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O draw a straight line perpendicular to a given T straight line of an unlimited length, from a given point without it.

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HE angles which one straight line makes with an-
other upon one side of it, are either two right

angles, or are together equal to two right angles.
- B 4 Iset

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or are together equal to two right angles.
For, if the angle CBA be equal to ABD, each of them is a

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* Per to right' angle; but, if not, from the point B draw BE at right

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angles" to CD ; therefore the angles CBE, EBD are two right angles"; and because CBE is equal to the two angles CBA, ABE together, add the to: EBD to each of these equals; therefore the angles CBE, EBD are equals to the three angles CBA, ABE, EBD. Again, because the angle DBA is equal to the two angles DBE, EBA, add to these equals the angle ABC ; therefore the angles DBA, ABC are equal to the three angles DBE, EBA, ABC; but the angles CBE, EBD have been demonstrated to be equal to the same three angles; and things that are equal to the same are equal 4 to one another; therefore the angles CBE, EBD are equal to the angles DBA, ABC; but CBE, EBD are two right angles; therefore DBA, ABC are

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ABC, ABD equal together to
two right angles. BD is in the

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Because the straight line AE makes with CD the angles CEA, AED, these angles are toge- C ther equal “to two right angles. Again, because the straight line DE makes with AB the ang. A AED, DEB, these also are together equal * to two right D angles; and CEA, AED have - been demonstrated to be equal to two right angles; wherefore the angles CEA, AED are equal to the angles AED, DEB. Take away the common angle AED, and the remaining angle CEA is equal” to the remaining angle DEB. In the same b 3. Ax, manner it can be demonstrated that the angles CEB, AED are equal. Therefore, if two straight lines, &c. Q.E.D.

CoR. 1. From this it is manifest, that, if two straight lines cut one another, the angles they make at the point where they cut, are together equal to four right angles.

CoR. 2. And consequently that all the angles made by any number of lines meeting in one poino are together equal to four right angles. \

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IF one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Let ABC be a triangle, and let its fide BC be produced to D, the exterior angle ACD is greater than either of the interior opposite angles CBA, BAC.

Bised AC in E, join BE A
and produce it to F, and make F
EF equal to BE; join also
FC, and produce AC to G.

Because AE is equal to
EC, and BE to EF; AE, EB
are equal to CE, EF, each to
each; and the angle AEB is
equal"to the angle CEF, be-
cause they are : ver- B C ID
tical angles; therefore the
base AB is equal" to the base
CF, and the triangle AEB to G
the triangle CEF, and the -
remaining angles to the remaining angles, each to each, to which
the equal fides are opposite; wherefore the angle BAE is equal
to the angle ECF; but the angle ECD is greater than the angle
ECA; therefore the angle ACD is greater than BAE : In the
same manner, if the fide BC be bise&ted, it may be demonstrated
that the angle BCG, that is ", the angle ACD, is greater than
the angle ABC. Therefore, if one side, &c. Q. E. D.

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A NY two angles of a triangle are together less than
two right angles.
A.

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THE greater fide of every triangle is opposite to the greater angle.

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THE greater angle of every triangle is subtended by the greater fide, or has the greater side opposite to it.

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