COR. If DEF be any angle, the adjacent ▲ DEK formed by producing one of its arms FE is the adjacent L FEL formed by producing the other arm.

K

For they are opposite angles.

=

.. if an angle one of its adjacent angles it is also = the other.

..if one straight line is to another, the latter is 1 to the former.

Thus each of the arms of a right is to the other.

PROPOSITION VII.

It is always possible to draw a straight line perpendicular to a given straight line from a given point in the same.

Let BC be the given straight line,

and A the given point in it.

Then from A a straight line can be drawn LBC.

From AB, AC cut off equal parts AD, AE.

With D and E as centres, describe equal Os intersecting in F

Join AF.

·· DA is = EA, DF=EF, and AF common to the two

PROPOSITION VIII.

From a given point in a given straight line there cannot be drawn more than one straight line perpendicular to it.

let AF and AG be each of them 1 to BC.

..LS FAB, FAC are equal to one another, and also ▲ S GAB, GAC are equal to one another, which is impossible.

.. from a given point in a given straight line there cannot be drawn more than one straight line 1 to it.

COR. All right angles are equal to one another.

Let ABC, GHK be two right angles,

and let the ABC be applied to the ▲ GHK,
so that B may fall on H, and BC along HK,
then will BA fall along HG.

For if it fell in any other direction, as HX, then from H there would be drawn two straight lines HG, HX, 1 to HK, which is impossible;

and

C. G.

.. BA must fall along HG,

PROPOSITION IX.

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 the straight line AB make with CD upon one side of it the Ls ABC, ABD.

These shall either be two rights;

or shall together be two right 4 s.

If the ABC is the ▲ ABD;

But if not,

L

then each of them is a right 4.

through B draw BEL to CD;

A

then the ABC is the 4s CBE and EBA together; the LS ABC and ABD are together the 4s CBE, EBA and ABD;

..

of which CBE is a right

and EBA, ABD make up a right ▲ ;

LS ABC, ABD are together = two rights.

COR. If a number of straight lines BA, BC, BF, BE meeting in the point B form angles ABC, CBF, FBG, GBA which fill up the space about B, then these is are together = 4 right L s.

For produce CB to X.

C

G

Then all the 4S ABC, CBF, FBG, GBA are equal to thes on one side of CBX together with the s on the other side of CBX, and are,, four right 4 s.

=

PROPOSITION X.

If two straight lines, drawn from one extremity of a straight line on 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.

B

Let BC, BD drawn from B one extremity of AB make the adjacent s ABC, ABD together = two right angles.

Then shall BD be in the same straight line with BC.

For if not, if possible, let BE be in the same straight line with BC;

.. LS ABC, ABE are together two right angles; (1.9) and are ... = 4 s ABC, ABD;

.. LABE is = ▲ ABD, which is impossible.

(hyp.)

.. no other straight line than BD is in the same straight line with BC;

.. BD is in the same straight line with BC.