12. On parallelograms.—Learn Definitions 31-34 on page 7, and Definition 1 on page 113. Note that all the angles of a rectangle are right angles. Problem XV.—To construct a parallelogram given the two sides and the included angle. (See figure on page 72.) Draw an angle ACD of the given magnitude; and make the arms CD and AC of the given lengths. Through A draw a line I to CD, and through D drawn a line || to AC. These lines complete the parallelogram. (The diagonal CB is not wanted.) A similar construction enables us to describe a rectangle, square, or rhombus, as in all these figures opposite sides are parallel. Examples IX. 1. Draw a rectangle whose sides measure 3 and 4 cm., and measure both diagonals. 2. Draw a rhombus each of whose sides is 2" and whose acute angle is 60°. Measure the diagonals and show that they bisect one another at right angles. 3. Draw a parallelogram on a base 2", having the left angle = 50° and the adjacent side 1". Show by measurement that opposite angles are equal. 4. On base 1.5" describe above it an isosceles triangle, having each side 2", and another isosceles triangle below it having each side 2.5", and measure the line joining the vertices of the two triangles. Also show by measurement that this line bisects the base. 5. Draw a square of side 2" and measure each diagonal. Show that the diagonals bisect one another. 6. Draw any two straight lines AB and CD which bisect one another at a point 0, at any angle. Join AC, CB, BD, and DA. Show, by measurement, that the two angles DAC and ACB together measure 180°. 7. In the last example if AB were 6 cm. and CD 4 cm. and the angle COA 45°, measure all the other lines and angles. 8. On AB 2" long as diagonal draw a parallelogram whose side BC is 1.5" long and whose side AC is 2.5", and measure the angles of the parallelogram and those at the intersection of the diagonals. 9. Construct the quadrilateral ABCD whose sides are 1", 1", 1-5′′, and 1′′, and one of whose diagonals is 1.7". Measure the other diagonal. 10. Construct the quadrilateral ABCD. AB (1") parallel to DC (2′′) at a distance of 1" from one another, the angle BAD being 120°; measure AC. 11. Construct a quadrilateral ABCD. AB (1.5′′) parallel to CD (2′′). AD being 2", and the diagonal AC 3". Measure the angles. 12. Construct a rectangle ABCD, given diagonal AC = 2′′, and ▲ ACD = 40°. Measure its sides. 13. Draw two lines of lengths 1.4" and 2.4" bisecting each other at right angles. Measure the sides of the quadrilateral of which these lines are diagonals. 14. Draw a circle of radius 1.5". Draw two diameters cutting at an angle of 40°. Measure the angles of the quadrilateral of which these are the diagonals. 15. Draw a circle of radius 1.4". Draw two diameters at right angles. Measure the sides and angles of the quadrilateral of which these are the diagonals. 16. Describe a rectangle in which a diagonal is 2.4", and one side is 1.2′′. Measure the longer side. (Start with the given side.) 17. Knowing that the diagonals of a parallelogram bisect each other, construct a parallelogram whose longer side is 35 mm., and whose diagonals are 50 and 40 mm. respectively. Measure the other sides. 18. Knowing that the diagonals of a square are equal and bisect each other at right angles, construct a square of diagonal 2.8 inches. Measure its sides. III. 1. (a) Twice, (b) four times, (c) six times, (d) ten times. 2. PG. V. 1. 750, 7510, 29°. 2. 90°, 53°, 37°. 4. 120°, 22, 38°. 7. 120°, 34°, 26°. 3. 1.41 in., 1 in., 75°. VI. 1. 1 in., 1.73 in., 90°. 5. 25 mm., 25 mm., 60°. = 7. BC 9. B 10. AB 12. BC 13. AB = 42.4 mm., BC 45°. = 24.5 mm., AB 45°, AC = 29 mm., A 66", B = 134°, C 35°. 2. 2.45 in., 3.35 in., 60°. 4. 101 in., 1.56 in., 100°. 6. 25.7 mm., 32.9 mm., 60°. 20 mm., AC = 34.6 mm., B = 60°. 8. AB = 17.8 mm., A = 35°, B 105°, or BC = 2. 1.75". 15. 15 mm. - 90°, 6. 2.13". VII. 1. 1.22". VIII. 2. 