If one angle of an oblique-angled triangle is obtuse, the triangle is said to be OBTUSE-ANGLED. If all of the angles are acute, the triangle is said to be ACUTE-ANGLED. 26. Quadrilaterals are classified with reference to the relative directions of their sides. There are then two classes the first class embraces those which have no two sides par allel; the second class embraces those which have at least two sides parallel. Quadrilaterals of the first class, are called trapeziums. Quadrilaterals of the second class, are divided into two species: trapezoids and parallelograms. 27. A TRAPEZOID is a quadrilateral which has only two of its sides parallel. 28. A PARALLELOGRAM is a quadrilateral which has its opposite sides parallel, two and two. There are two varieties of parallelograms: rectangles and rhomboids. 1st. A RECTANGLE is a parallelogram whose angles are all right angles. A SQUARE is an equilateral rectangle. 21. A RHOMBOID is a parallelogram whose angles are all oblique. A RHOMBUS is an equilateral rhomboid. 29. SPACE is indefinite extension. 30. A VOLUME is a limited portion of space, combining the three dimensions of length, breadth, and thickness. AXIOMS. 1. Things which are equal to the same thing, are equa to each other. 2. If equals be added to equals, the sums will be equal. 3 If equals be subtracted from equals, the remainders will be equal. 4. If equals be added to unequals, the sums will be unequal. 5. If equals be subtracted from unequals, the remainders will be unequal. 6. If equals be multiplied by equals, the products will be equal. 7. If equals be divided by equals, the quotients will be equal. 8. The whole is greater than any of its parts. 9. The whole is equal to the sum of all its parts. 10. All right angles are equal. 11. Only one straight line can be drawn joining two given points. 12. The shortest distance from one point to another is measured on the straight line which joins them. 13. Through the same point, only one straight line can be drawn parallel to a given straight line. POSTULATES. 1. A straight line can be drawn joining any two points. 2. A straight line may be prolonged to any length. 3. If two straight lines are unequal, the length of the less may be laid off on the greater. 4. A straight line may be bisected; that is, divided into two equal parts. 5. An angle may be bisected. 6. A perpendicular may be drawn to a given straight line, either from a point without, or from a point on the line, 7. A straight line may be drawn, making with a given straight line an angle equal to a given angle. 8. A straight line may be drawn through a given point, parallel to a given line. NOTE. In making references, the following abbreviations are employed, viz. › A. for Axiom; B. for Book; C. for Corollary; D. for Definition; 1. for Introduction; P. for Proposition; Prob. for Problem; Post. for Postulate; and S. for Scholium. In referring to the same Book, the number of the Book is not given; in referring to any other Book, the number of the Book is given. PROPOSITION I. THEOREM. If a straight line meet another straight line, the sum of the adjacent angles will be equal to two right angles. Let DC meet AB at C: then will the sum of the angles DCA and DCB be equal to two right angles. At C, let CE be drawn perpendicular to AB (Post. 6); then, by definition (D. 12), the angles E D B C ECA and ECB will both be right angles, and conse quently, their sum will be equal to two right angles. The angle DCA is equal to ECA and ECD But, of the angles DCA + DCB = ECA + ECD + DCB; ECD+DCB is equal to ECB (A. 9); hence, DCA + DCB = ECA + ECB. The sum of the angles ECA and ECB, is equal to two right angles; consequently, its equal, that is, the sum of the angles DCA and DCB, must also be equal to two right angles; which was to be proved. Cor. 1. If one of the angles DCA, DCB, is a right angle, the other must also be a right angle. Cor. 2. The sum of the angles BAC, CAD, DAE, EAF, formed about a given point on the same side of a straight line BF, is equal to two right angles. For, their sum is equal to D E B -F A the sum of the angles EAB and EAF; which, from the proposition just demonstrated, is equal to two right angles. DEFINITIONS. If two straight lines intersect each other, they form four angles about the point of intersection, which have received different names, with respect to each other. are 1o. ADJACENT ANGLES those which lie on the same side of one line, and on opposite sides of the other; thus, ACE and ECB, or ACE and ACD, are adjacent angles. A B 2o. OPPOSITE, or VERTICAL ANGLES, are those which lie on opposite sides of both lines; thus, ACE and DCB, or ACD and ECB, are opposite angles. From the pro position just demonstrated, the sum of any two adjacent angles is equal to two right angles. PROPOSITION II. THEOREM. If two straight lines intersect each other, the opposite or vertical angles will be equal. of the adjacent angles ACE and ECB, is also equal to two right angles. But things which are equal to the same thing, are equal to each other (A. 1); hence, |