If one straight line stand upon another straight line, then the adjacent angles shall be either two right angles, or together equal to two right angles.

Let the straight line AB stand upon the straight line DC: then the adjacent angles DBA, ABC shall be either two right angles, or together equal to two right angles.

CASE I. For if the angle DBA is equal to the angle ABC, each of them is a right angle.

Def. 7. CASE II. But if the angle DBA is not equal to the angle ABC,

I. 11.

from B draw BE at right angles to CD. Proof. Now the angle DBA is made up of the two angles DBE, EBA;

to each of these equals add the angle ABC; then the two angles DBA, ABC are together equal to the three angles DBE, EBA, ABC.

Ax. 2.

Again, the angle EBC is made up of the two angles EBA,

ABC;

to each of these equals add the angle DBE. Then the two angles DBE, EBC are together equal to the three angles DBE, EBA, ABC.

Ax. 2.

But the two angles DBA, ABC have been shewn to be equal to the same three angles;

therefore the angles DBA, ABC are together equal to the angles DBE, EBC.

But the angles DBE, EBC are two right angles; therefore the angles DBA, ABC are together equal

right angles.

Ax. 1.

Constr.

to two Q. E. D.

DEFINITIONS.

(i) The complement of an acute angle is its defect from a right angle, that is, the angle by which it falls short of a right angle.

Thus two angles are complementary, when their sum is a right angle.

(ii) The supplement of an angle is its defect from two right angles, that is, the angle by which it falls short of two right angles.

Thus two angles are supplementary, when their sum is two right angles.

COROLLARY. Angles which are complementary or supplementary to the same angle are equal to one another.

EXERCISES.

1. If the two exterior angles formed by producing a side of a triangle both ways are equal, shew that the triangle is isosceles.

2. The bisectors of the adjacent angles which one straight line makes with another contain a right angle.

The internal and external bisectors of an angle are at right angles to one another.

3. Shew that the angles AOX and COY are complementary.

4.

Shew that the angles BOX and COX are supplementary; and also that the angles AOY and BOY are supplementary.

PROPOSITION 14. THEOREM.

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

E

B

At the point B in the straight line AB, let the two straight lines BC, BD, on the opposite sides of AB, make the adjacent angles ABC, ABD together equal to two right angles:

then BD shall be in the same straight line with BC. Proof. For if BD be not in the same straight line with BC, if possible, let BE be in the same straight line with BC. Then because AB meets the straight line CBE, therefore the adjacent angles CBA, ABE are together equal to two right angles.

Нур.

Ax. 11.

I. 13. But the angles CBA, ABD are also together equal to two right angles. Therefore the angles CBA, ABE are together equal to the angles CBA, ABD. From each of these equals take the common angle CBA; then the remaining angle ABE is equal to the remaining angle ABD; the part equal to the whole; which is impossible. Therefore BE is not in the same straight line with BC. And in the same way it may be shewn that no other line but BD can be in the same straight line with BC. Therefore BD is in the same straight line with BC. Q.E.D.

EXERCISE.

ABCD is a rhombus; and the diagonal AC is bisected at O. If O is joined to the angular points B and D; shew that OB and OD are in one straight line.

Obs. When two straight lines intersect at a point, four angles are formed; and any two of these angles which are not adjacent, are said to be vertically opposite to one another.

If two straight lines intersect one another, then the vertically opposite angles shall be equal.

Let the two straight lines AB, CD cut one another at the point E:

then shall the angle AEC be equal to the angle DEB, and the angle CEB to the angle AED Proof. Because AE makes with CD the adjacent angles CEA, AED,

therefore these angles are together equal to two right angles.

I. 13.

Again, because DE makes with AB the adjacent angles AED, DEB,

therefore these also are together equal to two right angles. Therefore the angles CEA, AED are together equal to the angles AED, DEB.

From each of these equals take the common angle AED; then the remaining angle CEA is equal to the remaining angle DEB.

Ax. 3.

In a similar way it may be shewn that the angle CEB is equal to the angle AED.

Q. E. D.

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

COROLLARY 2. Consequently, when any number of straight lines meet at a point, the sum of the angles made by consecutive lines is equal to four right angles.

PROPOSITION 16. THEOREM.

If one side of a triangle be produced, then the exterior angle shall be greater than either of the interior opposite angles.

Let ABC be a triangle, and let one side BC be produced to D then shall the exterior angle ACD be greater than either of the interior opposite angles CBA, BAC.

Construction.

Bisect AC at E:

I. 10. Join BE; and produce it to F, making EF equal to BE. 1. 3.

Join FC.

therefore the triangle AEB is equal to the triangle CEF in all respects:

I. 4.

so that the angle BAE is equal to the angle ECF. But the angle ECD is greater than its part, the angle ECF; therefore the angle ECD is greater than the angle BAE; that is, the angle ACD is greater than the angle BAC. In a similar way, if BC be bisected, and the side AC produced to G, it may be shewn that the angle BCG is greater than the angle ABC.

But the angle BCG is equal to the angle ACD: 1. 15. therefore also the angle ACD is greater than the angle ABC.

Q. E. D.