48. If two parallel lines are cut by a third straight line, then the alternate-interior angles are equal.

Suppose XY to be a line drawn through O, making

XOP=b.

Then

XY is parallel to CD.

843

[If two straight lines are cut by a third straight line, making the alternateinterior angles equal, the lines are parallel. |

49. COR. If two or more parallel lines are cut by a third

straight line, the corresponding angles are equal.

Hint. -Reduce to Proposition XII.

50. Remark.-It follows from the previous propositions and corollaries that if two lines are parallel and cut by a third straight line, as in the figure,

and any angle of the first set is supplementary to any angle of the second set.

PROPOSITION XIII. THEOREM

51. Two angles whose sides are parallel, each to each, are either equal or supplementary.

GIVEN the angles at O and O' with their sides OA and OB respectively parallel to CF and ED.

TO PROVE

the angle a=a', and a+b=2 right angles.

52. Remark. To determine when the angles are equal and when supplementary, we observe that every angle, viewed from its vertex, has a right and a left side. (Thus OA is the left side of a.) Now, if the two angles have the right side of one parallel to the right side of the other and likewise their left sides parallel, they are equal; whereas, if the right side of each is parallel to the left side of the other, they are supplementary. Or, briefly, if their parallel sides are in the same right-and-left order, they are equal, if in opposite order, supplementary.

Thus, a and EO'F, which have their sides parallel, right to right (OB to O'E) and left to left (OA to O'F), are equal, while a and EO ̊C, which have their sides parallel right to left (OB to O′E) and left to right (OA to O'C), are supplementary. The student can easily test and verify all the sixteen cases obtained by comparing each of the four angles about O with each of the four about '.

PROPOSITION XIV. THEOREM

53. Two angles whose sides are perpendicular, each to each, are either equal or supplementary.

GIVEN the angle NOM, or a, and the lines AB and CD intersecting

at O' and respectively perpendicular to ON and OM.

TO PROVE the angle a=a', and a+b=2 right angles.

836

At O, draw OA' parallel to AB and OC' parallel to CD. OA', being parallel to AB, is perpendicular to ON. $ 36 [f two straight lines are parallel, and a third straight line is perpendicular to one of them, it is perpendicular to the other.] For the same reason OC', being parallel to CD, is perpendicular to OM.

From each of the right angles A'ON and C'OM take away the common angle w.

This leaves

c=a.

c=a'.

Ax. 3

$51

[Having their sides respectively parallel, and in the same right and-left

But

54. Remark.-The angles are equal if their sides are perpendicular right to right and left to left, but supplementary if their sides are perpendicular in opposite right-and-left order.

Thus a and DO'B, which have their right sides (OM and O'D) perpendicular and their left sides (ON and O'B) perpendicular, are equal; etc., etc

TRIANGLES

55. Def.-A triangle is a figure bounded by three straight lines called its sides.

56. Def.—A right triangle is a triangle one of whose angles is a right angle.

5%. Def.—An equiangular triangle is one whose angles are all equal.

58. The sum of the three angles of any triangle is two right angles.*

Draw KH parallel to BC, and from O, any point of this line, draw OE and OD parallel respectively to the sides AB and AC.

*This was first proved by Pythagoras or his followers about 550 B.C.