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PROP. XII. THEOR. 13. 1 Eu. 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.

Let AB make with DC, on the same side of it, Ls DBA, ABC; then / DBA + L ABC = 2 rt. Ls.

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PROP. XIII. THEOR. 14. 1 Eu.
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.

At the point B in AB let BC, BD, on the oppo, sides of AB, make the adj. Ls ABC + LABD = 2 rt. Zs; then shall BD be in the same straight line with BC.

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or, less

greater, which


Neu absurd ; :. BE is not in the same str. line with BC.

In like manner it may be shown, that no other line but BD can be in the same str. line with BC.

Wherefore, if at a point, &c.

PROP. XIV. THEOR. 15. 1 Eu. If two straight lines cut one another, the ver

tical or opposite angles shall be equal. Let the str. lines AB, CD, cut one another in E; then AEC = Z DEB, and Z CEB = L AED.

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From these equals take away the common

LAED, :: remain. Z CEA = remain. _ DEB. Ax. 3. In the same manner it may be shown, that

Therefore, if two straight lines, &c.

Cor. 1.-If two str. lines cut one another, the Zs which they make at the pt. where they cut, are together equal to 4 rt. Zs.

Cor. 2.—All the angles made by any number of lines meeting in one pt., are together equal to 4 rt. Zs.

PROP. XV. THEOR. 16. 1 Eu. If one side of a triangle be produced, the ex

terior angle is greater than either of the interior opposite angles.

Let ABC be a A, and the side BC be prod. to D; then the ext. L ACD > L CBA or < BAC, the int. oppo. Zs.

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PROP. XVI. THEOR. 17. 1 Eu. Any two angles of a triangle are together less

than two right angles. Let ABC be any A, any two of its s are together less than 2 rt. Zs.

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add ACB to each, .: L ACD + L ACB > LABC + L ACB; Prop. 12. but LACD + L ACB = 2 rt. ZS,

:: L ABC + L ACB < 2 rt. Zs.

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