Ex. 50. If two sides of a triangle are produced their own lengths through the common vertex, a line joining their ends is parallel to the third side of the triangle.

Ex. 51. If in the diagram for Prop. V Z ABE = ▲ BEF, prove that

83. Two lines are parallel if a transversal to these lines makes the corresponding angles equal.

Hyp. CD and EF are intersected by AB in H and I respec

tively, and

To prove

ZAHD / HIF.

CD || EF.

HINT. Prove the equality of a pair of alternate interior angles.

84. COR. If a transversal is perpendicular to two lines, these lines are parallel.

Ex. 53. If in the diagram for Prop. VIZ AHC=60°, and ▲ HIE=60°, is CD parallel to EF?

Ex. 54. In the same diagram, if AHD is the supplement of EIH, CD is parallel to EF.

PROPOSITION VII. THEOREM

85. Two lines are parallel if a transversal to these lines makes the interior angles on the same side of the transversal supplementary.

Hyp. CD and EF are intersected by a transversal AB in II and I, respectively, and Z DHI + Z HIF = 2 rt. 4.

To prove
Proof.

CD EF

ZDHI is sup. to Z HIF,

ZEIH is sup. to Z HIF,

(Hyp.)

(if two adjacent angles have their ext. sides in a st. line, they are sup.).

:. / DHI = ▲ EIH,

(supplements of same Z are equal).

Hence

CDEF,

(alt. int. being equal).

Q.E.D.

86. REMARK. Lines are demonstrated to be parallel by proving that

(a) Two alternate interior angles are equal,

(b) Two corresponding angles are equal, or

(c) Two interior angles on the same side of a transversal are supplementary.

Ex. 55. Two lines are parallel if a transversal to these lines makes the exterior angles on the same side of the transversal supplementary.

Ex. 56. If in the diagram for Prop. VII 2 AHD = 2 HIF, and perpendiculars be erected upon CD and EF at H and I, respectively, the perpendiculars are parallel.

Ex. 57. In the same diagram, if ▲ AHD is the supplement of LEIH, CD is parallel to EF.

87. If two parallels are cut by a transversal, the alternate interior angles are equal.

Hyp. The parallel lines AC and DF are intersected by a transversal in B and E.

Proof. ABE and BEF are either equal or unequal. Suppose they are unequal, and let EF" be drawn so that BEF = LABE.

Then

But

EF" AB,

(two lines are || if a transversal makes the alt. int. ▲ equal).

EF AB.

(Hyp.) Therefore two intersecting lines EF and EF" would be parallel to AC, which is impossible.

(Ax. 11.)

Q.E.D.

88. COR. If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other also.

Ex. 58. If two parallels are cut by a transversal, the alternate exterior angles are equal.

Ex. 59. In the diagram for Prop. VII, if ZAHD = 40°, how many degrees are in & HIF, HIE, and EIB?

Ex. 60. If in the diagram for Prop. VIII the transversal through B and E, the figure will contain three angles equa Find these angles.

Ex. 61. If the opposite sides of a quadrilateral are parall be equal.

PROPOSITION IX. THEOREM

89. If two parallel lines are cut by a tr the corresponding angles are equal. [Converse of Prop. VI.]

Hyp. Two parallel lines CD and EF are inter AB in H and I, respectively.

Ex. 62. In the diagram for Prop. IX, if LAHD degrees are in & EIB, CHI, AIF, and EIA?

Ex. 63. In the same diagram, prove

A

Ex. 66. If three points A, B, and C be joined, and BC be produced to D,

LACD=LA + Z B.

HINT. - Draw EC BA (annexed

diagram).

B

E

Ex. 67. In the same diagram, if CE bisects ACD, then ZA=LB. Ex. 68. The bisectors of supplementary adjacent angles are perpendicular to each other.

PROPOSITION X. THEOREM

90. If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary.

Hyp. Two parallel lines CD and EF are intersected by AB in H and I, respectively.

To prove Proof.

But

ZDHI+Z HIF = 2 rt. .

ZDHI ZHIE,

=

(alt. int. of || lines).

ZHIE+ZHIF = 2 rt. 4,

(two adj. ▲ whose ext. sides are in a st. line are supplementary).

.. Z DHI + Z HIF = 2 rt. .

Q.E.D.

Ex. 69. In the diagram for Prop. VII, if CD is parallel to EF, prove that ZAHD+ 2 HIE = 2 rt. .

Ex. 70. If two parallel lines are cut by a transversal, the exterior angles on the same side of the transversal are supplementary.