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.

D

A

4.7

B

H

E

F

D'

E'

Let&B and E (Fig. 1) have their sides BA and E D, and BC and E F respectively, parallel and lying in the same direction from their vertices.

We are to prove the LB=LE.

Produce (if necessary) two sides which are not until they

Let B' and E' (Fig. 2) have B'A' and E' D', and B' C' and E' F' respectively, parallel and lying in opposite directions from their vertices.

We are to prove the LB'LE'.

Produce (if necessary) two sides which are not until they intersect, as at H'.

PROPOSITION XX. THEOREM.

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.

Let A B C and D E F be two angles having BC and E D 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 ▲ ABC and DEF supplements of each other.

Produce (if necessary) two sides which are not until they intersect as at H.

But BHD and ▲ BH E are supplements of each other, § 34

(being sup.-adj. 4).

ZABC and DEF, the equals of BHD and

LBH 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 Sealene 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.

HYPOTENUSE.

RIGHT.

OBTUSE.

A

ACUTE.

87. DEF. A Right triangle is one which has one of the angles a right angle.

88. DEF. The side opposite the right angle is called the Hypotenuse.

89. DEF. An Obtuse triangle is one which has one of the angles an obtuse angle.

90. DEF. An Acute triangle is one which has all the angles acute.

C

EQUIANGULAR.

B

D

91. DEF. An Equiangular triangle is one which has all the angles equal.

92. DEF. In any triangle, the angle opposite the base is called the Vertical angle, and its vertex is called the Vertex of the triangle.

93. DEF. The Altitude of a triangle is the perpendicular distance from the vertex to the base, or the base produced.

94. DEF. The Exterior angle of a triangle is the angle included between a side and an adjacent side produced, as ≤ CBD. 95. DEF. The two angles of a triangle which are opposite the exterior angle, are called the two opposite interior angles, as A and C.

B

C

96. Any side of a triangle is less than the sum of the other two sides.

Since a straight line is the shortest distance between two points,

ACAB+ B C.

97. Any side of a triangle is greater than the difference of the other two sides.

In the inequality ACAB+ BC,

take away A B from each side of the inequality.

Ex. 1. Show that the sum of the distances of any point in a triangle from the vertices of three angles of the triangle is greater than half the sum of the sides of the triangle.

2. Show that the locus of all the points at a given distance from a given straight line AB consists of two parallel lines, drawn on opposite sides of A B, and at the given distance from it.

3. Show that the two equal straight lines drawn from a point to a straight line make equal acute angles with that line.

4. Show that, if two angles have their sides perpendicular, each to each, they are either equal or supplementary.