Proposition 13. Theorem.

234. In the same circle, or in equal circles, angles at the centre are in the same ratio as their intercepted arcs. Hyp. Let AOB, AOC be any two

/s at the centres of two equal os, and AB, AC their intercepted

Proof. Take M, any common measure of AB and AC, and suppose it to be contained five times in AB and four times in AC.

B

Draw radii to the several pts. of division of the arcs AB, AC, dividing the AOB into five s and AOC into four Zs.

Theses are all equal.

In the same or in equal os equal arcs subtend equal Ls at the

Now, these two ratios being always equal while the common measure is indefinitely diminished, they will be equal when D moves up to and as nearly as we please coincides with C.

(233)

Q.E.D.

235. COR. In the same circle, or in equal circles, sectors are in the same ratio as their arcs; for sectors are equal when their arcs are equal.

(195) 236. SCH. Since the angle at the centre of a circle, and the arc intercepted by its sides increase and decrease in the same ratio (234), the numerical measure of the angle is the same as that of the arc. This theorem, being of very frequent use, is expressed briefly by saying that an angle at the centre is measured by its intercepted arc.* This means simply that an angle at the centre is the same part of the whole angular magnitude about the centre that its intercepted arc is of the whole circumference.

237. The circumference is divided, like the angular magnitude about the centre (28), into 360 equal parts called degrees. The degree is divided into 60 equal parts called minutes, and the minute into 60 equal parts called seconds. Hence the unit of angle and the unit of arc are both called a degree. When the angle becomes a right angle (20), the arc becomes a quarter of the circumference, or a quadrant. When the angle becomes a straight angle (21), the arc becomes a semi-circumference, and so on. A right angle and a quadrant are both expressed by 90°. Two right angles. and a semi-circumference are both expressed by 180°. Four right angles and a circumference are both expressed by 360°.

*Rouché et Comberousse, p. 64.

Proposition 14. Theorem.

238. An inscribed angle is measured by one-half the arc intercepted between its sides.

Hyp. Let BAC be an

inscribed in

arc BC.

the ABC and intercepting the

arc BC.

To prove BAC is measured by
CASE I. When the centre O is within

the BAC.

Proof. Join AO, produce it to D, and

join OB, OC.

In ▲ AOB,

The ext.

But, since

< BOD = OAB +≤OBA.

B

of a ▲ equals the sum of the opp. int. ≤8 (98).

OA = OB,

.. ZOAB ZOBA,

being opp. equal sides (111).

.. BOD = 2 / OAB.

COD=2/OAC.

BOC 2 / BAC.

(Radii)

(Ax. 2)

Similarly,

.. whole

But

/ BOC is measured by arc BC.

The

at the centre is measured by the intercepted arc (236).
.. 2 BAC is measured by arc BC.

CASE II. When the centre O is without the BAC.

Proof. Join AO, produce it to D, and join OB, OC. Then,

and

DOB = 2/DAB, (Case I)

/DOC 2/DAC. (Case I)

../DOC/DOB = 2/ DAC-2/ DAB.

But

/ BOC is measured by arc BC.

.. BAC is measured by arc BC.

(236)

Q.E.D.

NOTE. This theorem is equally true when the angle at the centre is greater than two right angles, as the student may show.

239. COR. 1. All angles inscribed in the same segment are equal; for each is measured by one-half the same arc AFB.

A

D

B

240. COR. 2. Every angle AHB, inscribed in a semicircle, is a right angle; for it is measured by one-half a semi-circumference, or by a quad- A rant (237).

241. COR. 3. Every angle BAC, inscribed in a segment greater than a semicircle, is an acute angle; for it is measured by one-half the arc BDC, which is less than a quadrant.

B

Every angle BDC, inscribed in a segment less than a semicircle, is an obtuse angle; for it is measured by onehalf the arc BAC, which is greater than a quadrant.

242. COR. 4. The opposite angles of an inscribed quadrilateral are supplementary; for the sum of the s A and D is measured by one-half the Oce, which is the measure of two rights (237); therefore the s are supplementary (25).

Proposition 15. Theorem.

243. An angle formed by a tangent and a chord from the point of contact is measured by one-half the intercepted arc. Hyp. Let AC be a tangent to the

OBHE at B, and BD any chord of the

E

D

The to a tangent at the pt. of contact passes through the centre

ABD is measured by

are BED.

Q.E.D.

Similarly, 244. SCH. This proposition is a particular case of Prop. 14. Thus, let the side BD remain fixed, while the side BH turns about B, as in (28), until it becomes the tangent BC at the point B. In every position of the chord BH, the inscribed angle HBD is measured by half the intercepted arc HD. Therefore, when the chord BH becomes the tangent BC, the angle CBH is measured by half the arc BHD.

EXERCISES.

1. If the angle BAC at the circumference of a circle be half that of an equilateral triangle, prove that BC is equal to the radius of the circle.

2. If a hexagon be inscribed in a circle, show that the sum of any three alternate angles is four right angles.

3. If two circles intersect in the points A, B, and any two lines ACD, BFE, be drawn through A and B, cutting one of the circles in the points C, E, and the other in the points D, F, the line CE is parallel to DF.