What will be the direction of each line, and the value of each angle, if e be 45? if e be 90?

d.) A subtraction, like that above, is actually employed in finding the angle between two lines with a theodolite or compass. First, by taking sight, we find the direction of each line, as measured in degrees upon the instrument from a zero point, or from a north and south line; and then, we subtract one direction from the other to obtain the angle between them.

If in surveying we find the direction of one side of a field to be 25° upon the theodolite, and that of the adjoining side to be 85°, what is the angle between them? What is the angle, if the two directions are 45° and 160°? 60° and 179°?

§ 52. II. Without having recourse to more formal demonstration, it is evident that, if two straight lines have the same direction, they must differ equally in direction from a third straight line, and consequently must make equal angles with it; and conversely, that, if two straight lines differ alike in direction from a third, they must have the same direction with each other, or, in other words, must be parallel. — In applying this summary demonstration, the distinction in § 6. 2 must be regarded.

From A, one of the angular points of the triangle, draw AD

parallel to BC, the opposite side.

What lines make angles with these parallels?

What like angles are made? Of which species?
What unlike angles are made? Of which species?
Then,

aBCA n aCAD?

42.

46. C.

.. BCA+CAB+ ABC ≈≈≈ CAD+CAB+ABC? 20. d. But, CAD+CAB+ABC (or DAB+ABC)=hmL? 46. e. BCA + CAB + ABC = hm L?

20. a.

Show the same, drawing the parallel from A to the left hand, instead of the right.

Show the same, drawing the parallel from B; from C.

How many different ways can you draw the parallel, and show the same?

§ 54. THEOR. VII. The three angles of any triangle are together equal to two right angles.

[Proved by drawing from one of the points a parallel to the opposite side, and applying Theor. V.]

a.) How many degrees do the three angles of any triangle together contain?

29.1.

If two angles of a triangle are 600 and 800, what is the third angle? What is it, if the other two are 1000 and 200 ? 900 and 450? 500 and 250 ?

If one angle of a triangle is 40°, and the other two are equal, hm does each of them contain?

If the three angles of a triangle are equal, hm does each contain ?

If, in the triangle OPQ, aP is twice as great as aO, and aQ three times as great as aP, hm does each contain?

If two angles of a triangle are given, how do you find the third?

b.) How is each angle of a triangle related to the sum of the other two? 30.g.

c.) If two angles of one triangle are equal to two angles

of another, how does the third angle of the one compare with the third angle of the other?

d.) How many right angles can a triangle have? How many obtuse? How many acute?

If one of the angles of a triangle is a right or an obtuse angle, what must each of the other two be? To what kind of an angle, in each case, are the other two together equal?

If one of the angles of a triangle is equal to the sum of the other two, what must that angle be?

If two angles of a triangle are equal, what kind of an angle must each be?

e.) Through A, draw DE || BC.

XVIII.

Show that the three angles of B---------tABC are equal to the three angles at pA, and, therefore, = 2L.

f.) Produce BC to D, and draw CE || BA.

Show that the three angles of tABC are= = the three angles at pC, and, therefore, = 2L.

A

E

N

XIX.

ہے

§ 55. DEFINITIONS. I. A POLYGON is termed equilateral", when its sides are all equal; and equiangular, when its angles are all equal. A TRIANGLE is termed isosceles", when it has two equal sides; and scalene, when no two of its sides are equal. It is also termed right-, or obtuseangled, when it has one right, or one obtuse angle (§ 54. d); and acute-angled, when all its angles are acute.

(a) Lat. æquilateralis, from æquus, equal, and latus, side. (b) Gr. icoσxsλs, from ros, equal, and oxínos, leg. (c) Gr. σxaλnvés, limping, like a man whose legs are unequal.

NOTE. (a) Polygons which are both equilateral and equiangular are termed regular. (b) Polygons whose angles are all oblique are termed oblique-angled. (c) The shorter forms, right triangle, oblique triangle, &c., are sometimes used for right-angled triangle, oblique-angled triangle, &c.

Draw triangles of each of the different classes above mentioned.

§ 56. II. The side upon which a triangle is regarded as resting is termed its BASE, and the opposite angular point, its VERTEX or SUMMIT.

NOTE. (a) In distinction from the base, the other two sides are termed the legs, or sometimes simply the two sides. (b) In an isosceles triangle, the term base is specially applied to the side lying between the two equal sides.

In the triangles above, lying as they do, name the base, vertex, and legs of each.

§ 57. III. In a triangle, the side which is opposite to an angle is said to subtende the angle.

In the triangles above, name the side subtending the angle A; E; I; C; D; H. Name the angle subtended by AC; DE; HI; BC; DF; GH..

§ 58. IV. In a right-angled triangle, the side subtending the right angle is termed the HYPOTENUSE; and of the other two sides, one is often termed the BASE (§ 56), and the other the PERPENDICULAR.

In the right-angled triangle GHI, which side is the hypot

(d) Gr. Báois, foundation. (e) Lat. subtendo, to stretch beneath. (f) Gr. UTOτsivovσa, subtending. The term legs is sometimes applied to the other two sides of a right triangle.

enuse? As the triangle lies, which side is the base, and which the perpendicular?

59. V. The angle between a side of a polygon and an adjoining side produced is termed an EXTERIOR ANGLE of the polygon; as GIK, in the figure above.

a.) In taking the sum of the exterior angles of a polygon, only one side is produced from each angular point of the polygon, as in Fig. XXI. (§ 62).

If, at any point, both sides were produced, how many exterior angles would be made? How would these compare with each other?

b.) In distinction from the exterior angles, the angles within the polygon (commonly called simply the angles of the polygon) are termed interior.

c.) aGIK+aGIH = hm L ? = hm° ?

29.2. 30.

How, then, are these angles related to each other? REMARK. An exterior angle is the supplement of its adjacent interior.

In an acute-angled triangle, what kind of an angle is each exterior angle? In an obtuse-angled triangle? In a rightangled triangle?

§ 60. VI. A straight line connecting opposite angles of a polygon is termed a DIAGONALS.

Name the diagonals in Fig. XXIII. (§ 65).

(g) Lat. diagonālis, from Gr. Sá, through, and ywvía, angle.