GEOMETRICAL EXERCISES. DEFINITIONS. One angle is called the complement of another, when the two together make up a right angle. One angle is called the supplement of another, when the two together make up two right angles. When two straight lines are drawn from a point, they form an angle less than two right angles, and also an angle greater than two right angles. The latter is called a reflex angle, and together with the former makes up four right angles. When two sides of a polygon form a re-entrant angle, the reflex angle, not the smaller angle, is considered as one of the angles of the polygon. A scalene triangle is one contained by three unequal straight lines. An oblong has all its angles right angles, but has not all its sides equal. A rhombus has all its sides equal, but its angles are not right angles. A rhomboid has its opposite sides equal to each other, but all its sides are not equal nor its angles right angles. A trapezoid is a four-sided figure which has two of its sides parallel, and the remaining two not parallel. N.B. This figure is sometimes called a trapezium; but Euclid calls all quadrilateral figures trapeziums which are not parallelograms. If a straight line AB be divided in P and Q so that AP: AB as PQ: QB, then AB is said to be divided harmonically. Figures whose areas are equal are said to be equivalent to one another. A tetrahedron is a solid contained by four planes. A polyhedron is a solid contained by more than four planes. A dihedral angle is formed by the intersection of two planes. I. PLANE GEOMETRY. BOOK I. IF on the same base and on the same side of it two isosceles triangles are drawn, the vertex of one triangle must fall within the other. 2. If on the same base and on opposite sides of it two isosceles triangles are drawn, the straight line joining their vertices shall bisect the base. 3. Let the equal sides AB, AC of an isosceles triangle be produced to F, G making AF = AG: join FC, BG intersecting in H, then AH will bisect the angle BAC. 4. Prove that in the previous figure the lines bisecting the s at F, G will meet AH in the same point. 5. If the angles of one triangle are not severally equal to those of another, neither are their sides equal. 6. Divide a given angle into four equal angles. Describe a right-angled triangle whose hypothenuse shall be equal to one straight line, and one of its sides equal to another straight line. 8. Draw a straight line through a given point cutting off equal parts from the arms of a given angle. 9. Find a point in a straight line equidistant from two given points without it. 10. Trisect a given right angle. II. Construct a triangle having given the base, one of the angles at the base, and the sum of the sides. I2. Draw a straight line through a given point equally inclined to two given straight lines. 13. Find a point in the base of a triangle equidistant from the two sides. 14. Describe an isosceles ▲ having each of the sides double the base. 15. Describe an isosceles triangle having each of the sides three times the size of the base. 16. Find a point in a straight line at a given distance from a given point. 17. Find a point in the circumference of a circle at a given distance from a given point. 18. The lines drawn from the angular points of a s through any point within it to the opposite sides are together greater than the semi-perimeter of the A. 19. The difference between any two sides of a triangle is less than the third side. 20. Find the shortest path from a given point in one of two straight lines to the other, and then back again to the former straight line. 21. Given the base and one of the sides of an isosceles triangle, construct it. 22. Prove that the sum of the distances of any point from the three angles of a triangle is greater than half the perimeter. 23. Determine the shortest path from one point to another subject to the condition that it shall meet a given straight line. 24. Determine the shortest path from one point to another subject to the condition that it shall meet two given straight lines. 25. The four sides of a quadrilateral figure are together greater than the two diagonals. 26. Prove that the straight lines drawn through the extremities of the base of an isosceles triangle equally inclined to it and terminated by the sides, are equal to one another. 27. If straight lines be drawn to the angular points of a triangle from a point within it, then of the three angles thus formed there cannot be more than one acute angle. 28. Trisect a given straight line. 29. Describe a triangle equiangular to a given triangle and having its perimeter equal to a given straight line. 30. Find a point situated at a given distance from a given point, and also at a given distance from a given straight line. Is the problem always possible? 31. Through two given points draw two straight lines forming with a given straight line an equilateral triangle. 32. Through two given points draw two straight lines forming with a given straight line a triangle equiangular to a given triangle. 33. Describe a rhombus having each of one pair of opposite angles double each of the other pair. 34. If the two diameters of a parallelogram be drawn, it will be divided into four equal parts. 35. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram. 36. If the opposite angles of a quadrilateral are equal, the figure is a parallelogram. 37. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram. 38. Draw lines through the angular points of a parallelogram which shall form another parallelogram double the former. |