40°, 90°, ·77′′, ·77′′. 5. All right angles. 7. 35". 17. AC = 30°. 3.78 cm., C = 34°. 3. 126°. 4. 41°. 8. 27.6 mm. 8. 24 mm. IX. 1. 5 cm. each. 2. 2", 3.46". 9. 28.1 mm. 3. There are two angles of 50° and two of 130°. 4. 4.24". 5. Each 2.83". 7. Two sides measure 4.63", and the other two 2 12". Two angles measure 5910, and the other two 12030. 8. Angles at C and D each 53°; at A and B each 127°; angles at the inter section of diagonals are 56° and 124°. 9. 1.45". 10. 1.74". 12. 1.29", 1.53". 13. Each 1.39". 16. 2.08". 11. A, 83°; B, 111°; C, 69°; D, 97°. 18. 1.98". EUCLID was a famous Greek mathematician. He was born about 330 B.C., and taught at Alexandria during the reign of Ptolemy I., King of Egypt. The Elements' of Euclid is a work on elementary mathematics, and is divided into thirteen books. Books I., II., III., IV., and VI. treat of Plane Geometry; Book V. treats of Proportion; Books VII., VIII., and IX. treat of the properties of numbers; Book X. treats of surd quantities; Books XI. and XII. treat of Solid Geometry; and Book XIII. contains miscellaneous Propositions in Plane and Solid Geometry. Modern editions of Euclid's 'Elements' usually include Books I.-IV., part of Book V., Books VI. and XI., and sometimes one or two propositions from Book XII. Geometry (Gk. yî, earth; μéтpov, measure) treats of the measurement of lines, surfaces, and solids, with their various properties and relations. ECC. B EUCLID'S ELEMENTS. BOOK I. DEFINITIONS. 1. A point is that which has position, but no magnitude. 2. A line is that which has length without breadth. 3. The extremities of a line are points. 4. A straight line is a line which lies evenly between its extreme points. 5. A surface is that which has length and breadth, but no thickness. 6. The extremities of a surface are lines. 7. A plane surface is a surface in which any two points being taken, the straight line that joins them lies wholly in that surface. 8. A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction. 9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. NOTES ON THE DEFINITIONS. Each Book of Euclid begins with Definitions. A Definition is an explanation of the exact meaning of some important word or term used in the book. 1 and 2. In Euclid a point is supposed to have no size, and a line is supposed to have no breadth. But we cannot actually mark a point without marking some small area; nor can we draw a line without giving it some breadth. 3. A line may be considered as being formed by a number of points placed close beside one another. Therefore, if two lines meet or cut each other, they will meet or cut in a point; and if a point moves about, it will describe or pass along a line. 4. A right line is the same as a straight line. 5. A superficies is the same as a surface. 6. A surface may be considered as being formed by a number of lines placed close together. Therefore, if two surfaces meet or cut each other, they will meet or cut in a line or lines; and if a line moves or turns about, it will describe a surface. 7. A plane surface is often called simply a plane. A flat surface is a plane. 8 and 9. A plane angle is never used in Euclid. supposed to be contained by straight lines, that is, are rectilineal angles. An angle is usually denoted by three letters. Thus BOC, COD, and BOD are three different angles at the point 0. If there is only one angle at a point, the angle may be denoted by a single letter. The vertex is the point where the lines meet and form the angle. The arms of the angle are the lines which form the angle. A corner is an angle, and the size of an angle does not depend on the length of the arms. EXERCISES. 1. What dimensions has a point, a line, a square, a circle, and a marble? 2. How could you tell with a piece of string whether a certain piece of wood was quite flat? 3. How many edges and how many plane surfaces has an ordinary brick ? |