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letting D fall upon B. DS, the prolongation of DF, will fall upon BO, the prolongation of BE (Axiom 9). DC will fall upon BA (Theo. I, Cor. 2), and the given right angles are shown to be equal (Axiom 10).

Sch. 1.-The truth of Theorem V may be inferred from Theorem I; but this proof is given as applying to any two right angles.

Sch. 2.-The sum of all the angles, which can be drawn in a plane about a given point, is represented numerically by 360°. Hence, one right angle is equal to 90°.

THEOREM VI.

If the adjacent angles, made by one straight line with two others, are together equal to two right angles, these two lines are in one straight line.

A

-B'

B

The line DC, meeting AC and BC, makes the angles ACD and DCB, whose sum, by hypothesis, is two right angles. Then, are AC and CB in one straight line. If CB is not the prolongation of AC, some other line, as CB', must be. If ACB' is a straight line, ACD + DCB′ = 2 L's. .. ACD +DCB' ACD + DCB.

=

Subtracting ACD (Axiom 3), and DCB' = DCB; that is, the whole is equal to one of its parts. This is un

true (Sec. III, Def. 2, Cor. 2), and the error arose from taking some other line than CB as the prolongation of AC. This error will not appear when we assume CB as the prolongation of AC; therefore, it is such prolongation.

THEOREM VII.

If a perpendicular and oblique lines are drawn from the same point to a given line, the perpendicular is shorter than any oblique line.

BH is a perpendicular and BC an oblique line from B to AO. Then, BH is shorter than BC. Continue BH till HB' shall equal BH; and

draw B'C. Rotate the upper

B

H

C

A

B

BH will at

part of the figure upon AO as an axis. length fall upon HB', as the angles at H are right angles. The point B will fall upon B', since HB' = BH. And the line BC will coincide with and be equal to B'C (Axiom 8). Now, the broken line BCB' is longer than the straight line BB', joining the same points (Axiom 8). Therefore the perpendicular BH, which is half of BHB', is shorter than BC, which is half of BCB' (Axiom 7); that is, a perpendicular is shorter than an oblique line, if both are drawn to a line from any point outside that line.

THEOREM VIII.

If two oblique lines drawn from a point meet the given line at equal distances from the foot of the perpendicular, they are equal.

E

B

F

The oblique lines BA and BC meet the line EF at the points A and C, so that AD CD.

Rotate the left side of the figure upon the perpendicu

lar BD as an axis. ED will fall upon DF (Theo. V). The point A will fall upon C. Therefore (Axiom 8), AB CB.

=

THEOREM IX.

Every point in a line which is perpendicular to another line at its center, is equally distant from the two ends of the line.

E

G

By construction, FE = GE, and BE is perpendicular to FG.

If by rotation about BE as an axis the point F be made to fall upon G, then

will FB = GB (Axiom 8); FC = GC, and FD = GD.

SECTION V.-TRIANGLES.

DEFINITIONS.

1. A plane figure is a portion of a plane bounded on all sides by lines.

2. A polygon is a plane figure bounded by straight lines.

3. A triangle is a polygon of three sides.

If one of its angles is a right angle, the triangle is called a right-angled triangle; and the side opposite the right angle is called the hypotenuse.

Thus the plane figure ABC is a right-angled triangle. AB is the hypotenuse. CA and CB are sometimes called the legs.

An isosceles triangle has two equal sides.

An equilateral triangle has three equal sides.

Thus, ABD is isosceles, and ABC is equilateral. An equilateral triangle is always isosceles.

A scalene triangle has no two sides equal.

Can one side of a triangle be equal to the sum of the other two? Can it be greater?

Thus, the triangle ABC is scalene.

4. Any side of a triangle may be considered as its base, or that upon which it is supposed to rest, and the opposite angle as the vertex; but in an isosceles triangle the base is the side that is not equal to any other side.

Thus, with reference to BC as the base, A is the vertex. When we regard BA as the base, C is the vertex; similarly with AC and B.

When two sides of a triangle

A

have been named in a demon- B
stration, the third is often called
the base.

The altitude of a triangle is

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the perpendicular let fall from the vertex on the base, or the base produced.

E. G.-3.

A

B

D

Thus, BD is the altitude of the triangle ABC.

5. A polygon is equiangular when all its angles are equal.

6. Two polygons are called mutually equilateral, or mutu

ally equiangular, if the sides or angles of the one are equal to the sides or angles of the other, each to each, taken in the same order.

7. When the sum of two angles is two right angles, or 180°, they are termed supplementary. When their sum is one right angle, or 90°, they are termed complementary.

Cor. 1.-The supplements of equal angles are equal?

Cor. 2.-The complements of equal angles are equal? 8. An exterior angle is formed by a side of a polygon, and an adjacent side produced. It is the supplement of the adjacent interior angle (Theo. 1).

THEOREM X.

The three interior angles of a triangle are together equal to two right angles.

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EBD +CBE+CBA=CAB+ ACB + CBA (Axiom 2).

The first member of this

equation is equal to two right angles (Theo. I, Cor. 1). Therefore the second member, which is the sum of the three interior angles, is equal to two right angles.

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