THEOREM 2.

The sum of all the angles made by any number of lines taken consecutively which meet at a point will be four right angles.

Let any number of lines OA, OB, OC, OD, OE meet at O, then AOB+BOC+COD + DOE+ EOA will make up four right angles.

Proof. For the sum of these angles is an angle of one revo

lution, which is equal to four right angles.

B

Def. 17. The opposite angles made by the intersection of two straight lines are called vertically opposite angles.

THEOREM 3.

Vertically opposite angles will be equal to one another.

Let AOD, BOC be vertically opposite angles. They will be equal.

Proof. For since the same quan- A

tity of turning of AB round O which would make OB coincide with OC,

D

B

would also make OA coincide with OD, it follows that the angle AOD=the angle BOC.

Similarly AOC=BOD.

Def. 18. Angles are expressed arithmetically as multiples of some known angle. For this purpose a right angle is divided into 90 equal angles which are called degrees, and written thus, 45o. A degree is subdivided into 60 equal parts called minutes and written thus, 15'. And a minute is subdivided into 60 seconds (60").

EXERCISES ON ANGLES.

I. If two straight lines intersect at a point, and one of the four angles is a right angle, prove that the other three are right angles.

2. If five lines meet at a point and make equal angles with one another all round that point, each of the angles will be four-fifths of a right angle. Express this in degrees.

3. If the four angles made by four straight lines which meet at a point are all right angles, prove that the four lines form two straight lines.

4. Two straight lines meet at a point. Are the angles at that point together equal to four right angles?

5. Prove that the bisectors of adjacent supplementary angles are at right angles to one another.

6. Find the angle between the bisectors of adjacent complementary angles.

7. Of two supplementary angles the greater is double of the less; find what fraction the less is of four right angles. 8. Twelve lines meet at a point so as to form a regular twelve-rayed star: find the number of degrees in the angle between consecutive rays.

9.

If A is the number of degrees in any angle, prove that 90° + Ao is supplementary to 90° – Ao; and that 45° + A° is complementary to 45° – Ao.

10. Find the supplement and complement of 21° 35′ 45′′.

II.

If four straight lines OA, OB, OC, OD meet at a point, and AOB = COD, and BOC=DOA, prove that AOC, BOD are straight lines.

12. Prove that the bisectors of the four angles which one straight line makes with another form two straight lines. at right angles to one another.

In the first section a single straight line was defined as a line which has the same direction at all parts of its length.

We now proceed to consider the relations of two or more straight lines in one plane in respect of direction.

Ax. 6. Two different straight lines may have either the same or different directions.

Ax. 7. Two different straight lines which meet one another have different directions.

Ax. 8. Two straight lines which have different directions would meet if prolonged indefinitely.

Thus A and B in the figure have the same direction; and C and D which meet have different directions; and E and F which have different directions would meet if produced far enough.

Def. 19. Straight lines which are not parts of the same straight line, but have the same direction, are called parallels.

From this definition, and the axioms above given, the following results are immediately deduced:

(1) That parallel lines would not meet however far they were produced.

For if they met, they would have different directions by Ax. 7.

(2) That lines which are parallel to the same line are parallel to one another.

For they each have the same direction as that line, and therefore the same direction as the other.

Def. 20. straight lines it makes with them eight angles which have received special

When a straight line intersects two other

names.

In the figure 1, 2, 7, 8, are called exterior angles, and 3, 4, 5, 6, are called interior angles.

5/6

1/2

3/4

Again, 1 and 5 are said to be corresponding angles; so also are 2 and 6, 7 and 3, 8 and 4: and 3 and 6 are said to be alternate angles, so also are 4 and 5.

Ax. 9. An angle may be conceived as transferred from one position to another, the direction of its arms remaining the same.

THEOREM 4.

If two lines are respectively parallel to two other lines, the angles made by the first pair will be equal or supplementary to the angles made by the second pair; equal, if both are taken in the same or both in the opposite direction; supplementary, if one is taken in the same and one in the opposite direction.

Let AOB, COD be respectively parallel to EKF, GKH.

Then will the angle AOD be equal to EKH or GKF, and supplementary to EKG and HKF.

Proof. For conceive the angle AOD transferred to K,

the direction of its arms being unaltered. Ax. 9.

Then

since, by hypothesis, OA and OD have the same direction as KE and KH, they would then coincide with KE and KH; and the angle AOD would coincide with the angle EKH.

Therefore the angle AOD = EKH.

But EKH=GKF (Th. 3), and therefore AOD = GKF Also GKE or FKH is supplementary to EKH (Th. 1), and therefore GKE or FKH is supplementary to AOD.

COR. If a straight line intersect two parallel lines, it will make the corresponding angles equal, the alternate angles equal, and the interior angles on the same side of the intersecting line supplementary.

Let AB, CD be parallels, and let EFGH intersect them.

Then will the angles at G be equal to the corresponding angles at F

Proof. For conceive the angles

E

F

B

A

G

C

D

H

at G, transferred to F, the direction of the lines being unaltered. Each angle would then coincide with its corresponding angle, and is therefore equal to it.