PROPOSITION XLVII. THEOREM.

159. The exterior angles of a polygon, made by producing each of its sides in succession, are together equal to four right angles.

Let the figure ABCDE be a polygon, having its sides produced in succession.

We are to prove the sum of the ext. s = 4 rt. §.

Denote the int. As of the polygon by A, B, C, D, E ;

In like manner each pair of adj. = 2 rt. 4;

.. the sum of the interior and exterior as many times as the figure has sides,

2 rt. taken

=

taken as many times as the

figure has sides less two, 2 rt. 4 (n − 2),

EXERCISES.

1. Show that the interior angles of a hexagon are equal to eight right angles.

2. Show that each angle of an equiangular pentagon is g of a right angle.

3. How many sides has an uiangular polygon, four of whose angles are together equato seven right angles?

4. How many sides has the polygon the sum of whose interior angles is equal to the sum is exterior angles?

5. How many sides has the on the sum of whose interior angles is double that of its eter angles?

6. How many sides has the polygon the sum of whose, exterior angles is double that of its interior angles?

21.4

7. Every point in the bisector of an angle is equally distant from the sides of the angle; and every point not in the bisector, but within the angle, is unequally distant from the sides of the angle.

8. BAC is a triangle having the angle B double the angle A. If BD bisect the angle B, and meet AC in D, show that BD is equal to A D.

9. If a straight line drawn parallel to the base of a triangle bisect one of the sides, show that it bisects the other also; and that the portion of it intercepted between the two sides is equal to one half the base.

10. ABCD is a parallelogram, E and F the middle points of AD and BC respectively; show that BE and DF will trisect the diagonal A C.

11. If from any point in the base of an isosceles triangle parallels to the equal sides be drawn, show that a parallelogram is formed whose perimeter is equal to the sum of the equal sides of the triangle.

12. If from the diagonal BD of a square ABCD, BE be cut off equal to BC, and EF be drawn perpendicular to BD, show that D E is equal to E F, and also to FC.

13. Show that the three lines drawn from the vertices of a triangle to the middle points of the opposite sides meet in a point.

160. DEF. A Circle is a plane figure bounded by a curved line, all the points of which are equally distant from a point within called the Centre.

161. DEF. The Circumference of a circle is the line which bounds the circle.

162. DEF. A Radius of a circle is any straight line drawn from the centre to the circumference, as O A, Fig. 1.

163. DEF. A Diameter of a circle is any straight line passing through the centre and having its extremities in the circumference, as A B, Fig. 2.

By the definition of a circle, all its radii are equal. Hence, all its diameters are equal, since the diameter is equal to twice the radius.

164. DEF. An Arc of a circle is any portion of the circumference, as A M B, Fig. 3.

165. DEF. A Semi-circumference is an arc equal to one half the circumference, as A M B, Fig. 2.

166. DEF. A Chord of a circle is any straight line having its extremities in the circumference, as A B, Fig. 3.

Every chord subtends two arcs whose sum is the circumference. Thus the chord A B, (Fig. 3), subtends the arc AMB and the arc A D B. Whenever a chord and its arc are spoken of, the less arc is meant unless it be otherwise stated.

A

167. DEF. A Segment of a circle is a portion of a circle enclosed by an arc and its chord, as A M B, Fig. 1.

168. DEF. A Semicircle is a segment equal to one half the circle, as A DC, Fig. 1.

169. DEF. A Sector of a circle is a portion of the circle enclosed by two radii and the arc which they intercept, as A C B, Fig. 2.

170. DEF. A Tangent is a straight line which touches the circumference but does not intersect it, however far produced. The point in which the tangent touches the circumference is called the Point of Contact, or Point of Tangency.

171. DEF. Two Circumferences are tangent to each other when they are tangent to a straight line at the same point. 172. DEF. A Secant is a straight line which intersects the circumference in two points, as A D, Fig. 3.

173. DEF. A straight line is Inscribed in a circle when its extremities lie in the circumference of the circle, as A B, Fig. 1. An angle is inscribed in a circle when its vertex is in the circumference and its sides are chords of that circumference, as ZABC, Fig. 1.

A polygon is inscribed in a circle when its sides are chords of the circle, as A ABC, Fig. 1.

A circle is inscribed in a polygon when the circumference touches the sides of the polygon but does not intersect them, as in Fig. 4.

174. DEF. A polygon is Circumscribed about a circle when all the sides of the polygon are tangents to the circle, as in Fig. 4. A circle is circumscribed about a polygon when the circumference passes through all the vertices of the polygon, as in Fig. 1.

E

M

175. DEF. Equal circles are circles which have equal radii. For if one circle be applied to the other so that their centres coincide their circumferences will coincide, since all the points of both are at the same distance from the centre. 176. Every diameter bisects the circle and its circumference. For if we fold over the segment A M B on A B as an axis until it comes into the plane of AP B, the arc A M B will coincide with the arc APB; because every point in each is equally distant from the centre 0.

A

P

B

chord.

PROPOSITION I. THEOREM.

177. The diameter of a circle is greater than any other

Let A B be the diameter of the circle

AMB, and A E any other chord.

M

Substitute for C E, in the above inequality, its equal CB.