the tree and the distance wanted? Why did he choose the angle 45° ? 275. How many ways of recognizing equal angles without measurement have you learned thus far? Give them all. 276. How many ways of recognizing equal lines without measurement do you know? 277. Give the three tests for recognizing equal triangles. 278. Make a triangle (A B C, Fig. 47). Make M N = A B. With M as centre and with A C as radius describe With N as centre and B C as radius describe an an arc. arc cutting the first arc at P and Q. Complete the triangles M P N and MQ N. Clearly these triangles have the same three sides that triangle A B C has. Is there any reason for thinking that AM NP has the same angles as AMNQ? If only one angle is the same in both, the triangles must be equal in all respects. Why? Let us aim to show that MPN=/ MQ N. Join P Q. What kind of triangle is AP MQ? Why? ... / M P Q = Z...... Why? Likewise N P Q = .... ... / M PN = ...... Why? MP Ĺ Why? .. AMPN=AMQN. Why? Should you place ▲ A B C in the plane of the paper with A B upon its equal M N, where would C have to fall? Why? Hence ▲ A B C = MNP = ^ MQ N. Hence a new principle for recognizing equal triangles: Two triangles are equal in all respects, if the three sides of one are equal to the three sides of the other respectively. 279. In Fig. 47 can you tell the the angle between M N and PQ? number of degrees in Give reasons. 280. If, in ABC, A B is 7 in. long, and A C is 5 in. long, how short can B C be? How long can it be? What reasons can you give for your answers? SECTION XII. QUADRILATERALS. 281. Construct quadrilaterals with your compasses and ruler as follows: 1st. A quadrilateral no two of whose sides are parallel. Such a quadrilateral is called a trapezium. 2d. A quadrilateral only two of whose sides are parallel. This figure is a trapezoid. 3d. A quadrilateral whose opposite sides are parallel. What is this figure called? 282. If one line crosses two others, what angles must be alike to make the lines parallel? Is there more than one pair of angles whose equality will make the lines parallel? 283. How many degrees must there be in the sum of the four angles of a quadrilateral? See 106-110. 284. Can you make a trapezium with two of its sides alike? with three of its sides alike? with all four of its sides alike? In the last case, draw a line joining two opposite corners, dividing the quadrilateral into two triangles, and, from your study of these triangles, decide whether the figure may be a trapezium or must be another kind of quadrilateral. 285. Can you make a trapezium with two angles alike? with three angles alike? with four angles alike? Can you make a trapezium with three of its angles 60° each? Why? Can there be three angles of 70° in a trapezium? 286. Can you make a trapezoid with two sides alike? with three sides alike? with two angles alike? with three angles alike? A: D from = 70°, 287. Draw a trapezoid, A B C D, making A B and C D antiparallel. It is possible to fix the value of the value of A. Do you see how? If what is D? If / B = 25°, what is C? help you to find / C? Does A 288. If, in making a trapezoid A B C D, you first make [ A = 70° and B = 40°, can you make C and D of any size that you please, or are they fixed in value? Give a reason for your answer. 289. How many angles of a parallelogram can you make before all of the angles are fixed in value ? 290. The most interesting of the quadrilaterals are the parallelograms, of which there are several varieties. Those with their sides all alike and with their angles 90° each are called squares. Those that have their sides alike but their angles oblique are called rhombuses. Those that have right angles but adjacent sides unequal are called rectangles. Those that have oblique angles and adjacent sides unequal are called rhomboids. 291. Construct models of each kind of parallelogram, describing the steps in the order in which you take them and at the time at which you take them. 292. Select examples of the different kinds of parallelogram from the school-room. What kind of parallelogram is a sheet of writing-paper? a diamond? a window-pane? 293. Can you make a rhomboid with three of its angles alike? Why? 294. Is any proof needed to show that the opposite sides of a parallelogram are parallel, pair by pair? Why? 295. Can you draw a parallelogram whose opposite sides are not equal? Does your answer to this need proof? Why? 296. Can you see any reason why a parallelogram should not be defined in the first place as a quadrilateral with its opposite sides parallel and equal? NOTE. It is important at this point to make clear, if possible, the danger of imposing too many conditions upon lines and angles, and also, in the same line of thought, to impress upon the pupil the importance of drawing the figures exactly according to the conditions, except in rare cases. If the condition imposed is that the lines be made parallel, they should be made parallel, and not equal, although it may be possible to prove that they are parallel when made equal. The subject should not be left until, by varied illustration, some impression is produced upon. the pupil. It will generally be found necessary to return to the subject at short intervals. Abundant opportunity for testing the pupil's understanding of the subject may be found in questions about the line bisecting the vertical angle of an isosceles triangle. 297. If a parallelogram really cannot be drawn with its opposite sides unequal, there must be some reason for the fact based upon the nature of lines and angles, or upon principles already established. Draw an angle, D A B; complete a parallelogram by making D C↑↑ A B and B C↑↑ A D, taking pains to make A B and A D unequal. Draw a line, D B, joining two opposite vertices (the line is called a diagonal), and study the triangles thus formed. It should be a simple matter for you to prove AABD = A B C D. Arrange your work in a vertical line, putting no more than one fact and its reason on a line. Follow this model : I. I make D C ↑↑ A B. II. I make B C ↑ ↑ A D, completing a parallelogram. To prove A B D C and A D B C. = = III. Draw diagonal D B and compare AA BD and D B C. IV. / .... .... (Insert reason.) = 298. It may help you to understand 297 to cut a parallelogram something like A B C D from stiff paper, and to cut the parallelogram into two parts along the diagonal D B. Can you place the two triangles formed by cutting the paper parallelogram so that they will not form a parallelogram? Is a quadrilateral whose diagonal divides it into two equal triangles necessarily a parallelogram? 299. Can you place the two paper triangles of 298 in more than one position so that they will form a parallelogram? 300. Does it make any difference which diagonal you draw? Will the parallelogram be divided into two equal triangles in each case? 301. How does a rhomboid differ from a rhombus ? 302. How does a rhombus differ from a square ? 303. How does a rectangle differ from a square ? 304. Notice that, although squares, rhombuses, and rectangles are not rhomboids, they differ from the rhomboid merely in having special characteristics in addition to those of the rhomboid. Hence any general principle that is proved about the rhomboid is applicable to the rhombus, the square, and the rectangle; but it is dangerous to assume that a principle proved true of a square, a rhombus, or a rectangle can be applied to a rhomboid, because the principle may depend upon the special characteristics of the former figures. Therefore, when trying to prove a principle that will apply to all parallelograms, be careful to draw a rhomboid for illustration, or to have a rhomboid in your imagination. 305. Into what kind of triangles does a diagonal divide a square? a rectangle? a rhombus ? 306. Can you construct a parallelogram with one angle 90° and another 80° ? Why? |