The annexed diagram illustrates how, by sliding the set-square along the ruler, lines may be drawn parallel to each other; and also how a line may be drawn perpendicular to another from any point, whether the point be without or on the latter line. In drawing a perpendicular to a line by placing an edge of the right angle of set-square against the latter, we are often unable to bring the perpendicular up to the line by reason of the right angle of the set-square having become rounded through use. Sometimes a convenient way of drawing a perpendicular through a point is to draw a perpendicular in any position, and then a parallel to this through the given point, the former perpendicular being afterwards erased. M Draw a number of equal and perpendicular lines AB, BC, CD, • . MN, and finally draw A perpendicular to AB, and NO perpendicular to MN, as indicated in the figure. The accuracy of the series of con- B. structions may be tested by the conditions that AO is both A equal and perpendicular to NO. Exercises. 1. What is meant by the distance of a point from a line? 2. AOB is any angle and OC bisects it. What do you observe as to the distances of any point in OC from OA and OB? Give proof. What do you observe as to the angles which OC makes with these distances to OA and OB? Give proof. 3. AB and CD intersect in 0. Bisect the angles AOC and BOD by OE and OF. What position do OE and OF occupy with respect to each other? Give proof. Bisect the angle AOD by OG. What position do OG and OE or OF occupy with respect to each other ? Give proof. 4. Construct a triangle ABC, and find a point in the base BC such that the perpendiculars from it on AB and AC are equal. 5. Taking any two points, A and B, in the plane of the paper, draw a line such that the distances from any point on it to A and B are equal. 6. Find a point equidistant from two given points A and B, and one inch from a third given point C. Is it always possible to do this? 7. In a given straight line find a point which is equidistant from two given points not lying in the line. 8. Take two points, A and B, in the plane of the paper, and describe a circle of radius two inches which shall pass through A and B. 9. Construct a triangle ABC with sides 4, 33 and 3 inches. Bisect the angles ABC, ACB by BD, CD meeting in D. What do you observe as to the distances of D from the three sides? Give proof. 10. Construct a triangle ABC. Bisect AB in D, and AC in E. Join DE, producing it both ways. In the base BC, or BC produced, take points F, G, H, . ; and join AF, AG, AH . What do you observe as to the division of the lines AF, AG, AH by DE or DE produced ? 11. Draw two straight lines AB, AC, and with the set-square draw any two lines at right angles to them. What relation do you observe between the acute angle between the lines and the acute angle between the perpendiculars? Give proof. 12. Construct a triangle one of whose angles is a right angle. How many degrees do you find in the sum of the other two angles ? Give reason. 13. Construct a triangle ABC with C a right angle. At C make the angle BCD equal to the angle CBA, CD meeting AB in D. What do you note as to the magnitudes of the lines DA, DB, DC? Give reason. 14. ABC is an isosceles triangle with AB=AC. Produce BA to D, making AD equal to AB'or AC. Join CD. What is the magnitude of the angle BCD? Give reason. 15. AB and CD are any two straight lines. Find a point E such that EAB and ECD are both isosceles triangles. 16. With ruler and compasses (i.e., not using set-square) draw from a point at the extremity of a given line another line at right angles to it, without producing the given line. 17. Construct an equilateral triangle ABC, with side two inches. Draw AD to the bisection of the base BC. How many degrees are there in each of the angles of the triangle ABD ? 18. Construct an equilateral triangle ABC, and draw AD perpendicular to the base BC. On AD describe another equilateral triangle EAD. How many degrees does each of the sides of EAD make with each of the sides of ABC ? 19. At the points A and B in the line AB draw equal lines AC, BD at right angles to AB and on the same side of it. Join CD, and produce it and AB both ways. From other points E, F, G in AB draw perpendiculars EK, FL, GM to CD. Compare the lengths of EK, FL, GM with AC or BD. What are the angles at E, F, G ? 20. Construct a triangle ABC, and bisect the sides BC, CA, AB at D, E and F respectively. Join DE, EF, FD. What relations exist between the lengths of the sides of the triangles ABC and DEF? What relations exist between the angles of the two triangles? Make three different figures, the triangles being of different shapes, and examine whether the same relations exist in the three cases. CHAPTER V. Respecting Angles of a Triangle. 1. We have seen (Ch. III., 4) that the sum of the three angles of any triangle is two right angles, or 180°. Definition: In any rectilineal figure, an exterior angle is an angle contained by any side and an adjacent side produced. Produce the side BC to D. With the bevel or protractor lay off the angle ABE equal to the angle at A. Using the bevel, examine now the relation existing between the angle EBC, which is equal to the sum of the angles A and B of the triangle, and the exterior angle ACD. Repeat the construction and examination in the case of a triangle of different shape, say one in which the angle at B is an obtuse angle. If the constructions and measurements have been accurately made, it will be found that the exterior angle (ACD) is equal to the sum of the two interior and opposite angles at A and B. We may show that this is always the case as follows: The three angles of the triangle make up 180°; but the angles ACB, ACD also make up 180'; hence the angle ACD must be equal to the sum of the angles A and B. 2. Lay off about a point, and adjacent to one another, angles equal to the three exterior angles of the triangle; or, with the protractor, measure the number of degrees in each of these angles. What is their sum ? Give reasons for this sum being what it is. 3. Of course, since the exterior angle of any triangle is equal to the sum of the two interior and opposite angles, it follows that the exterior angle of any triangle is greater than either of the interior and opposite angles. Also, since the sum of the three angles of any triangle is equal to two right angles, the sum of any two angles of a triangle is less than two right angles. We may, however, by elongating the triangle, make the sum of two of its angles but little short of two right angles. Thus the sum of A and B will be found, on using the pro B с tractor, to be but little less than 180°; and, by still further removing C, we may still further increase their sum. 4. Construct a triangle with sides 50, 70 and 90 millimeters; and, by adjusting the bevel to the angles, find out which is the greatest angle, which is next in magnitude, and which is least. Repeat the same examination in the case of the triangle whose sides are 2, 4 and 5 inches. |