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are not symparallel because they fail to answer all the conditions. Write the principle given above, using the notation explained in 121, scholium.

145. Make a line A B, and choose any point C outside of the line. Draw through C a line symparallel to A B. Suggestion: Draw C A as a help line, and use the principle of 144.

146. Make four lines symparallel to a line A B drawn at pleasure. Will these lines be symparallel to each other?

147. Make a three-sided figure. Through each corner draw a line symparallel to the opposite side.

148. Make a quadrilateral with its opposite sides parallel. Such a figure is called a parallelogram.

149. Can you discover any truth about the angles of a parallelogram without actual measurement? Record the principle in the "Summary."



FIG 33.


150. Make a triangle A B C, and through the vertex C (Fig 33) draw DE ↑↑ A B. Select pairs of equal angles, telling in each case how you know that the angles are equal. What can you say about the sum of the angles with vertices at C? What can you say about the sum of the angles of the triangle? Compare the result obtained here with the result of your experiments in 106. Notice the difference in the methods of reaching the conclusion. In this case the conclusion is reached after studying one triangle, in 106 after experimenting with several triangles; in this case what we know about the angles depends upon the nature of angles, while in 106 our knowledge is based upon our measurements. We might have some doubt as to whether the truths learned about the few triangles with which we experimented in 106 would apply to all triangles; but we are absolutely sure that the principle discovered in


150 applies to all triangles, because it is based on the nature of angles and lines. In this case the principle is proved; in 106 it is merely tested by experiment. Record the principle in the "Summary."


151. Make a triangle A B C with B A C = 70°, and A B C 43°. How large is ACB? Why? Through each vertex, A, B, and C, in turn draw a line parallel to the opposite side. Four new triangles will thus be formed. Give the value of each angle in degrees without using your protractor.

152. One angle of a triangle is 35°, another 47°; find the third angle. If two angles of a triangle are 48° and 95° respectively, what is the third angle?

153. Can you make a triangle whose angles are 90°, 40°, and 60°? Why? Can you make a triangle with two obtuse angles in it? Can you make a triangle two of whose angles shall be right angles?

154. Make any triangle. Can you make a second triangle which shall have two angles like two of the first triangle, but which shall have its third angle unlike the third angle of the first triangle?

155. If the angles of Fig. 34 are two angles of a triangle, how can you find the third angle without using your protractor? Can you think of more than one way of solving this problem?

156. Two angles of a triangle are said to determine the


FIG. 34.


third angle. Give illustrations of the meaning of this


x (Fig.

157. If one angle of a parallelogram is equal to 34), how can you find the other three angles without constructing the parallelogram? See 149.

158. One angle of a parallelogram is 72°. What are the other angles?

159. Find all the angles of a parallelogram, if one angle

is 90°; if one angle is 120°; if one angle isy (Fig. 34).

160. How many angles of a parallelogram are required to determine the remaining angles? Put your answer in the form of a principle, and record it in the "Summary."

161. In the parallelograms whose angles you have studied thus far, can you discover a principle that applies always to the adjacent angles?

162. Extend side C D of the parallelogram A B C D to R. What can you say about the respective directions of the sides of RDA and DA B? What follows from the directions of the sides of the angles?

163. Make an angle A B C. At some point P on A B between A and B draw (on the opposite side of A B from B C) a line, P R, that shall diverge from P B as much as B C diverges from B P. What is the geometrical name for angles R P B and PBC (121 Scholium)? Which of the following sentences is true? P R ↑↑ B C or PR↑ BC? Write a translation of the correct sentence. Is PR parallel to B C ?

164. Make a line A B and choose a point C outside of A B. Draw a line through C parallel to A B, without prolonging your help line A C beyond C. Will the line be antiparallel or symparallel to A B?

165. Repeat the problem of 164 three times, placing A, B, and C in different positions each time.

166. The last four exercises lead to the following principle: If two lines are crossed by a third in such a way that the alternate interior or alternate exterior angles are equal, the two lines are parallel.

167. Combining the principles of 144 and 166, we can say that two lines are parallel, if a third line crosses them in such a way as to make any pair of corresponding angles equal, or any pair of alternate interior or alternate exterior angles equal.

168. A practical way of drawing parallels by making the



corresponding angles equal is to slide a triangular piece of wood or stiff paper along a ruler until the edge passes through any desired point. For instance, in Fig. 35 it is desired to draw a line through P parallel to A B. Place your ruler in such a position that one edge, x, of your triangle will have the direction of A B; then slide the triangle along the ruler until the same edge, x, passes through P, and draw M N. The ruler represents a line crossing M N and A B in

FIG. 35.

such a way as to make the corresponding angles equal. 169. Make a parallelogram by the method described in 168.

170. Mark off six equal distances on a line A B; then, by the method of 168, draw parallels through the points of division.

171. Make a triangle; then through each vertex draw a line parallel to the opposite side, using the method of 168.


172. Make a triangle A B C (Fig. 36). The sign for triangle is A, and for triangles A. What is the difference between A B C and ABC? Give as full an answer as you can to this question, naming all the respects in which they differ. Are they alike in any respect?

173. What is the difference between ▲ A B C and ДВАС? What is the difference between A B C and BAC? In how many ways can you name the ▲ A B C,

and in how many ways can you name the A B C, using the letters A, B, C each time?

174. Write a short sketch of the ▲ A B C, telling how many parts it has, what it is by nature, what you have


FIG. 36.

learned about its angles, and what fact about its sides you can infer from 124. Also write all you can about A B C. In writing these sketches, take great pains with your English.


175. After making A B C (Fig. 36), mark off, on a long line, M N equal in length to A B; then make / NM R equal to B A C; then mark off on M R a length M P A C; and, finally, join P N, thus completing M N P. Notice that you made the lines M N and M P and the angle N M R to match the corresponding parts of the AABC, but that, after taking these steps, you had no control over the line P N, because you were obliged to draw it between two fixed points P and N. With your protractor compare MPN with AC B, and / M N P with ABC; and, with your compasses, compare P N with C B, recording your results by means of the sign = or the sign (unequal) as the case may be. If you find that P N is not equal to C B, write P NC B, but, if you find that PN is equal to C B, write P N C B, etc. Record your work from the beginning according to the model which follows, taking pains to record each step as you take it, not waiting to give your description after all the work is done. Put your figure at the upper right cor

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