PART SECOND.

§ 24. DEFINITIONS. Angles are termed ADJACENT", when they lie side by side at the same point; VERTICAL, when they are made by the intersection of two lines, and lie opposite to each other.

A

Thus, AEC and AED are adjacent angles, of which AE is the common, or inner, side, and CE and ED, the outer sides. AEC and DEB, or AED and CEB, are vertical angles.

C

D

E

B

How many pairs of adjacent angles can you find in the figure? Name them. How many pairs of vertical angles can you find? Draw other angles upon the board, and show which are adjacent, and which vertical.

§ 25. If one straight line meeting another makes the two adjacent angles equal, these angles are termed RIGHT ANGLES; and the lines are said to be PERPENDICULARd to each other. If it makes the adjacent angles unequal, both the angles and the lines are termed OBLIQUE. An oblique angle less than a right angle is termed ACUTE; and one greater, OBTUSE".

(a) Lat. adjaceo, to lie contiguous, or close by. (b) So called from their being simply joined at the vertex. (c) So called, because one line stands upright upon the other (regarded as lying horizontally). (d) L. perpendicu⚫ lāris, plumb, from perpendiculum, plumb-line. (e) L. obliquus, slanting. (f) L. acutus, sharp. (g) L. obtusus, blunt.

Draw various angles, and show which are right, and which are oblique; which are acute, and which obtuse. Show which of the lines employed are perpendicular to each other, and which are oblique.

PROPOSITION I.

§ 26. Given, any right angles ABC, ABD, EFG, EFH. Required, ABC ≈

ABD EFG EFH (that is, the compara

tive magnitude of the angles ABC, ABD, EFG,

EFH; see § 13).

Place the first figure upon the second, with the point B upon the point F, and the line BC upon the line FG. As CD and GH are straight lines, where must BD fall? 6. c.b Does CD now form one and the same straight line with GH? As pB' lies upon pF, do AB and EF both meet this line at the same point, making the adjacent angles equal? Where then must AB fall?

With what angle must ABC coincide; and with what, ABD?

ABC EFG, and ABD EFH?

17. a.

(h) Numbers and letters thus placed at the end of a line refer to sections and their parts. (i) See § 14. e.

But, as ABC and ABD are right angles, ABC≈≈≈ ABD? 25. ABC ABD EFG EFH?

20. a.

§ 27. THEOREM I. All right angles are equal.

[Proved by superposition.]

§ 28. a.) From any point C in any straight line AB, draw CD above, and CE below, each perpendicular to AB (§ 25). Then, DCA ≈≈≈ DCB ~ ECA ECB?

Into how many equal angles is the divergence around C divided?

How many degrees does each of

these angles contain (§ 10. a)?

How many degrees must every right angle contain?

B

27.

b.) The divergence between CB and its reverse CA (reckoned either way, § 11. c) = hmƖ*? = hm?

If this divergence (on either side of AB) be divided into 3 equal angles, how many degrees will each contain? How many, if it be divided into 4 equal angles? into 5? into 6? into 9 into 10? into 18? into 45? into 60?

Into how many angles of 90° each can the divergence between CB and its reverse CA be divided? Into how many of 600? of 450? of 300? of 360? of 200? of 100?

The sum of any angles made at the point C on one side of AB = =hmL? = hm?

c.) The whole sum of any number of angles made around the point Chm L?

How many obtuse angles can lie about a point? How many acute?

29. d.) COROLLARIES. (1.) A right angle contains 900. (2.) The angles made upon one side of a straight line at any point are together equal to two right angles, or contain 180°. (3.) All the angles around any point are together equal to four right angles.

(k) See $$ 12, 13. c.

G

A

30. e.) DEFINITION. When one straight line meets another, each of the adjacent angles is termed the SUPPLEMENT1 of the other. Thus, ABC is the supplement of ABD; and, conversely, ABD is the supplement of ABC. f.) An angle and its supplement If the number of degrees in an angle is given, how do you obtain the number of degrees in its supplement?

B

= hmL? hm? 29.2.

What is the supplement of an angle of 600? of 90° ? of 1100 of 450? of 200? of 1700 ?

If an angle is a right angle, what kind of an angle must its supplement be? If acute? If obtuse?

g.) In general, any angle is called the supplement of another, however they may be situated, if the two are together equal to two right angles, or contain 1800; and angles so related are termed supplementary.

h.) If two angles are equal, how do their supplements compare?

If two angles are unequal, how does the supplement of the greater compare with the supplement of the less?

If the difference of two angles is 100, what is the difference of their supplements?

Is the supplement of an angle of 250 greater or less than the supplement of an angle of 500, and how much?

Equal angles have equal supplements; greater, less; and less, greater.

(1) Lat. supplementum, from suppleo, to fill up, supply.

is, the relative position of CB and BD; see

§ 13).

Producem CB to E. Then, ABC + ABE = hmL?

ABC+ABE≈≈≈ ABC + ABD?

ABE ABD?

Upon what line then must BE fall?

29.2.

20. a.

20. f.

.. As CB is in the same straight line with BE, CB BD?

§ 32. THEOR. II. If two adjacent angles are together equal to two right angles, the outer sides form a straight line.

[The converse" of § 29. 2.]

a.) If two adjacent angles are both right angles, do the outer sides form a straight line? Do they form a straight line, if the two angles are together greater, or together less, than two right angles? If one angle is 600, and the other 1200? If the two angles are 50° and 1300? 90° and 700 ? 900 and 1000? 600 and 800? 750 and 1500?

b.) If the angle ABC contains 1200, how many degrees must ABD contain, in order that CB and BD may form a straight line? How many, if ABC contains 1100? 1250 ? 1300 ?

(m) See § 5. c. (n) Lat. conversus, turned about. Of two propositions or sentences, each is said to be the converse of the other, when the condition of the first is the conclusion of the second, and the conclusion of the first is the condition of the second; or when, in like manner, subject and predicate change places.