(iii) Lastly, the LEGB = the GHD;

add to each the BGH;

Proved.

then the L'EGB, BGH together the angles BGH, GHD.

=

=

But the adjacent LEGB, BGH together two right angles; .. the two interior ▲ BGH, GHD together two right angles.

=

Q.E.D.

PARALLELS ILLUSTRATED BY ROTATION.

The direction of a straight line is determined by the angle which it makes with some given line of reference.

Thus the direction of AB, relatively to the given line YX, is given by the angle APX.

Suppose AB to rotate about P through the LAPX, so as to take the position XY. Thence let it rotate about Q the opposite way through the equal LXQC: it will now take the position CD. Thus AB may be brought into the position of CD by two rotations which, being equal and opposite, involve no final change of direction.

HYPOTHETICAL CONSTRUCTION. In the above diagram let AB be a fixed straight line, Q a fixed point, CD a straight line turning about Q, and YQPX any transversal through Q. Then as CD rotates, there must be one position in which the LCQX the fixed LAPX.

=

Hence through any given point we may assume a line to pass parallel to any given straight line.

Obs. If AB is a straight line, movements from A towards B, and from B towards A are said to be in opposite senses of the line AB.

THEOREM 15. [Euclid I. 30.]

Straight lines which are parallel to the same straight line are parallel to one another.

Let the straight lines AB, CD be each parallel to the straight line PQ.

It is required to prove that AB and CD are parallel to one another.

Draw a straight line EF cutting AB, CD, and PQ in the points G, H, and K.

Proof. Then because AB and PQ are parallel, and EF meets them,

... the AGK = the alternate GKQ.

And because CD and PQ are parallel, and EF meets them, .. the exterior GHD = the interior opposite ▲ GKQ.

.. the AGH = the GHD ; and these are alternate angles; ... AB and CD are parallel.

Q.E.D.

NOTE. If PQ lies between AB and CD, the Proposition needs no proof; for it is inconceivable that two straight lines, which do not meet an intermediate straight line, should meet one another.

The truth of this Proposition may be readily deduced from Playfair's Axiom, of which it is the converse.

For if AB and CD were not parallel, they would meet when produced. Then there would be two intersecting straight lines both parallel to a third straight line: which is impossible.

Therefore AB and CD never meet; that is, they are parallel.

EXERCISES ON PARALLELS.

1. In the diagram of the previous page, if the angle EGB is 55°, express in degrees each of the angles GHC, HKQ, QKF.

2. Straight lines which are perpendicular to the same straight line are parallel to one another.

3. If a straight line meet two or more parallel straight lines, and is perpendicular to one of them, it is also perpendicular to all the others.

4. Angles of which the arms are parallel, each to each, are either equal or supplementary.

5. Two straight lines AB, CD bisect one another at O. Shew that the straight lines joining AC and BD are parallel.

6. Any straight line drawn parallel to the base of an isosceles triangle makes equal angles with the sides.

7. If from any point in the bisector of an angle a straight line is drawn parallel to either arm of the angle, the triangle thus formed is isosceles.

8. From X, a point in the base BC of an isosceles triangle ABC, straight line is drawn at right angles to the base, cutting AB in Y, and CA produced in Z: shew the triangle AYZ is isosceles.

9. If the straight line which bisects an exterior angle of a triangle is parallel to the opposite side, shew that the triangle is isosceles.

10. The straight lines drawn from any point in the bisector of an angle parallel to the arms of the angle, and terminated by them, are equal; and the resulting figure is a rhombus.

11. AB and CD are two straight lines intersecting at D, and the adjacent angles so formed are bisected: if through any point X in DC a straight line YXZ is drawn parallel to AB and meeting the bisectors in Y and Z, shew that XY is equal to XZ.

12. Two straight rods PA, QB revolve about pivots at P and Q, PA making 12 complete revolutions a minute, and QB making 10. If they start parallel and pointing the same way, how long will it be before they are again parallel, (i) pointing opposite ways, (ii) pointing the same way?

THEOREM 16. [Euclid I. 32.]

The three angles of a triangle are together equal to two right angles.

B

x

=

Let ABC be a triangle.

It is required to prove that the three L ABC, BCA, CAB together = two right angles.

Produce BC to any point D; and suppose CE to be the line through C parallel to BA.

Proof. Because BA and CE are parallel and AC meets them, .. the LACE = the alternate CAB.

Again, because BA and CE are parallel, and BD meets them. .. the exterior ECD = the interior opposite ABC.

. the whole exterior LACD = the sum of the two interior opposite LS CAB, ABC.

then the

To each of these equals add the ▲ BCA;

BCA, ACD together the three * BCA, CAB, ABC. But the adjacent 4 BCA, ACD together two right angles. ..the BCA, CAB, ABC together two right angles.

=

Q.E.D.

Obs. In the course of this proof the following most important property has been established.

If a side of a triangle is produced the exterior angle is equal to the sum of the two interior opposite angles.

Namely, the ext. 4 ACD = the CAB + the 2 ABC.

INFERENCES FROM THEOREM 16.

1. If A, B, and C denote the number of degrees in the angles of a triangle,

then A+B+C = 180°.

2. If two triangles have two angles of the one respectively equal to two angles of the other, then the third angle of the one is equal to the third angle of the other.

3. In any right-angled triangle the two acute angles are complementary.

4. If one angle of a triangle is equal to the sum of the other two, the triangle is right-angled.

5. The sum of the angles of any quadrilateral figure is equal to four right angles.

EXERCISES ON THEOREM 16.

1. Each angle of an equilateral triangle is two-thirds of a right angle, or 60°.

2. In a right-angled isosceles triangle each of the equal angles is 45°.

:

3. Two angles of a triangle are 36° and 123° respectively deduce the third angle; and verify your result by measurement.

4. In a triangle ABC, the LB=111°, the LC=42°; deduce the LA, and verify by measurement.

5. One side BC of a triangle ABC is produced to D. If the exterior angle ACD is 134°; and the angle BAC is 42°; find each of the remaining interior angles.

6. In the figure of Theorem 16, if the LACD=118°, and the LB 51°, find the LA and C; and check your results by measurement.

7. Prove that the three angles of a triangle are together equal to two right angles by supposing a line drawn through the vertex parallel to the base.

8. If two straight lines are perpendicular to two other straight lines, each to each, the acute angle between the first pair is equal to the acute angle between the second pair.