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PROP. XIII. THEOR.

A

B

GEN. ENUN.—The angles, which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles.

Part. Enun.—Let the st. line AB make with cd, upon one side of it, Fig. 1. the Z S CBA, ABD; then these are either two rt. Zs, or their sum=two rt. Zs.

If the Z CBA = Z ABD,
as in fig. 1, then each of them o
is a rt. 2 (Def. 10). But, if not,
CONST.–From the pt. B

Fig. 2.
(fig. 2) draw be at rt. Zs to
CD (Prop. XI.), so that the 2
CBE, EBD are two rt. Zs.

DEMONST.-1. Now since the Z CBE Z CBA + ABE, add to each the 2 EBD; Z CBE + EBD = L CBA + ABE + EBD (Ax. 2).

2. Again, since the Z DBA = Z DBE + EBA, add to each the Z ABC; .. Z DBA + ABC = DBE + EBA + ABC (Ax. 2).

3. .; Z DBA + ABC = CBE + EBD (Ax. 1) = two rt. Zs.

Wherefore the < s which one st. line, &c.— Q. E. D.

It is clear from this Proposition that the sum of all the

A

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B

AB =

L', which are made by any number of st. lines drawn to the same pt. in another st. line, on the same side of it, two rt. Zs.

The Proposition is of great practical use in Trigonometry and Astronomy.

We are now in a position to furnish a very simple demonstration of the equality of the 2 s above and below the base of an isosc. A. Thus:Part. Enun.-Let ABC be an isosc. A, having the side

the side AC; and let the sides AB, AC be produced to D and E; then the | ABC =

L ACB,

A and the LDBC = 2 ЕСв.

Const.-Bisect the BAC by the st. line AF.

DEMONST.-1. Because AB = AC, AF is common to AS ABF, ACF, and L

L cAF; .. the As are =, and the L ABC L ACB: i. e, the /s above the base are =.

2. Again, the < $ CBA + CBD = two rt. 28 = 48 BCA + BCE (Prop.XIII.); but ABC = 2 ACB; .. remain. ing / DBC = remaining / ECB: i. e. 28 below the base

.-Q. E. D.

B

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D

E

are

PROP. XIV. THEOR.

GEN. ENUN.-If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line.

Part. Enun.–At the pt. B in the st. line AB, let the two st. lines BC, BD make the adjacent z ABC, ABD = twort. Zs; then bc and bd are in the same st. line.

A

G

B

DEMONST. (Ad absurd.).

For, suppose that bd be not in the same st. line with BC, and that be is in the same st. line with it.

1. Then, because Ab makes with cbe, upon one side of it, the < $ CBA, ABE ; ... the Zs CBA + ABE = two rt. 28. (Prop. XIII.

2. But the ZS CBA + ABD = two rt. Zs. (Hyp.)

3. .. ZS CBA + ABE = Zs CBA + ABD (Ax. 1).

4. .-. (by subtracting the common Z CBA) the Z ABE =

2 ABD (Ax. 3): i. e. the < = >, which is impossible.

5. ... BE is not in the same st. line with Bc.

6. In like manner it may be proved that no other can be in the same st. line with it but BD, which is ... in the same st. line with it.

Wherefore, if at a pt. &c.-Q. E. D. This Proposition is the converse of the preceding. It is necessary that the two lines BC, BD should meet from opposite sides of AB; for it is very possible that on the same side two lines may form 48 = to two rt. 4 $ with it, without being in the same st. line. Thus :

Const.-Let c be any pt. in a st. line AB. Draw cp. at rt. Zs to AB. (Prop. XI.) Bisect the 2 ACD by the st. line ce. (Prop. IX.) In cd take any pt. D, and through d draw de at rt. Zs to cp. Produce Ed to F, and make DF = ED. (Prop. III.) Join CF.

DEMONST.-Then in A' CDE, CDF, ED, DC = FD, DC, each to each, and the

rt. FDC; ... base ce = base cr (Prop. IV.); and the L ECD = L FCD = half a rt. 2 (Const.): .. the L ACF = a rt. L + half a rt. L, and the L ACF + ACE

A

E

с

D

B

rt. 2 EDC =

E

= two rt. 28; and they are formed by two st. lines EC, FC, meeting Ab in the same pt. The said lines, however, are not in the same st. line.

PROP. XV. THEOR.

B

E

Gen. Enun.—If two straight lines cut one another, the vertical, or opposite, angles shall be equal.

Part. Enun.- Let the two st. lines AB, CD cut one another in the pt. c E; then the < AEC shall be Z DEB, and the ZA CEB shall be = 2 AED.

DEMONST.-1. Because the st. line AE makes, with

CD, the ZS CEA, AED; .. these <s are = two rt. Zs. (Prop. XIII.)

2. Also, because the st. line de makes, with AB, the ZS AED, DEB; .'. these Zs are also = two rt. 2$. (Prop. XIII.)

3. ... the ZS CEA + AED = %$ AED + DEB (Ax. 1).

4. .. (by subtraction) the < CEA = DEB.

5. In like manner it may be shown that Z CEB = 2 AED.

Wherefore, if two st. lines, &c.-Q. E. D.

Cor. 1.-From this it is manifest that, if two straight lines cut one another, the Z8 which they make at the pt. of intersection are together = to four rt. Zs.

Cor. 2.—Consequently, the sum of all the <s made by any number of st. lines intersecting at one pt. is = to four rt. Zs.

DEB=

As the student should take nothing for granted, he will do well to demonstrate the equality of the 4 $ CEB and AED, as stated in the Proposition.-Though capable of a rigorous demonstration, the Proposition is nevertheless but little more than an inference from the definition of an L. For if two straight lines, inclined at an L, intersect, their inclination must continue the same on both sides; or, in other words, the vertical (s will be With respect to the Corollaries, it is clear that ce makes, with AB, the 2 S AEC + CEB = two rt. 2$, and dE makes, with AB, the Z S AED +

two rt. 28 (Prop. XIII.); .. by addition, the < $ AEC + CEB + AED + DEB = four rt. 25. Again, as these four 2s are merely subdivided by any number of st. lines intersecting in E, the sum of four rt. _ s remains unaltered. The

practical application of the 15th Proposition to the measurement of 28 may be thus illustrated :

Let abcd be a graduated O (Obs. on Def. 18), having an object-glass, or telescope, moveable round the cr. E; let the glass be directed to the object s, and the degree marked to which it points on the Oce; then let it be turned round till it points to another object t, and the degree marked as before. The 2' distance between the objects, or the number of degrees which the glass has passed over, is clearly that which is contained in the L AEB, measured by the arc ab; or, which is the same thing, the number con. tained in the L CED, measured by the arc cd. Upon this principle the Theodolite, and several astronomical instruments are constructed.

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