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 ZABE = ▲ BEF, prove that the bisectors of CBE and

DEB are parallel.

Ex. 52. If in the annexed

A

B

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

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. VI / AHC=60°, and ▲ HIE=60°, is CD parallel to EF?

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

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

Hyp.

B

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

To prove Proof.

CD || EF.

ZDHI is sup. to ▲ HIF,

LEIH is sup. to Z HIF,

(Hyp.)

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

(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 ▲ EIH, 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,

EF AB.

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

(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 AHD = 40°, how many degrees are in HIF, HIE, and EIB?

Ex. 60. If in the diagram for Prop. VIII the transversal be produced through B and E, the figure will contain three angles equal to ZABE. Find these angles.

Ex. 61. If the opposite sides of a quadrilateral are parallel, they must be equal.

PROPOSITION IX. THEOREM

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

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

Ex. 62. In the diagram for Prop. IX, if ZAHD = 50°, how many degrees are in & EIB, CHI, AIF, and EIA?

Ex. 63. In the same diagram, prove

A

C

-F

-H

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

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.

[Converse of Prop. VII.]

E

-D

B

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

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

.. ZDHI+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. 4.

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