A triangle which has two of its sides equal is called an isosceles triangle (ἴσos equal, σκέλος a leg). A triangle which has all its sides equal is called an equilateral triangle (aequus equal, latus a side). A triangle which has no two of its sides equal is called a scalene triangle (okaλnvós lame or uneven). Ex. 103. Which of the triangles in Ex. 102 are isosceles, and which are equilateral ? Ex. 104. Make a triangle of strips of cardboard, its sides being 4 in., 5 in., 6 in. long. To do this, cut out strips about in. longer than the given lengths, pierce holes fig. 43. at the given distances apart and hinge the strips together by means of string, or gut with knots, or by means of “ eyes" such as a shoemaker uses. Can the shape of the triangle be altered without bending or straining the sides? Ex. 105. Make a quadrilateral of strips of cardboard, its sides being 3 in., 3·5 in., 4·5 in., 6 in. long. Can its shape be altered without bending or straining? Could it be made rigid by a strip joining two opposite corners? The straight line joining opposite corners of a quadrilateral is called a diagonal. Ex. 106. Repeat Ex. 105 with a pentagon each of whose sides is 3 in. long. How many additional strips must be put in to make the frame-work rigid? Ex. 107. Construct quadrilaterals ABCD to the following measurements: (i) AB = 2.3 in., BC= 2.1 in., CD = 3.3 in., BD = 3.4 in. (ii) ABCD = 6·4 cm., (iii) AB = AD = 1·9 in., BC = DA = 3.7 cm., DA = 1.5 in., BD = 5.7 cm. LB = 146°. (vi) AB = 5.3 cm., BC= 6·3 cm., CD = 67 cm., ▲ B=70°, 4C = 48°. (vii) AB = 27 cm., BC= 7.5 cm., AD = 8.4 cm., C = 98°, ▲ DBC = 28°. (viii) BC= CD = 2.4 in., BD = 1·9 in., ▲ ABD = 2 ADB = 67°. (ix) AB = 9.3 cm., BC = DA = 6·7 cm., ▲ A=111°, ▲ B = 28°. Ex. 108. Construct pentagons ABCDE to the following measurements : (i) AB = 2.0 in., BC= 2.2 in., CD = 1.7 in., DE=2.2 in., EA = 2.5 in., 2B=111°, C=149°. (ii) AB = 1.7 in., EA = 0.5 in., (iii) BC = 1.0 in., CD = 2.2 in., DE=3.4 in., LA = 126°, LB = 137°. AB = 5 cm., BC= 3.7 cm., CD=3.6 cm., DE=4.3 cm., EA = 3.8 cm., AC = 6.4 cm., AD = 6.7 cm. (iv) AB = BC = CD = DE = EA = 5.0 cm., AC = BE = 8.1 cm. PYRAMIDS. THE TETRAHEDRON. Figs. 44, 45 represent a tetrahedron, i.e. a solid bounded by four faces (TETρa- four-, ëdpa a seat, a base). Ex. 109. Make a tetrahedron of thin cardboard (or thick paper); fig. 46 represents what you will have to cut out (this will be referred to as the net of the tetrahedron); each of the small triangles is equilateral (their sides should be 4 in. long); the paper is to be creased (not cut) along the dotted lines, and the edges fastened with stampedging. Ex. 110. How many corners has a tetrahedron ? Ex. 111. Ex. 112. How many edges meet at each corner? What is the total number of edges? fig. 46. Ex. 113. Can you explain why the total number of edges is not equal to the number of corners multiplied by the number of edges at each corner? Ex. 114. What is the greatest number of faces you can see at one time? Ex. 115. Make sketches of your model in three or four different positions. Figs. 47, 48 represent a square pyramid (i.e. a pyramid on a square base). Ex. 116. Make a square pyramid (fig. 49 represents its net); make each side of the square 2 in. long and the equal sides of each triangle 2.5 in. long. Ex. 117. How many corners has a square pyramid ? Ex. 118. How many edges? fig. 49. Ex. 119. What is the greatest number of faces you can see at one time? Ex. 120. Make sketches of your model in three or four different positions. Ex. 121. Draw the net of a regular hexagonal pyramid, and make a rough sketch of the solid figure. TRIANGLES (continued). Ex. 122. What is the sum of the angles of a triangle? Ex. 123. Cut out a paper triangle; mark its angles; tear off the corners and fit them together with their vertices at one point, as in fig. 50. What relation between the angles of a triangle is suggested by this experiment? fig. 50. Ex. 124. Cut out a paper quadrilateral and proceed as in Ex. 123. Ex. 125. If two angles of a triangle are 54°, 76°, what is the third angle? Ex. 126. If two angles of a triangle are 27°, 117°, what is the third angle? Ex. 127. If two angles of a triangle are 23°, 31°, what is the third angle? Ex. 128. If two angles of a triangle are 65°, 132°, what is the third angle? Ex. 129. If the angles of a triangle are all equal, what is the number of degrees in each ? Ex. 130. If one angle of a triangle is 36°, and the other two angles are equal, find the other two angles. Ex. 131. Repeat Ex. 130 with the given angle (i) 90°, (ii) 132°, (iii) 108°. Ex. 132. In fig. 51, triangle ABC has ▲ A = 90°, AD is drawn perpendicular to BC. If B = 27°, find the angles marked x, y, z. Ex. 133. Repeat Ex. 132 with (i) ▲ B = 54°, (ii) ▲ B=33°, (iii) ▲ B = 45°. Ex. 134. A triangle ABC has ▲ A=75°, 4B = 36°; if AD is drawn perpendicular to BC, find each angle in the figure. Ex. 135. Would it be possible to have triangles with angles of (i) 90°, 60°, 30°, (iv) 135°, 22°, 22°, (ii) 77°, 84°, 20°, (v) 73°, 73°, 33°, (iii) 59°, 60°, 61°, (vi) 54°, 54°, 72° ? Ex. 136. (i) Give two sets of angles which would do for the angles of a triangle. (ii) Give two sets which would not do. Ex. 137. Construct a triangle ABC, having ▲ A=76°, 4 B=54°, BC= 2.8 in. What is C? First find BC, B and Measure drawing. C by calculation, then construct the triangle as though A; this will be a means of testing the accuracy of your |