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Moreover, the corresponding values of the two variables, namely,

LAC D and

ZACB are equal, however near these variables approach their limits.

LA C' B' ... their limits

and

are equal. ZACB

arc A' D arc A B

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arc A' B' arc A B

§ 199

Q. E. D.

202. SCHOLIUM. An angle at the centre is said to be measured by its intercepted arc. This expression means that an angle at the centre is such part of the angular magnitude about that point (four right angles) as its intercepted arc is of the whole circumference.

A circumference is divided into 360 equal arcs, and each arc is called a degree, denoted by the symbol ().

The angle at the centre which one of these equal arcs subtends is also called a degree.

A quadrant (one-fourth a circumference) contains therefore 90° ; and a right angle, subtended by a quadrant, contains 90°.

Hence an angle of 30° is of a right angle, an angle of 45° is 4 of a right angle, an angle of 135° is of a right angle.

Thus we get a definite idea of an angle if we know the number of degrees it contains.

A degree is subdivided into sixty equal parts called minutes, denoted by the symbol ().

A minute is subdivided into sixty equal parts called seconds, denoted by the symbol (").

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

B

B

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In the circle P AB (Fig. 1), let the centre C be in one

of the sides of the inscribed angle B.

We are to prove

< B is measured by 1 arc P A.

LA,

Draw CA.

CA=CB,
(being radii of the same O).
... ZB

$ 112
(being opposite equal sides).
2 Р С А ZB+ ZA.

$ 105 (the exterior 2 of a A is equal to the sum of the two opposite interior é).

Substitute in the above equality Z B for its equal Z A.
Then we have ZPCA = 2 2 B.

§ 202

But Z PC A is measured by A P,
(the < at the centre is measured by the intercepted arc).

::2 Z B is measured by A P.
..Z B is measured by 1 A P.

CASE II.

In the circle BAE (Fig. 2), let the centre Cfall

within the angle EBA.
We are to prove ZEBA is measured by } arc E A.

Draw the diameter BCP.
ZPB A is measured by arc P A, (Case I.)

ZPB E is measured by 1 arc P E, (Case I.) ..ZPBA + Z PBE is measured by 1 (arc P A + arc P E).

..Z E B A is measured by 1 arc E A.

CASE III.

In the circle B FP (Fig. 3), let the centre C fall with

out the angle A BF.
We are to prove Z A B F is measured by 1 arc A F.

Draw the diameter B CP.
< P B F is measured by 1 arc P F, (Case I.)

ZPBA is measured by arc P A, (Case I.) i. Z PBF-2 PBA is measured by 1 (arc P F arc P A).

.. Z A B F is measured by į arc A F.

Q. E. D.

204. COROLLARY 1. An angle inscribed in a semicircle is a right angle, for it is measured by one-half a semi-circumference, or by 90°.

205. COR. 2. An angle inscribed in a segment greater than a semicircle is an acute angle; for it is measured by an arc less than one-half a semi-circumference; i. e. by an arc less than 90°.

206. CoR, 3. An angle inscribed in a segment less than a semicircle is an obtuse angle, for it is measured by an arc greater than one-half a semi-circumference ; i. e. by an are greater than 90°.

207. Cor. 4. All angles inscribed in the same segment are equal, for they are measured by one-half the same arc.

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208. An angle formed by two chords, and whose vertex lies between the centre and the circumference, is measured by one-half the intercepted arc plus one-half the arc intercepted by its sides produced.

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Let the ZAOC be formed by the chords A B and C D.

We are to prove

ZA OC is measured by 1 arc A C + 1 arc B D.

Draw A D.

§ 105

LCOA= LD + LA, (the exterior 2 of a A is equal to the sum of the two opposite interior ). But Z D is measured by į arc A C,

$ 203 (an inscribed Lis measured by } the intercepted arc) ;

and ZA is measured by 1 arc B D,

§ 203 ..Z C O A is measured by į arc AC + 1 arc B D.

Q. E. D.

Ex. Show that the least chord that can be drawn through a given point in a circle is perpendicular to the diameter drawn through the point.

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209. An angle formed by a tangent and a chord is measured by one-half the intercepted arc.

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Let HAM be the angle formed by the tangent OM

and chord A H.

We are to prove

Z HA M is measured by 1 arc A EH.

Draw the diameter AC F.

§ 186

Z FA M is a rt. Z,
(the radius drawn to a tangent at the point of contact is I to it).

Z FAM, being a rt. 2, is measured by the semi-circumference A EF.

ZFA H is measured by ] arc FH, $ 203

(an inscribed Z is measured by } the intercepted arc) ; iiZ FAM Z FAH is measured by ] (arc A EF arc H F).

..Z HAM is measured by 1 arc A E II.

Q. E. D.

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