Proposition XI

76. Draw two parallel lines; also a transversal. Name the pairs of corresponding angles. How do the angles of any pair compare in size? Theorem. If two parallel straight lines are cut by a transversal, the corresponding angles are equal.

77. Draw two lines; also a transversal. In what direction do the lines extend with reference to each other, if the corresponding angles are equal?

Theorem. Two straight lines cut by a transversal are parallel, if the corresponding angles are equal. (Converse of Prop. XI.)

Proposition XIII

78. Draw two parallel lines; also a transversal. How does the sum of the two interior angles on the same side of the transversal compare with a right angle?

Theorem. If two parallel straight lines are cut by a transversal, the sum of the two interior angles on the same side of the transversal is equal to two right angles.

Data: Any two parallel straight

lines, as AB and CD, cut by a transversal, as EF.

To prove the sum of the two in

terior angles on the same side of the transversal, as t and s, equal to two right angles.

F

S

E

Ex. 21. If two parallel lines are cut by a transversal, what is the sum of the two exterior angles on the same side of the transversal ?

Ex. 22. The straight lines AB and CD are cut by EF in G and H respectively; angle EHD = 38°. What must be the value of the angle EGB in order that AB and CD may be parallel?

Ex. 23. A transversal cutting two parallel lines makes an interior angle of 50°. What is the value of the other interior angle on the same side of the

transversal ?

Ex. 24. Two parallel lines are cut by a third line making one interior angle 35°. What is the value of each of the other interior angles? How many degrees are there in the sum of the interior angles upon the same side of the transversal ?

Ex. 25. How do lines bisecting any two alternate interior angles, formed by two parallel lines cut by a transversal, lie with reference to each other? Ex. 26. The straight lines AB and CD are cut by EF in G and H respectively; angle EHD = 40°. What must be the value of the angle AGF,

if AB and CD are parallel?

MILNE'S GEOM. 3

Proposition XIV

79. Draw two lines; also a transversal. In what direction do the lines extend with reference to each other, if the sum of the two interior angles on the same side of the transversal is equal to two right angles?

Theorem. Two straight lines cut by a transversal are parallel, if the sum of the two interior angles on the same side of the transversal is equal to two right angles. (Converse of Prop. XIII.)

Ex. 27. AB and CD are two lines cut in G and H, respectively, by EF; ZBGF = 123°, and ▲ GHD = 62°. Are the lines AB and CD parallel?

Ex. 28. If two lines are cut by a transversal and the sum of the two exterior angles on the same side of the transversal is equal to 180°, are the lines parallel?

Ex. 29. Two parallel lines are cut by a transversal so that one exterior angle is 105°. How many degrees are there in the sum of each pair of alternate interior angles ?

Ex. 30. The bisectors of two adjacent angles are perpendicular to each other. What is the relation of the given angles to each other?

Ex. 31. Two lines are cut by a transversal. In what direction do they extend with reference to each other, if the alternate exterior angles are equal?

Proposition XV

80. Draw a straight line; also two other lines each parallel to the given line. In what direction do these two lines extend with reference to each other?

Theorem. Straight lines which are parallel to the same straight line are parallel to each other.

Proof. Draw any transversal, as KL, cutting the lines AB, CD,

Ex. 32. The straight lines AB, CD, and EF are cut in G, H, and J respectively, by KL; angle KGB = 37°; angle _KHC = 149°; angle FJL = 143°. Are the lines AB and CD parallel? AB and EF? CD and EF? Ex. 33. Can two intersecting straight lines both be parallel to the same straight line?

Ex. 34. How many degrees are there in the angle formed by the bisectors of two complementary adjacent angles ?

Ex. 35. If the line BD bisects the angle ABC, and EF is drawn through B perpendicular to BD, how do the angles CBE and ABF compare in size?

Ex. 36. If a straight line is perpendicular to the bisector of an angle at the vertex, how does it divide the supplementary adjacent angle formed by producing one side of the given angle through the vertex ?

Proposition XVI

81. 1. Construct two angles whose corresponding sides are parallel. How do the angles compare in size, if both corresponding pairs of sides extend in the same direction from their vertices? If both pairs extend in opposite directions from their vertices?

2. Discover whether it is possible for the angles to have their sides parallel and yet not be equal.

Theorem. Angles whose corresponding sides are parallel are either equal or supplementary.

Data: AB parallel to DE, and BC parallel to HF, forming the angles r, s, t, s', and t'.

r

บ

B

G

Proof. 1. Produce BC and ED, if necessary, to intersect as

Hence, § 32, Zr and Zt are supplementary;

also, since, § 59, ≤t=≤t', ≤r and Zt' are supplementary. Therefore, etc.

Q.E.D.

82. Scholium. The angles are equal, if both corresponding pairs of sides extend in the same or in opposite directions from their vertices; they are supplementary, if one pair extends in the same and the other in opposite directions.

Ex. 37. If two straight lines are perpendicular each to one of two parallel straight lines, in what direction do they extend with reference to each other?

Ex. 38. How do lines bisecting any two corresponding angles, formed by parallel lines, cut by a transversal, lie with reference to each other?