23. When one of the angles of an isosceles triangle is known, the others can be found; if one of the angles at the base is go, the other will be the same, and the vertical angle 180° 29°; if the vertical angle is to, the others will together be 180° — to, therefore each 90° — 1ť°.

II. THE DEPENDENCE OF ANGLES AND ARCS.

24. An angle having its vertex at the centre of a circle, and its legs being radii, is called an angle at the centre; by superposition it may be shewn that equal angles at the centre of the same circle or of equal circles stand on equal arcs and conversely, and that equal arcs are subtended by equal chords.

The circle is like the whole revolution divided into 360°, so that an angle at the centre contains the same number of degrees as the arc on which it stands; we therefore say that the angle at the centre is measured by its arc.

An angle at the circumference of a circle has its vertex in the circumference, and its sides are chords; it contains half as many degrees as the arc on which it stands.

a) If the one leg of the angle passes through the centre, a radius is drawn to the extremity of the other leg; we then have

As y is measured by the arc AB, x will be measured by half the arc.

b) If one leg lies at each side of the centre, this case is made to refer to the preceding one by drawing a diameter from the vertex; we then have

~AB; y

~BC,

(24, a)

c) In the same manner we get, if both legs lie at the same

Thus the proposition holds good in all cases.

25. As the proposition in 24 holds good however small the one chord becomes, it must also hold when the chord grows infinitely small, that is, when it (produced) becomes a tangent. Therefore an angle contained by a chord and a tangent is measured by half the arc which the chord cuts off.

26. From these propositions it follows that an angle at the circumference, which stretches over a diameter, is a right angle, for the arc on which it stands is 180°, and that the opposite angles of an inscribed quadrilateral are supplementary angles as they together stand on the whole circumference, which is 360°.

27. An angle, having its vertex inside the circle, is measured by half the sum of the arcs, which it and its vertical angle intercept.

B

28. An angle, having its vertex outside the circle, is measured by half the difference of the arcs, which it intercepts.

The legs of the angle can be two secants, a secant and

a tangent, or two tangents.

In the last case, instead of B, we can put 360° — b, from

Therefore an angle contained by two tangents is measured by subtracting the number of degrees in the smaller arc from 180°.

From 27 and 28 it follows that of any angles standing on an arc AB, those which have their vertices outside the circle are less, and those having their vertices inside, greater, than an angle at the circumference standing on AB.

Note. If the legs of an angle C pass, the one through A, the other through B, a circle can always be described through A and B, so that C shall lie within this circle. Describe a semicircle on AB; if C> R., C will lie within this semicircle; if not then with the middle point of the circumference of the semicircle as centre, describe a circle through A and B; the angles at the circumference of this circle, standing on AB, are R.; if <C>R., then C will lie within this second circle; but if not we proceed in this manner, and shall in succession get circles, the angles at the circumferences of which are R., R., &c., we must then at last get a circle in which the angles at the circumference standing on AB are less than C, and this circle will pass outside C.

29. Two tangents, drawn from any one point to a circle, are equal, reckoned from this point to the points of contact. For the triangle formed by joining the points of contact is isosceles, as the angles at the base are measured by the same arc (25). (See the last figure to 28).

30. A tangent is perpendicular to the radius to the point of contact, for the angle they contain is a right angle, as the arc is 180°. (25).

x ༢

III. PARALLEL LINES.

31. If a straight line cuts two other straight lines and makes the exterior angle equal to the interior, opposite angle, on the same side of the line (as x and y), the two lines are said to be parallel. Parallel to" is denoted thus . When the angles formed by the intersection of one straight line with the parallel lines are equal, they will be equal for all straight lines.

For as

then

y

и

32. At each of the points of intersection there are four angles; when the lines are parallel, each of the acute angles in the one group is equal to each of the acute angles in the other group, but is the supplement of each of the obtuse angles. Conversely, when one of these conditions is fulfilled, the lines are parallel.

33. Parallel lines can never cut each other.

For if they did, we would, by drawing a line cutting them both, get a triangle, in which the angle adjacent to one angle of the triangle would be equal to one of the other angles of the triangle, which is in opposition to 19.

Note. Conversely, we see that the lines must meet if x and y are not equal; for let A be a point in the one line, B a point in the other, and AB the line joining them; we can then always (28, Note) in the line through A, within a certain finite circle find a point C, so that ACB y-x (or x -y); CB must therefore just be the line through B, as its angle with AB becomes y; that is, the two lines meet in a point C at a finite distance.

-

34. By observing the angles formed by an intersecting line, it is evident that through one given point, there can only be drawn one line parallel to a given line, and that

When two lines are parallel to the same third line, they are parallel to one another.

35. Angles with parallel legs are equal, when the legs both extend in the same direction or both in opposite directions.

If one of the legs of the one angle be produced till it cuts a leg of the other angle, we get

1. In a triangle one angle is 75°, another 42°; how great is the third?

2. In a triangle the two angles are each 42° 12′ 42"; how great is the third?

3. In a triangle one angle is A, the other B; how great is the third?

4. In an isosceles triangle the vertical angle is 60°; how great is the angle at the base?

5. How great is the angle between perpendiculars on two of the sides of an equilateral triangle?

6. O is a point within a triangle ABC; prove that ZAOB > Z ACB.