Show the same, taking a different angle in the place of AED. Show, in like manner, AED≈≈ CEB.

§ 34. THEOR. III. Vertical angles are equal.

[Because each of two vertical angles makes, with one of the intermediate angles, the same sum, viz. 180°.]

a.) If AEC contains 500, hm does each of the other angles in the figure contain? If it contains 600? 750? 450? 900?

If one of the angles made by two intersecting lines is a right angle, what must they all be? If one is acute, what must the rest be? If one is obtuse?

A

VIII.

E

C

B

§35. b.) Given, at the point E four angles, of which those that are opposite are equal; i. e. AEC = DEB, and AED = CEB. Required, CE ED, and AE 》 EB.

AECAED≈≈≈ DEB + CEB?

20. d.

But

AEC+ AED + DEB + CEB

=

hm L?

29.8.

And

AEC+ AED =hm L?

CEED?

AED + DEB = hm L? .. AE EB?

82.

If, of four angles at the same point, those which are opposite are equal, they are formed by the intersection of two straight lines.

§ 36. DEFINITIONS. A plane figure inclosed by straight lines is termed a POLYGON° (or simply, a RECTILINEAL FIGURE). The lines which in

(0) Gr. #oλúywvos, many-angled, from woλús, many, and ywvia, corner, angle.

close it are termed its SIDES; and the sum of these lines, its PERIMETER'.

§ 37. A polygon of three sides is termed a TRIANGLE; of four, a QUADRILATERAL'; of five, a PENTAGON; of six, a HEXAGON'; of seven, a HEPTAGON"; of eight, an OCTAGON'; of ten, a DECAGON"; &c.

§ 38. The sides and angles of a polygon are termed its PARTS; and those which lie next to each other (taking sides and angles alternately) are termed ADJACENT PARTS.

Thus, in Fig. IX. (§ 39), the side AB is adjacent to the angles A and B, and the angle A is adjacent to the sides AB and AC.

What are the parts adjacent to the side BC? to AC? to the angle B? C? to the side EF? to the angle D?

How many parts has a triangle? a quadrilateral? a hexagon? a decagon?

In every polygon, how does the number of the sides compare with the number of the angles?

PROPOSITION IV.

§ 39. Given, two triangles, ABC, DEF, agreeing in three adjacent parts (i. e. having three adjacent parts of the one equal to three adjacent

(p) Gr. EpiμET pos, measure round, from epi, around, and μérgov, measure. (q) Lat. triangulus, three-cornered, from tres, three, and angulus. (r) L. quadrilaterus, four-sided, from quatuor, four, and latus, side. (s) Gr. #evTáywvos, five-cornered, from EVTs, five, and ywvía. (t) Gr. E, six. (u) Gr. ixτá, seven. (v) Gr. ixτú, eight. (w) Gr. Sixx, ten.

= aD.

I. Let the three parts be two sides and the included angle; e. g. ABDE, AC = DF, and aA Place tABC upon tDEF, with pA upon pD, and the side AB upon DE.

Then, as AB = DE, where will pB fall?

As aA aD, where will AC fall?

=

=

As AC DF, where will pC fall?

Where, then, will BC fall?

Do the two triangles coincide throughout?

6. C.

How then do they compare in every part? 17. a, 16. c. What side is equal to BC? What angle to aB? to aC?

II. Let the three parts be a side and the two adjacent angles; e. g. BC= EF, aB = aE, and aC = aF.

Place tABC upon tDEF, with B upon E, and BC upon EF. Then, as BC= EF, where will C fall?

As aB

As aC

= aE, where will BA fall?

=

aF, where will CA fall? Where, then, must pA fall?

Are the two triangles identical?

What side is equal to AB? to AC? What angle is equal to aA?

§ 40. THEOR. IV. Triangles agreeing in three adjacent parts are identical.

[Proved by superposition.]

In identical triangles, how do those angles compare which are opposite to equal sides? How do those sides compare which are opposite to equal angles?

In the identical triangles GHI and KLM (Fig. X.), mention each pair of equal parts.

REMARK. In identical triangles, equal angles are opposite to equal sides.

§ 41. DEFINITIONS. When parallels are crossed by a single straight line, or by other parallels, the angles which are made are termed, with respect to each other, LIKE, or UNLIKE. They are termed LIKE, if they are turned either the same way or directly opposite ways; but otherwise, UNLIKE.

Thus, AGE, CHG, BGH, and DHF, form one class of like angles about the parallels AB

and CD, being all turned either up or
down; and EGB, GHD, AGH, and
CHF, form a second class of angles
which are like to each other,
but un-
like the first class, being all turned
either to the right hand or to the
left.

F

XI.

E

G

§ 42. Different pairs of angles about parallels have also received special names according to their situation. Thus,

a.) LIKE ANGLES. Two angles turned the same way, the one without and the other within the parallels, are termed exterior-interiory angles; as EGB and GHD, or AGE and CHG. Two angles turned opposite ways, and both within

(x) Lat., outer. (y) Lat., inner.

the parallels, are termed alternate-interior (or often simply alternate) angles; as, AGH and GHD, or BGH and GHC. Two angles turned opposite ways, and both without the parallels, are termed alternate-exterior angles; as, EGB and CHF, or AGE and DHF.

b.) Of UNLIKE ANGLES, we distinguish the interior angles upon the same side, as AGH and GHC, or BGH and GHD ; and the exterior angles upon the same side, as AGE and CHF, or EGB and DHF.

NOTE. The above may be termed different orders, or species, of like and unlike angles.

Draw two parallels crossing two other parallels, as in Fig. XIV. (§ 45); and mention each pair of exteriorinterior angles, of alternate-interior angles, of alternateexterior angles, of interior angles upon the same side, and of exterior angles upon the same side. Mention all the angles of each of the two classes of like angles (§ 41).

Let the figure AEFC be folded over upon the figure BEFD (or, in other words, be placed upon it with EF of the one figure upon EF of the other).

As aAEF =

aBEF (§ 25), where will EA fall?

Does EA now form the same line with EB?

As FC has the same direction with EA, and FD with EB, have now FC and FD the same direction with each other? Where, then, must FC fall?

(z) Lat. alternatus, from alterno, to vary, interchange.