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PROPOSITION XVII. THEOREM. 75. Two straight lines which are parallel to a third straight line are parallel to each other.

Let A B and C D be parallel to E F.
We are to prove AB || to C D.

Draw HK I to E F.
Since C D and E F are II, H K is I to CD, § 67
(if a straight line be I to one of two lls, it is I to the other also).
Since A B and E F are II, H K is also I to AB, $ 67

..Z HOB= 2 HP D,

(each being a rt. Z).
.:. A B is II to CD,

§ 72 (when two straight lines are cut by a third straight line, if the ext. -int. $ be equal, the two lines are II).

Q. E. D.

PROPOSITION XVIII. THEOREM. : 76. Two parallel lines are everywhere equally distant from each other.

E M H ,

A

B

AK
Let A B and C D be two parallel lines, and from any

two points in A B, as E and H, let EF and HK
be drawn perpendicular to A B.
We are to prove E F = H K.
Now E F and H K are I to CD,

$ 67
(a line I to one of two lls is I to the other also).
Let M be the middle point of E H.

Draw MPI to A B. On MP as an axis, fold over the portion of the figure on the right of M P until it comes into the plane of the figure on the left.

M B will fall on MA,
(for 2 PM H= 2 PM E, each being a rt. 2);

the point H will fall on E,
(for MH= ME, by hyp.);

H K will fall on EF,
(for Z M H K = Z M E F, each being a rt. 2);
and the point K will fall on E F, or E F produced.

Also, P D will fall on PC,
(2 M PK = Z MPF, each being a rt. 2);

and the point K will fall on P C.
Since the point K falls in both the lines E F and PC,
it must fall at their point of intersection F.
. H K = E F,

3.18 (their extremities being the same points).

Q. E. D.

PROPOSITION XIX. THEOREM. 77. Two angles whose sides are parallel, two and two, and lie in the same direction, or opposite directions, from their vertices, are equal.

A D

Fig. 1.

/ H EL

DI

Fig. 2.
Let s B and E (Fig. 1) have their sides BA and E D,

and BC and EF respectively, parallel and lying
in the same direction from their vertices.
We are to prove the Z B= 2 E.

Produce (if necessary) two sides which are not ll until they intersect, as at H; then ZB=DHC,

§ 70 (being ext.-int. 6), and ZE= DHC,

§ 70 ..LB= 2 E.

Ax. 1

Let Is B' and E' (Fig. 2) have B' A' and E' D', and B'C'

and E' F' respectively, parallel and lying in oppo-
site directions from their vertices.
We are to prove the B' = _ E'.

Produce (if necessary) two sides which are not || until they
intersect, as at H.
Then
Z B = 2 E' H' C",

§ 70 (being ext.-int. ), and

ZE = E' H'C',

(being alt.-int. A ) ;
.. 2 B'= E',

Ax. 1.
Q. E. D.

§ 68

[blocks in formation]

78. If two angles have two sides parallel and lying in the same direction from their vertices, while the other two sides are parallel and lie in opposite directions, then the two angles are supplements of each other.

[blocks in formation]

Let A B C and D E F be two angles having B C and ED

parallel and lying in the same direction from their vertices, while E F and B A are parallel and lie in opposite directions.

We are to prove Z A B C and Z D E F supplements of each other.

Produce (if necessary) two sides which are not || until they intersect as at II. 2 A B C = B IH D,

$ 70 (being ext.-int. £). Z DEF= 2 B II E,

§ 68 (being alt.-int. 4). But Z BI D and Z B II E are supplements of each other, § 34

(being sup.-adj. ). .. LA B C and Z D E F, the equals of Z BII D and ZBH E, are supplements of each other.

Q. E. D.

On TRIANGLES.

79. DEF. A Triangle is a plane figure bounded by three straight lines.

A triangle has six parts, three sides and three angles.

80. When the six parts of one triangle are equal to the six parts of another triangle, each to each, the triangles are said to be equal in all respects.

81. DEF. In two equal triangles, the equal angles are called Homologous angles, and the equal sides are called Homologous sides.

82. In equal triangles the equal sides are opposite the equal angles.

SCALENE.

ISOSCELES.

EQUILATERAL.

83. DEF. A Scalene triangle is one of which no two sides are equal.

84. DEF. An Isosceles triangle is one of which two sides are equal.

85. DEF. An Equilateral triangle is one of which the three sides are equal.

86. DEF. The Base of a triangle is the side on which the triangle is supposed to stand.

In an isosceles triangle, the side which is not one of the equal sides is considered the base.

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