Let ABCD be a quadrilat. fig. in O ABCD: then any two of its opp. Ls shall be together equal to 2 rt. Zs.

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In the same or equal circles, equal angles at

the centre or the circumference stand upon equal arcs.

Let ABG, CDH be equal Os, and the equal angles AEB, CFD at their centres, and

AGB, CHD at their circumferences; then shall arc AB = arc CD.

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For, let str. line AE be applied to str. line CF, so that pt. A may coincide with pt. C, and AE fall upon CF.

Then : AE=CF
:: pt. E coincides with pt. F.

(AE coincides with CF,
And ;' / AEB = L CFD,

i EB falls upon FD. And :: EB = FD

.. pt. B coincides with pt. D; Also ::: every point in arc AB is the same dist. from E, as every point in arc CD, is from F; and the points A, B, coincide with points C, D,

i arc AB coincides with arc CD,

i. e. arc AB = arc CD; ... the equal _s at the Cr. AEB, CFD, as also the _s at the Oce AGB, CHD which are also equal (Ax. 7) stand upon equal arcs AB and CD.

Wherefore, in the same, &c.

PROP. LIX. THEOR. 31. 3 En. The angle in a semicircle is a right angle.

Let BADC be a semi O, of which the diam. is BC and centre E; also BAC an L contained in the semi O.

Then _ BAC=rt. L

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PROP. LX. THEOR. 32. 3 Eu. If a straight line touch a circle, and from

the point of contact a straight line be drawn
cutting the circle, the angles which this line
makes with the line touching the circle,
shall be equal to the angles which are in the
alternate segments of the circle.

Let str. line EF touch O ABCD in B; from
B draw BD, cutting the O: then the s

which BD makes with the touching line EF shall be equal to the 2s in the alternate segments of the O; that is, ZDBF = L in seg. DAB; and _DBE = L in seg. DCB.

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... rem. DBE -Srem. ZBCD, in the

| altern, seg. DCB. Wherefore if a str. line, &c.

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Upon a given straight line to describe a segment of a circle, which shall contain an angle equal to a given rectilineal angle.

Let AB be a given str. line, C the given L; it is required to descr. on AB, a seg. of O, that shall contain an L=LC. Ist. If LC be a rt. L.


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