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210. An angle formed by two secants, two tangents, or a tangent and a secant, and which has its vertex without the circumference, is measured by one-half the concave arc, minus one-half the convex arc.

0

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CASE I. Let the angle 0 (Fig. 1) be formed by the two secants

O A and O B.

We are to prove
20 is measured by į arc A B – 1 arc E C.

Draw C B.
LACB=LO+ LB,

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

20=2ACB-LB,
But LAC B is measured by } arc A B,

(an inscribed Z is measured by ż the intercepted arc).
and < B is measured by 1 arc C E,

..2 O is measured by 1 arc A B

§ 203

§ 203

1 arc C E.

CASE II.

Let the angle 0 (Fig. 2) be formed by the two tan

gents 0 A and O B.

We are to prove
2 O is measured by 1 arc A M B arc A S B.

Draw A B.
LABC 20+20 AB,

$ 105 (the exterior 2 of a A is equal to the sum of the two opposite interior &). By transposing,

20 =2 A B -2 0 4 B. But Z A B C is measured by : arc A M B, § 209 (an 2 formed by a tangent and a chord is measured by the in'. cepted arc),

and 20 A B is measured by 1 arc A S B. $ 209 .. 2 O is measured by ļ arc A M B 1 arc A S B.

CASE III.

Let the angle 0 (Fig. 3) be formed by the tangent

OB and the secant 0 A.

We are to prove

20 is measured by 1 arc A DS – 1 arc C E S.
Draw CS.
LACS=20+ LCSO,

§ 105 (the exterior 2 of a A is equal to the sum of the two opposite interior 6). By transposing,

Lo=LACS-LCSO.
But LACS is measured by 1 arc A DS,

§ 203
(being an inscribed 2).
and
LCSO is measured by 1 arc C ES,

$ 209 (being an Z formed by a tangent and a chord). i. 2 O is measured by 1 arc A DS – 1 arc C E S.

Q. E. D.

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211. Two parallel lines intercept upon the circum. ference equal arcs.

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Let the two parallel lines C A and B F (Fig. 1), inter

cept the arcs C B and A F.

We are to prove

arc CB

= arc A F.

Draw A B.

$ 68

LA= LB,
(being alt.-int. $ ).

But the arc C B is double the measure of Z A.

and the arc A F is double the measure of Z B.

.. arc CB= arc A F.

Ax. 6. Q. E. D.

212. SCHOLIUM. Since two parallel lines intercept on tho circumference equal arcs, the two parallel tangents M N and O P (Fig. 2) divide the circumference in two semi-circumferences A C B and A Q B, and the line A B joining the points of contact of the two tangents is a diameter of the circle.

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213. If the sum of two arcs be less than a circumference the greater arc is subtended by the greater chord; and conversely, the greater chord subtends the greater arc.

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P
In the circle ACP let the two arcs A B and BC to-

gether be less than the circumference, and let
A B be the greater.
We are to prove

chord A B > chord BC.

Draw A C.

In the A ABC
Z C, measured by 1 the greater arc A B,

§ 203
is greater than 2 A, measured by ] the less arc BC.
::. the side A B > the side BC,

§ 117 (in a A the greater Z has the greater side opposite to it).

CONVERSELY : If the chord A B be greater than the chord BC.

We are to prove

arc AB > arc B C.
In the A ABC,
AB > BC,

Нур. . .:.2C> A,

$ 118 (in a A the greater side has the greater Z opposite to it). .. arc A B, double the measure of the greater Z C, is greater than the arc B C, double the measure of the less < A.

Q. E. D.

PROPOSITION XX. THEOREM.

214. If the sum of two arcs be greater than a circumference, the greater arc is subtended by the less chord; and, conversely, the less chord subtends the greater arc.

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§ 213

In the circle BCE let the arcs A EC B and BA EC

together be greater than the circumference, and
let arc A ECB be greater than arc B A EC.
We are to prove

chord AB < chord BC. From the given arcs take the common arc A EC; we have left two arcs, C B and A B, less than a circumference,

of which C B is the greater.

... chord C B > chord A B, (when the sum of two arcs is less than a circumference, the greater arc is

subtended by the greater chord). ... the chord A B, which subtends the greater arc AEC B, is less than the chord B C, which subtends the less arc B A EC.

CONVERSELY : If the chord A B be less than chord BC.
We are to prove

arc A ECB > arc B A E C.
Arc A B + arc A ECB the circumference.

Arc B C + arc B A E C: the circumference.
... arc A B + arc A ECB = arc B C + arc B A EC.
But
arc A B < arc B C,

$ 213
(being subtended by the less chord).
... arc A ECB > arc B A E C.

Q. E. D.

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