namely,

Moreover, the corresponding values of the two variables,

are equal, however near these variables approach their limits.

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 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 (").

PROPOSITION XIV. THEOREM.

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

In the circle PAB (Fig. 1), let the centre C be in one of the sides of the inscribed angle B.

(the exterior of a ▲ is equal to the sum of the two opposite interior 4).

Substitute in the above equality B for its equal ≤ 4.

(the ▲ at the centre is measured by the intercepted arc).

.. 2B is measured by A P.

.. Z B is measured by 1⁄2 A P.

CASE II.

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

:: Z PBA + Z PBE is measured by 1⁄2 (arc PA + arc P E). ..ZEBA is measured by arc E A.

CASE III.

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

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 arc 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.

PROPOSITION XV. THEOREM.

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.

Let the

AOC be formed by the chords A B and C D.

We are to prove

ZAOC is measured by arc AC + 1⁄2 arc B D.

But

(the exterior of a ▲ is equal to the sum of the two opposite interior

ZD is measured by arc A C,

§ 105 ).

$ 203

..ZCOA is measured by arc A C+ 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.

PROPOSITION XVI. THEOREM.

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

Let HAM be the angle formed by the tangent OM

(the radius drawn to a tangent at the point of contact is to it).

FAM, being a rt. 2, is measured by the semi-circum

ference A EF.

ZFAH is measured by arc FH,
(an inscribed is measured by the intercepted arc);

$ 203

../FAM-FAH is measured by

(arc A EF-arc HF).

..ZHA M is measured by

arc A E H.

Q. E. D.