345. COROLLARY. In the same circle, or in equal circles, sectors are to each other as their arcs. [The proof is analogous

to that of the Proposition, substituting sector for angle.]

346. SCHOLIUM.

If two diameters are

drawn perpendicular to each other, four right
angles are formed at the center of the circle.
By § 267, the circumference is divided into A
four equal arcs called quadrants.

D

M

If one of these right angles were divided into any number of equal parts, it could be shown by § 267, that the quadrant subtending the right angle is also divided into the same number of equal parts. If, for example, the right angle at the center were divided into. four equal parts, the arcs intercepted by the sides of these angles would each be one fourth of a quadrant; and conversely, radii intercepting an arc that is one fourth of a quadrant, form an angle at the center which is one fourth of a right angle. If any angle as DOM be taken at random and compared with a right angle,

i.e. the angle DOM is the same part of a right angle that its intercepted arc is of a quadrant.

In this sense an angle at the center is said to be measured by its intercepted arc.

347. SCHOLIUM. A quadrant is usually conceived to be divided into ninety equal parts, each part called a degree of arc. The angle at the center that is measured by a degree of arc is called a degree of angle.

The degree is divided into sixty equal parts called minutes, and each minute is again subdivided into sixty equal parts called seconds.

Degrees, minutes, and seconds are designated by the symbols '," respectively. Thus, 49 degrees, 27 minutes, and 35 seconds, is written 49° 27' 35".

351. EXERCISE. Multiply 13° 27′ 35′′ by 3, and add the product to one half of 12° 15' 10".

352. EXERCISE.

How many degrees are there in each angle of an

isosceles right-angled triangle?

353. EXERCISE. Express in degrees, minutes, and seconds the value of one angle of a regular heptagon (a seven-sided polygon). ·

354. DEFINITION. An inscribed angle is an angle whose vertex is in the circumference and whose sides are chords.

B

The symbol ~ is used for the phrase is measured by. Thus, ABC~ arc AC is read: The angle ABC is measured by the arc 4C.

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A segment is that part of a circle which is included between an arc and its chord. [ACB and ADB are both segments.]

An angle is inscribed in a segment when its vertex is in the arc of the segment and its sides terminate in the extremities of that arc. [ABC and ADC are inscribed in the segment AmC.]

m

D

PROPOSITION XX. THEOREM

355. An inscribed angle is measured by one half of the arc intercepted by its sides.

CASE I

A

Let ABC be an inscribed angle having a diameter for one of its sides.

.. B, which is one half 1, is measured by one half the

356. COROLLARY I. Angles inscribed in the same segment are equal.

A

in a semicircle are right angles.

[<1ABC. But of the arc ABC is a quadrant. Therefore, by § 346, 1 is a right angle.]

358. COROLLARY III. An angle inscribed in a segment that is greater than a semicircle

is acute.

BE

359. COROLLARY IV. An angle inscribed in a segment that is less than a semicircle is obtuse.

360. COROLLARY V. The opposite angles of an inscribed quadrilateral are supplementary.

[Show that the sum of the measures of 1 and 2 is a semicircumference, or two quadrants.]

B

361. EXERCISE. The sides of an inscribed angle intercept an arc of 50°. What is the size of the angle ?

362. EXERCISE. How many degrees in an arc intercepted by the sides of an inscribed angle of 40° ?

363. EXERCISE. If the opposite angles of a quadrilateral are supplementary, a circle may be circumscribed about it.

(Converse of Cor. V.) Then show that

[Pass a circumference through three of the vertices. the fourth vertex can fall neither without nor within the circumference.]

364. EXERCISE. Show by § 355 that the sum of the angles of a triangle is two right angles.

365. EXERCISE.

rectangle.

Any parallelogram inscribed in a circle is a

366. EXERCISE. Two circles are tangent at A. AD and AE are drawn through the extremities of a diameter BC.

Prove that DE is also a diameter.

367. EXERCISE. Prove the preceding exercise when the two circles are tangent externally.

368. EXERCISE.

and two.

369. EXERCISE.

370. EXERCISE.

E

The angles of an inscribed trapezoid are equal two

Prove § 355, Case I, using the figure of § 322.

Two chords AB and CD intersect in a circle at the

point E.

Their extremities are joined by the lines AC and DB.

Prove

the ACE and BDE mutually equiangular.

371. EXERCISE. The sum of one set of alternate angles of an inscribed octagon is equal to the sum of the other set.

SANDERS' GEOM.-8