triangles, and have three acute angles. 6. The acute angles in a right-angled triangle are comple mentary to one another. |