An angle may be designated by the letter at the vertex when no other angle has the same vertex; as, the angle O: y C B If the but, when two or more angles have the same vertex, they must be designated by letters in the opening, or by three letters, the two on the sides, and the one at the vertex between them; as, A angle x or AOB, angle y or BOC. Angles may be added; as, AOB+ BOC AOC. DEF. 2. The sign stands for angle; as, ZAOB. line OB be turned about the point O towards OA, the angle AOB will gradually diminish and become zero when OB corresponds with, or has the same direction as, OA. If OB be turned in the opposite direction about the point O, the angle AOB will increase. When OB lies in the opposite direction of OA, the angle AOB is called an extended angle. B The two sides of an extended angle lie in the same straight line. B' A DEF. 3. A right angle is half an extended angle. It is designated simply by the letter R; as, ZAOB' = B'OB = R. DEF. 4. When one straight line meets another so as to form a right angle, the lines are said to be perpendicular to each other; as OB' is perpendicular to OA, which may be expressed by the sign OB' OA; also OA LOB'. B DEF. 5. In all other positions, OB is called an oblique DEF. 7. An angle greater than a right angle is called an obtuse angle; as, AOB. B DEF. 8. Two angles which have the same vertex, and a common side between them, are called contiguous angles; as, ZAOB' and B'OB. When the two outer sides, OA and OB, lie in the same straight line, they are called adjacent angles. B A COR. 1. All extended angles are equal; for they may be so laid upon each other that their sides will correspond.. COR. 2. All right angles are equal (3, 3). COR. 3. From a point in a straight line, only one perpendicular can be drawn to that line; for, in the revolution of the line OB' about the point O, there can be but one position in which AOB' = AOB = R. By the term angle, we will understand an angle less than an extended angle. IV. Theorem. Two adjacent angles are together equal to two right angles. For the two adjacent angles, AOB′ and B'OB, are together equal to the extended angle AOB = 2R for every position of the line OB'. COR. 1. If one of two adjacent angles is a right angle, the other is also a right angle. If two adjacent angles are equal, each is a right angle. COR. 2. The sum of all the angles at a point on the same side of a straight line are together equal to two right angles; for AOC+ COD + DOE + EOB = AOB = 2R. B E D A COR. 3. The sum of all the angles that may be formed at the point O, upon the other side of AB, is also equal to two right angles. Hence all the angles formed about a point are together equal to four right angles. D V. DEF. 1. Two angles are said to be complements of each other when their sum is equal to one right angle; as, ZAOC and ZCOD. DEF. 2. Two angles are said to be supplements of each other when A their sum is equal to two right angles; as, ZAOC and COB. B COR. 3. It is clear that the complements of equal angles are equal to each other, and that the supplements of equal angles are equal to each other. VI. Theorem. If the sum of two contiguous angles is equal to two right angles, the two outer sides form one straight line. HYPOTHESIS. AOC+COB=2R. B A TO PROVE. BO and OA form one straight line. PROOF. The angle AOC is the supplement of COB (5, 2).* But the angle which CO makes with the prolongation of BO is also the supplement of COB (4), and is therefore equal to the angle AOC (5, 3). Hence BO and OA form one straight line. * Let the student quote every reference in full, and show its application. VII. Theorem. If two straight lines cut each other, the opposite or vertical angles are equal. HYPOTH. The line AB cuts CD. TO PROVE. AEC DEB. PROOF. AEC+AED=2R (4); also D E hence DEB+AED=2R (4) C AEC+AED DEB+AED (Ax. 1); B COR. If one of the angles is a right angle, the three remaining angles will be right angles also. PARALLEL LINES. VIII. A DEF. 1. Straight lines which have the same direction are called parallel lines. The sign || stands for parallel; as, AB || CD. C B D AXIOM. Two parallel lines lie in the same plane, and cannot meet, however far both be produced. COR. 1. Through the same point, A, only one line, AB, can be drawn parallel to a given line, CD; for any other line, AE, A must have a different direction from C AB, and, consequently, from CD also. E B D COR. 2. If a straight line cuts one of two parallel lines, it will cut the other also, if sufficiently produced; for, if not, it would have the same direction, and be parallel to it. COR. 3. Two straight lines which are parallel to a third line are parallel to each other. If A DEF. 2. Two straight lines that are not parallel are said to converge in the direction they approach each other, and diverge in the opposite direction. E IX. DEF. 1. When two straight lines, AB and CD, are cut by a third line, EF, eight angles are formed, four exterior angles, 1, 2, 3, 4; and four interior angles, 5, 6, 7, 8. They are divided into the following pairs : — -B 1\2 A 5 6 78 3 D DEF. 2. Corresponding angles are an exterior and interior angle on the same side of the cutting line, but not adjacent; as, 2 and 8, 6 and 4, 1 and 7, 5 and 3. DEF. 3. Alternate angles are two exterior angles or two interior angles on opposite sides of the secant line, but not adjacent; as, 1 and 4, 2 and 3, 5 and 8, 6 and 7. DEF. 4. Opposite angles are two exterior or two interior angles upon the same side of the cutting line; as, 1 and 3, 2 and 4, 5 and 7, 6 and 8. DEF. 5. The cutting line EF is called a secant. X. Theorem. If two parallels are cut by a third line, 1. The corresponding angles are equal. 2. The alternate angles are equal. 3. The sum of two opposite angles is equal to two right |