4. When 2 lines pass beyond the junction-points, IV. Three straight lines meeting at 3 points. 1. When no line passes a junction-point, (fig. 29.) 1 R. 2 ac. 1 ob. 2 ac. 3 ac. 2. When 1 line passes 1 point, (fig. 30.) 2 R. 2 ac. 1 R. 1 ob. 2 ac. 2 ac. 2 ob. 1 ob. 3 ac. 3. When 1 line passes 2 points, (fig. 31.) 2 R. 1 ob. 2 ac. 1 R. 2 ob. 2 ac. 3 ob. 2 ac. 2 ob. 3 ac. 3-2. When 2 lines pass through the same point, (fig 31-2.) 4 R. 2 ac. 1 R. 2 ob. 3 ac. 2 ob. 4 ac. 3 ob. 3 ac. 4. When 2 lines pass each 1 point, (fig. 32.) 2 R. 1 ob. 2 ac. 1 R. 2 ob. 2 ac. 3 ob. 2 ac. 2 ob. 3 ac. 5. When 1 line passes 2 points and another passes 1, (fig. 33.) 2 R. 2 ob. 2 ac. 3 ob. 3 ac. 1 R. 3 ob. 3 ac. 4 ob. 3 ac. 4 R. 1 ob. 2 ac. 2 R. 2 ob. 3 ac. 3 ob. 4 ac. 6. When 2 lines pass 2 points each, (fig. 34.) 4 R. 2 ob. 2 ac. 2 R. 3 ob. 3 ac. 4 ob. 4 ac. 7. When 3 lines pass 1 point each, (fig. 35.) 8. When 1 line passes 2 points, and 2 lines pass 1 9. When 2 lines pass 2 points each, and 1 line passes 1 point, (fig. 37.) 4 R. 3 ob. 3 ac. 2 R. 4 ob. 4 ac. 5 ob. 5 ac. 10. When each line passes 2 points, (fig. 38.) 4 R. 4 ob. 4 ac. 6 ob. 6 ac. 34. If there be only one angle at a point, we designate it by the letter at the vertex, as the point where the sides meet and form the angle is called. If there be more than one angle at the same point, we make use of three letters, as, (fig. 39,) BAC, CAD, &c., the letter at the vertex of the angle being placed in the middle; or we sometimes use one letter placed within the vertex; as the angle BAC may also be called the angle x. Now let us suppose the side AC of the acute angle BAC to depart more and more from AB, so as successively to reach AD, AE, and AF. By this movement, the mutual inclination of the sides AB and AC will be changed, and the angle BAC will become greater and greater thus BAD is a right angle; BAE is an obtuse angle; BAF is in fact no angle at all, though it is sometimes considered as equal to two right angles; for BAD is a right angle; therefore DAF must be a right angle; since by the definition of a right angle they are equal. If we wish to make more than two angles at the point A, on the same side of the line BF, it can only be done by dividing one or both of these right angles, and the whole taken together will only be equal to 2 R. A. In like manner the sum of all the angles which can be made on the other side of the line BF, at the point A, is equal to 2 R. A.; therefore the sum of all the angles, which can be made about the point A, or about any point, is equal to 4 R. A. Let us suppose the side AC to be moved forward until it coincides with AG; then BAG can be considered as an angle; this angle, which is greater than 2 R. A., is called a convex angle. All angles not convex are concave angles. Obtuse, right, and acute angles are Wherever there is a concave angle, all concave angles. there will also be a convex angle. 35. We will now seek how many concave and convex angles may be formed by straight lines going out, or RADIATING from 1 point. Two straight lines may form 2 angles, viz. : 1 convex and 1 concave. Three straight lines may form 6 angles, viz.: 3 convex and 3 concave. The whole number of angles formed at a point is double the number of the concave angles. For as there is a convex wherever there is a concave angle, we may reckon two angles at each vertex. The number of simple and compound angles always equals the number of concave and convex, for the sum of each is the greatest possible number of angles. Four straight lines. Each straight line, or ray, may form 2 angles with each of the other rays; therefore each ray 32 angles; 4 rays 4 x 3 x 2 angles. But each angle has 2 sides; therefore the product 4 × 3 × 2 must 4×3×2 2 be divided by 2; thus we have angles. =4X3=12 We may proceed in a similar manner with any number of rays. We shall find this general rule. To find the number of angles, both concave and convex, made by a given number of straight lines radiating from one point; Multiply the number of lines by the same number, less 1. 2 straight lines give 2 x 12 angles. 36. Suppose a given number of simple angles to be formed about a point, let us consider what kind of angles they may be. 1. Of two angles, one will be a concave angle less than 2 R. A.; the other a convex angle greater than 2 R. A. As much as the one is less than 2 R. A., by just so much will the other be greater than 2 R. A., since the sum of both is equal to 4 R. A. For the same reason there can only be one convex angle about a point. 2. Three angles. There may be 3 concave, or 2 concave and 1 convex. If the 3 concave angles are equal, each is an obtuse angle; if they are not equal, there may be either 3 obtuse, or 2 obtuse and 1 R. A., or 2 ob. and 1 ac. 3. Four angles. There may be 4 concave, or 3 conThe 4 concave angles may be as cave and 1 convex. follows: 37. We will now consider the kind of concave angles, which different polygons may have. 1. The Quadrilateral. |