Circumscribe. A circle is circumscribed about a polygon when it passes through every vertex; the polygon is said to be inscribed in the circle.

A polygon is circumscribed about a circle when its sides are tangents to the circle; the circle is said to be inscribed in the polygon.

Coincide. All points in one figure fall upon corresponding points in a second figure.

Commensurable. Exactly divisible by a common unit of measure. Complement. An angle is the complement of a second angle when their sum equals a right angle.

Composition. See Proportion.

Congruent. Two figures that have their corresponding parts equal. They have the same form and size, and can be made to coincide. Consequent. See Proportion.

Constant. A quantity whose position, form, or size does not change during the discussion.

Converse theorem is formed when the condition and conclusion of a theorem change places.

When the condition and conclusion are made negative and change places, the converse of the opposite is formed.

Corresponding. Similarly placed.

Corollary. A dependent theorem.

D

Degree is of a circumference.

Determine. A line is determined in position when it is the only line that can be drawn under the given conditions.

Diameter. A chord through the center of a circle.

Distance between two points is represented by a straight line; between a point and a straight line, by a perpendicular; between a point and a circle is the external part of the straight line from the point to the center of the circle.

Division. A line can be divided into two parts internally or externally:

Internally when the point of division is in the line.

Externally when the point of division is in the line extended.

In either case the parts made by the division point are the distances from this point to the ends of the given line. In external division the parts are algebraic not geometric.

Division. See Proportion.

E

Equal figures have the same size.

In the case of straight lines, angles, and circles, they will coincide if superposed.

Equiangular triangle has all angles equal.

Equilateral triangle has all sides equal.

Exterior angle of a triangle is formed by the extension of one side. Extreme and mean ratio. A line is divided in extreme and mean ratio when one part of the line is a mean proportional between the whole line and the other part.

Extremes. See Proportion.

Figure. See Geometrical figure.

G

Geometrical figure. Any combination of points, lines, surfaces, and

solids.

Geometry is the science of space. It treats of the position, form, and size of geometrical figures.

H

Homologous. See Corresponding.

Hypotenuse.

The side opposite the right angle in a right triangle. Hypothesis. The given conditions.

I

Incommensurable. Not exactly divisible by a common unit of

measure.

Inscribe. See Circumscribe.

Inversion. See Proportion.

Isosceles triangle has two sides equal.

L

Limit. See Variable.

Line. The boundary of a surface. It has length but no width or thickness.

Locus of points under given conditions is the line or lines that contain all points under these conditions and no others.

Measurement of a quantity is the process of finding the numerical value or measure of that quantity.

A unit of measure can be anything but it must be of the same kind as the quantity to be measured. Thus, inch, etc., are units of measure for straight lines; square inch, etc., for surfaces; degree, etc., for revolutions.

The number of times a unit of measure is contained in a given quantity is the numerical measure of that quantity.

Median. A straight line from a vertex of a triangle to the middle of the opposite side.

Mid-line. A straight line joining the mid-points of the two nonparallel sides of a trapezoid.

Obtuse angle. An angle greater than a right angle but less than a straight angle.

Opposite theorem is formed when the condition and conclusion of a theorem are both made negative.

P

Parallel lines. Two straight lines, in the same plane, that never meet no matter how far extended.

Parallelogram. A quadrilateral with opposite sides parallel.
Perimeter. The sum of the sides of a closed figure.

Perpendicular. A straight line, meeting another straight line, making two adjacent angles equal. Each angle is called a right angle. The ratio between a circumference of a circle and its diameter. Its value is approximately 34.

Plane surface is such that a straight line joining any two points therein lies wholly in the surface.

Point is the boundary or end of a line. It has position but no length, width, or thickness.

Polygon. A portion of surface bounded by lines.
Postulate. A geometrical axiom of construction.
Problem. A statement of a figure to be constructed.

Projection of a point upon a line is the foot of the perpendicular from the point to the line. The projection of a line upon a line is the line through the feet of all the perpendiculars from the first line to the second line; that is, the locus of the projections of its points.

Proportion. An equality between two ratios. Thus

a

b

A proportion may be changed in form by algebraic operations:

=

C

ď

= ad.

6. Product of the means equals the product of the extremes, bc 7. If the product of two quantities equals the product of two other quantities, either pair may be made the means of a proportion of which the other pair are the extremes. (Converse of 6.)

8. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

The first and fourth terms of a proportion are called the extremes; the second and the third, the means. When the second and third terms are equal, either is called the mean proportional, and the last term is the third proportional.

The first and third terms are called antecedents; the second and fourth, consequents.

Proposition. A theorem or problem.

Q

Quadrilateral. A portion of surface bounded by four lines.

R

Radius. In a circle, a straight line from the center to the circumference. In a regular polygon, the straight line from the center to any vertex.

Ratio of two quantities is found by dividing their numerical measures. Rectangle. A parallelogram with all angles right angles.

Rectilinear. Made with straight lines.

Regular polygon. A polygon with all sides equal and all angles equal.

Revolution. The moving of a straight line in a plane, about its end as an axis, until it reaches its original position.

Rhombus. An equilateral parallelogram.

Right angle. Half a straight angle. See Perpendicular.
Right triangle. A triangle having one angle a right angle.

ន

Scalene triangle. A triangle having no sides equal.

Secant. A straight line cutting a circle in two points.

Sector. Part of a circle between two radii and their intercepted arc. Segment. Part of a circle between a chord and its arc, or between two parallel chords.

Series of equal ratios. See Proportion.

Similar polygons have corresponding angles equal and corresponding sides in proportion. (They have the same form.)

Solid.

thickness.

An enclosed portion of space. It has length, width, and

Square. An equilateral rectangle.

Straight angle. An angle with its sides in a straight line and on opposite sides of the vertex.

Straight line. Any part will coincide with any other equal part, if superposed.

Subtend. To stretch beneath.

Supplement. One angle is the supplement of a second angle, if their sum equals a straight angle.

Surface is the boundary of a solid. It has no thickness, only length and width.

Synthetic proof. A series of statements beginning with the given condition and ending with the desired conclusion. One statement leads to another, and every statement depends upon a previous definition, axiom, or theorem.