THEOREM VIII.

56. Any point without the perpendicular erected at the middle point of a straight line is unequally distant from the extremities of that line.

57. Two oblique lines drawn from a point in a perpendicular are equal if they cut off equal distances from the foot of the perpendicular.

On CD as an axis, revolve CDE till it falls in the plane

THEOREM X.

58. The sum of two lines drawn from a point to the extremities of a straight line is less than the sum of two other lines similarly drawn and enveloping them.

Let AP and BP be two lines drawn from P to the extremities of AB, and let AC and BC be two lines drawn similarly and enveloping AP and BP.

Then AP + BP + PD < AC + CD + DB + PD.

Substitute BC for its equal CD + DB, and subtract PD from each member.

Then

AP+BPAC + BC.

Q. E. D.

THEOREM XI.

59. Of two oblique lines drawn from the same point, that is the greater which terminates at the greater distance from the foot of the perpendicular.

Let CO be to AB, and CE and CD oblique lines drawn so that EO> DO.

Then, as in (52), CD DF, and CE EF.

But

or

=

=

CO, and draw EF

(58)

Q. E. D.

60. COR. 1.-Two equal oblique lines terminate at equal distances from the foot of the perpendicular.

61. COR. 2.-Only two equal straight lines can be drawn from a point to a line; and of two unequal oblique lines, the greater terminates at the greater distance from the foot of the perpendicular.

IGHT

PARALLEL STRAIGHT LINES.

THEOREM XII.

62. If two parallel lines are cut by a third line

I. The corresponding angles are equal;

II. The alternate interior angles are equal;

III. The sum of the interior angles on the same side of the secant equals two right angles.

Let the lls AB and CD be cut by the line EF.

Since OB and PD are II, they have the same direction and open equally from the line EF;

La=Lb.

Likewise we can prove that c = L d.

Likewise we can prove that e = L f.

(48)

(Case I.)

(Ax. 1)

63. COR.-If a straight line lies in the same plane with two parallels, and is perpendicular to one of them, it is perpendicu lar to the other also.

THEOREM XIII.

64. If two straight lines are cut by a third line, these two lines are parallel

I. If the corresponding angles are equal;

II. If the alternate interior angles are equal;

III. If the sum of the two interior angles on the same side of the secant equals two right angles.

Let the straight lines AB and CD be cut by EF.

I. To prove that AB and CD are || if ▲ a = b.

If ▲ a=Lb, OB and PD open equally from EF, and

hence have the same direction, or are II.

(28)

II. To prove that AB and CD are || if ≤ d = ▲ b.