114. Historical Note. Exercise 69 is known as the pons asinorum, or bridge of asses, since it has proved difficult to many beginners in geometry. This proposition on and the proof here suggested are due to Euclid, a great mathematician who wrote the first systematic text-book geometry. In this work, known as Euclid's Elements, the exercise here given is the fifth proposition in Book I. Of the life of Euclid there is but little known except that he was gentle and modest and 66 was a Greek who lived and taught in Alexandria about 300 B.C." To him is attributed the saying, "There is no royal road to geometry." His appreciation EUCLID of the culture value of geometry is shown in a story related by Stobaeus (which is probably authentic). "A lad who had just begun geometry asked, 'What do I gain by learning all this stuff?' Euclid called his slave and said, 'Give this boy some coppers, since he must make a profit out of what he learns.""" Ex. 69. By using the accompanying diagram prove that the base angles of an isosceles triangle are equal. HINT. Prove A ABE = A DBC. Then prove A ACEA DAC. Ex. 70. (a) If equal segments measured from the vertex are laid off on the arms of an isosceles triangle, the lines drawn from the ends of the segments to the foot of the bisector of the vertex angle will be equal. (b) Extend (a) to the case in which the equal segments are laid off on the arms prolonged through the vertex. Ex. 71. (a) If equal segments measured from the ends of the base are laid off on the arms of an isosceles triangle, the lines drawn from the ends of the segments to the foot of the bisector of the vertex angle will be equal. (b) Extend (a) to the case in which the equal segments are laid off on the arms prolonged through the ends of the base. Ex. 72. (a) If equal segments measured from the ends of the base are laid off on the base of an isosceles triangle, the lines joining the vertex of the triangle to the ends of the segments will be equal. (b) Extend (a) to the case in which the equal segments are laid off on the base prolonged (Fig. 1). A FIG. 2. FIG. 1. Ex. 73. (a) If equal segments measured from the ends of the base are laid off on the arms of an isoscles triangle, the lines drawn from the ends of the segments to the opposite ends of the base will be equal. (b) Extend (a) to the case in which the equal segments are laid off on the arms prolonged through the ends of the base (Fig. 2). Ex. 74. Triangle ABC is equilateral, and AE BF CD. = Prove triangle EFD equilateral. A B C 115. Measurement of Distances by Means of Triangles. The theorems which prove triangles equal are applied practically in measuring distances on the surface of the earth. Thus, if it is desired to find the distance between two places, A and B, which are separated by a pond or other obstruction, place a stake at some point accessible to both A and B, as F. Measure the distances FA and FB; then, B keeping in line with F and B, measure CF equal to FB, and, in line with F Lastly and 4, measure FE equal to FA. C E measure CE, and the distance from A to B is thus obtained, since AB is equal to CE. Can this method be used when A and B are on opposite sides of a hill and each is invisible from the other? R Ex. 75. Show how to find the distance across a river by taking the following measurements. Measure a convenient distance along the bank, as RT, and fix a stake at its mid-point, F. Proceed at right angles to RT from T to the point P, where F, S, and P are in line; measure PT. T Ex. 76. An army engineer wished to obtain quickly the approximate distance across a river, and had no instruments with which to make measurements. He stood on the bank of the river, as at A, and sighted the opposite bank, or B. Then without raising or lowering his eyes, he faced about, and his line of sight struck the ground at C. He paced the distance, C B AC, and gave this as the distance across the river. Explain his method. Ex. 77. Tell what measurements to make to obtain the distance between two inaccessible points, R and S (Fig. 1). Ex. 78. The fact that a triangle is determined if its base and its base angles are given was used as early as the time of Thales (640 B.C.) to find the distance of a ship at sea; the base of the triangle was usually a lighthouse tower and the base angles were found by observation. Draw a figure and explain. H G FIG. 1. finding SR (Fig. 2). F Ex. 79. Explain the following method of Place a stake at S, and another at a convenient point, Q, in line with S and R. From a convenient point, as T, measure TS and TQ. Prolong QT, and make TF equal to QT. Prolong B ST, and make TB equal to ST. Then keep in line with F and B, until a point is reached, as G, where T and R come into line. is equal to the required distance, RS. Then BG Ex. 80. In an equilateral triangle, if two lines are drawn from the ends of the base, making equal angles with the base, the lines are equal. Is this true of every isosceles triangle? G FIG. 2. R PROPOSITION V. THEOREM 116. Two triangles are equal if the three sides of one are equal respectively to the three sides of the other. A B 13 214 CR S Given ABC and RST, AB = RS, BC= ST, and CA TR. To prove ▲ ABC=▲ RST. T ARGUMENT REASONS 1. Place A RST so that the 1. Any geometric figure may Ex. 81. (a) Prove Prop. V, using two obtuse triangles and applying the shortest side of one to the shortest side of the other. (b) Prove Prop. V, using two right triangles and applying the shortest side of one to the shortest side of the other. 117. Question. Why is not Prop. V proved by superposition? SUMMARY OF CONDITIONS FOR EQUALITY OF TRIANGLES a side and the two adjacent angles 118. Two triangles are equal if two sides and the included angle three sides a side and the two adjacent angles of one are equal respectively to two sides and the included of the other. angle three sides Ex. 82. The median to the base of an isosceles triangle bisects the angle at the vertex and is perpendicular to the base. Ex. 83. In a certain quadrilateral two adjacent sides are equal; the other two sides are also equal. Find a pair of triangles which you can prove equal. |