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c. To prove that a line is the locus of points under certain

conditions, prove:
1. Any point fulfilling the conditions is on the line.
2. Any point on the line fulfills the conditions.

Or, one of these, and the opposite of the other. Polygons, regular: 231. a. All sides are equal and all angles equal. b. It is formed by chords or tangents at the points of equal

division of a circle. c. To inscribe a regular polygon in a circle, place its central

angle at the center of the circle. Polygons, similar: 298. a. Their corresponding angles are equal and their correspond

ing sides are in proportion. b. They are composed of the same number of triangles, similar

each to each and similarly placed. Straight lines determined: 28. a. Only one straight line can be drawn through two given

points. b. Through a given point, only one straight line can be drawn,

making a given angle with a given straight line, and

having this angle on a given side of each line. c. Through a given point, only one straight line can be drawn

perpendicular to a given straight line. d. Through a given point, only one straight line can be drawn

parallel to a given straight line. Straight lines equal: 85. a. The corresponding parts of congruent triangles are equal. b. If two angles of a triangle are equal, the opposite sides are

equal.
118. C. Opposite sides of a parallelogram.

d. Diagonals of a parallelogram bisect each other.
e. Diagonals of a rectangle are equal.
f. Parallel lines cutting off equal parts on one transversal cut

off equal parts on any other transversal. 186. g. Equal arcs are subtended by equal chords.

h. Chords equally distant from the center are equal.
i. Two tangents from an outside point are equal.
j. A radius perpendicular to a chord bisects the chord.

Straight lines parallel: 105. a. They lie in the same plane and do not meet no matter how

far extended. b. Alternate-interior angles equal. C. Corresponding angles equal. d. Each perpendicular to the same line. e. Each parallel to the same line. f. Two interior angles on the same side of the transversal

supplementary. g. Opposite sides of a parallelogram. h. The line joining the middle points of the non-parallel sides

of a trapezoid or of a triangle is parallel to the base. 105. A quadrilateral is a parallelogram if:

1. The opposite sides parallel.
%. The opposite sides equal.
3. Two sides equal and parallel.

4. The diagonals bisect each other. 255. i. A line dividing two sides of a triangle proportionally is par

allel to the third side.

Straight lines perpendicular:
128. a. They form equal adjacent angles.

b. They form a right angle.
c. A line perpendicular to one of two parallel lines is perpen-

dicular to the other.
d. Two points, each equidistant from the ends of a straight line,

determine the perpendicular bisector of that line. e. The diagonals of a rhombus (or a square) bisect each other at

right angles. 145. f. The shortest line from a point to a straight line is the per

pendicular. 175. g. A tangent to a circle is perpendicular to the radius drawn

to the point of tangency. 218. h. An angle inscribed in a semicircle is a right angle.

Straight lines in proportion: 254. a. Parallel lines cut off proportional parts on two transversals. b. The bisector of an angle of a triangle divides the opposite

side into parts proportional to the other two sides.

279. c. The corresponding sides of similar triangles are in proportion. d. The corresponding parts of two chords intersecting internally

or externally are in inverse proportion.

Straight lines in one straight line: 28. a. If the sum of two adjacent angles equals two right angles,

their exterior sides lie in one straight line.

Straight lines tangent to a circle:
175. a. The line touches the circle at only one point.

b. The line is perpendicular to a radius at its end.

Straight lines unequal:
144. a. The whole is greater than any one of its parts.

b. A straight line is the shortest line between two points.
c. If two angles of a triangle are unequal, the opposite sides are

unequal and the greater side is opposite the greater angle. d. If two lines are drawn from a point in a perpendicular, cut

ting off unequal distances from the foot of the perpen

dicular, the more remote is the greater. e. The perpendicular is the shortest line from a point to a line. f. The sum of two lines, drawn from a point to the ends of a

line, is greater than the sum of two other lines similarly

drawn but enveloped by them. g. If two triangles have two sides of the one equal respectively

to two sides of the other but the included angle of the first greater than the included angle of the second, the third

side of the first is greater than the third side of the second. 195. h. Unequal arcs are subtended by unequal chords.

i. The chord nearer the center is the greater.

Triangles congruent: 67. a. If two triangles have two sides and the included angle of the

one equal respectively to two sides and the included angle

of the other, the triangles are congruent. b. If two triangles have two angles and a side of the one equal

respectively to two angles and the corresponding side of the other, the triangles are congruent.

c. If two triangles have three sides of the one equal respectively

to three sides of the other, the triangles are congruent. d. If two right triangles have a side and other part, except

the right angle, of the one equal respectively to a corresponding side and other part, except the right angle, of the other, the triangles are congruent.

Triangles similar: 267. a. Two angles of the one equal respectively to two angles of

the other. b. Two pairs of sides in proportion and the included angles

equal. c. Three pairs of sides in proportion. d. In right triangles, an acute angle of the one equals an acute

angle of the other, or if two sides of the one are in proportion with two corresponding sides of the other.

DEFINITIONS

A

Acuto angle. An angle less than a right angle.

Adjacent angles. Two angles that have a common vertex and a common side between them.

Alternate-interior angles. When two lines in the same plane are crossed by a transversal, two interior angles on opposite sides of the transversal, but not adjacent.

Alternation. See Proportion.
Altitude. A perpendicular from a vertex to the opposite side.

Angle. Two lines that meet form an angle. An angle represents the amount of revolution of one side about the vertex.

Antecedent. See Proportion.

Apothom. The perpendicular from the center to the side of a regular polygon.

Arc. A part of a circumference.
Area. The ratio of a surface to a unit of surface.
Arm. A side adjacent to the right angle in a right triangle.
Axiom. A statement assumed to be true without proof.

B

Base. That side of a figure upon which it apparently rests.
Bisect. To divide into two equal parts.

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Central angle. In a circle, an angle between two radii; in a regular polygon, an angle between two successive radii.

Chord. A straight line joining two points in a circumference.

Circle. A figure fromed by a curved line, called a circumference, all points of which are equally distant from a point within called the center.

Circumference. See Circle.

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