SECTION II.

ANGLES AT THE CENTRE AND SECTORS.

DEF. 4 An arc is a part of a circumference. Two arcs, which together make the whole circumference, are said to be conjugate. The greater of the two is called the major conjugate arc and the smaller the minor conjugate arc.

DEF. 5. The conjugate angles formed at the centre of a circle by two radii are said to stand upon the coniugate arcs opposite them intercepted by the radii, the major angle upon the major arc, and the minor angle upon the minor arc.

DEF. 6. A sector is a figure contained by an arc and the radii drawn to its extremities. The angle of the sector is the angle

at the centre which stands upon the arc of the sector.

THEOR. 4. In the same circle, or in equal circles, equal angles at the centre stand on equal arcs, and of two unequal angles at the centre the greater angle stands on the greater

arc.

In the equal circles DEF, HKL, of which A and B are the respective centres, let the angle DAE be equal to the angle HBK:

ANGLES AT THE CENTRE AND SECTORS. 17

P

L

M

then shall the arc DE on which the angle DAE stands be equal to the arc HK on which the angle HBK stands.

Let the circle DEF be applied to the circle HKL, so that the centre A falls on the centre B,

then their circumferences will coincide, since the circles are equal. III. 3.

Let DEF be turned about the centre until AD falls on BH, then the circles continue to coincide, III. 3, Cor. 1.

and the point D falls on H.

Then, because the angle DAE is equal to the angle HBK,

Hyp.

therefore AE will fall on BK, and the point E on the point K; therefore the arc DE coincides with, and is therefore equal to, the arc HK.

Again, if the angle DAE is greater than the angle HBK: then shall the arc DE be greater than the arc HK.

For in this case, if the radius AD be brought to coincide with BH, AE will coincide with some radius BM beyond BK,

and the arc DE with the arc HM, which is greater than HK; therefore the arc DE is greater than the arc HK.

Q.E.D.

Cor. Sectors of the same, or of equal circles, which have equal angles are equal, and of two such sectors which have unequal angles the greater is that which has the greater angle.

Ex. 16. If in equal circles one angle at the centre is double another, the arc on which the first angle stands is double that on which the second stands.

Ex. 17. If in equal circles the angle of one sector is double that of another, then will the first sector be double the second.

THEOR. 5. In the same circle, or in equal circles, equal arcs subtend equal angles at the centre, and of two unequal arcs the greater subtends the greater angle at the centre.

In the equal circles DEF, HKL of which A and B are the respective centres, let DE and HK be arcs, subtending at the centre the angles DAE, HBK :

then according as the arc DE is greater than, equal to, or less than, the arc HK, so shall the angle DAE be greater than, equal to, or less than, the angle HBK.

For in the preceding Theorem it has been shown that if the angle DAE is greater than the angle HBK, then the arc DE is greater than the arc HK; if the angle DAE is equal to the angle HBK, then the arc DE is equal to the arc HK ; if the angle DAE is less than the angle HBK, then the arc DE is less than the arc HK.

ANGLES AT THE CENTRE AND SECTORS. 19.

Now of these hypotheses one must be true, and of the conclusions no two can be true at the same time; hence, by the Rule of Conversion, the converse of each of the above Theorems is true.

Q.E.D.

COR. Equal sectors of the same, or of equal circles, have equal angles, and of two unequal sectors the greater has the greater angle.

EXERCISES.

18. Bisect a given arc of a circle whose centre is given.

19. Bisect a given sector.

20.

Trisect a given semicircle.

I., Ex. 81.

21. Trisect a given quadrant.

22.

Trisect the circumference of a given circle whose centre is given.

SECTION III.

CHORDS.

DEF. 7. A chord of a circle is the straight line joining any two points on the circumference.

THEOR. 6. In the same circle, or in equal circles, equal arcs are subtended by equal chords; and of two unequal minor arcs the greater is subtended by the greater chord.

In the equal circles DEF, HKL let the arc DE be equal to the arc HK ·

E

F

H

then shall the chord DE be equal to the chord HK.

Let A and B be the centres of the circles;

join AD, AE, BH, BK.

Then, because the arc DE is equal to the arc HK,

therefore, whether DE and HK be minor or major arcs, the