88. To measure the number of degrees in a given angle. Let ABC be the given angle. From the centre C, with a radius equal to the chord of 60° on the line of chords, describe the arc AD. Lay the distance AD on the line of chords FIG. 20 D with the compasses; the number it extends to will show the number of degrees in the arc AD and in the angle B. If AD be greater than the line of chords, measure it in parts, and add. Problem 13. 89. At any distance from a straight line, to draw a straight line parallel to it. A FIG. 21 B E Let AB be the given straight line, and the length of the line C, the distance of the required parallel. From D and E, any two points in AB, with the radius C, describe two arcs on the same side of AB. Draw FG, just touching, but not cutting the arcs— a tangent (32) to both. FG will be parallel to AB. Problem 14. 90. To draw through a given point a straight line parallel to a given straight line. FIG. 22 B Let AB be the line, and C the point. From C, with any radius of sufficient length, describe an arc cutting AB in D. With the same radius, from D, cut AB in E, and from E, cut the arc in F. Join CF. CF will be parallel to AB. Or, take any point D in AB; join CD; and at C, in the line CD, make the angle DCF equal to the angle CDA, by Problem 8. CF will be parallel to AB. 91. Note 1.-This depends on the important geometrical truth that, if a line (CD) meeting two others (CF, AB) makes the alternate angles equal (ADC equal to DCF), these two lines are parallel. Note 2.-Usually a line is drawn parallel to another by means of the instrument called "The Parallel Ruler." This may also be done by the " Square," or "Triangle," used to draw a right angle. If, from B, GB were drawn perpendicular to AB, a perpendicular to GB at G would be parallel to AB. THE TRIANGLES. 27 DEFINITIONS-SECOND SERIES. Rectilineal, or Straight-lined Plane Figures. 92. There are three kinds of rectilineal plane figures -the triangle, the quadrilateral, and the polygon. The Triangle. 93. A TRIANGLE is a plane figure contained by three straight lines, which are called its sides; as, the triangle ABC. 94. There are seven things to be considered about a triangle-the three sides, the three angles, and the area, or extent of surface which it contains. 95. Any side is said to be adjacent B FIG. 23 to the angles which it aids in forming, and to subtend or be opposite to the other angle. AB is adjacent to the angles A and B. It subtends, or is opposite to C. 96. One side of a triangle is sometimes called the base; the angular point opposite to the base is then called the vertex of the triangle: the angle there is called the vertical angle. 97. The ALTITUDE, or height of a triangle, is the perpendicular from one angular point on the opposite side, or on that side produced, which is then called "the base." BD is the altitude in each of the triangles in fig. 24 ; in one, it falls without the triangle, the base AC being produced. The distances from the perpendicular to the ends of the base are called the segments of the base; that is, DA and DC in both figures. Varieties of Triangles. 98. AN ACUTE ANGLED TRIANGLE has all its angles acute. 99. AN OBTUSE ANGLED TRIANGLE has one obtuse angle. A triangle can only have one obtuse angle. Fig. 23 is an acute angled triangle. Both triangles, ABC, in fig. 24, are obtuse angled. E FIG. 25 A 100. A RIGHT ANGLED TRIANGLE has one right angle. triangle can only have one right angle. EDF is a right angled triangle. IOI. In a right angled triangle, the side opposite the right angle is called the hypotenuse. The sides containing the right angle are often called base and perpendicular. ED is the hypotenuse. 102. AN EQUILATERAL TRIANGLE has all its sides equal. 103. AN ISOSCELES TRIANGLE has two of its sides equal. The angle between the two equal sides is often called "the vertical angle," and the opposite side "the base." The Quadrilateral. 104. A QUADRILATERAL is a plane figure contained by four straight lines called its sides. 105. THE DIAGONAL of a quadrilateral is a straight line between two opposite angles. Every quadrilateral THE PARALLELOGRAM. 29 may have two diagonals; as, in fig. 26, a straight line from A to C, or from B to D. 106. There are three kinds of quadrilaterals-the parallelogram, the trapezoid, and the trapezium. The Parallelogram. 107. A parallelogram is a quadrilateral of which the opposite sides are parallel. The figure ABCD FIG. 26 sides of every parallelogram are equal; the opposite angles also are equal; and a diagonal divides it into two triangles equal in all respects. BC is equal to AD; AB to CD. The angles A and C are equal, also the angles B and D. A diagonal, as BD, divides ABCD into two triangles, ABD, CBD, equal in all respects. 109. There are four kinds of parallelograms-the square, the rectangle, the rhombus, the rhomboid. IIO. A SQUARE is a parallelogram having all its sides equal, and all its angles right angles. EFGH is a square. A square is spoken of as the square of any of its sides. The adjoining figure may be called the square of EH, or the square of EF, &c. And it may be expressed shortly EH2, or EF2. F FIG. 27 H In The square is a figure of great use in geometry. mensuration it is taken as the unit of measure for expressing the areas of figures. III. A RECTANGLE (or oblong) is a parallelogram |