362

o

oblate/prolate A curved surface similar to a sphere

but lengthened or shortened in one direction is called a

spheroid. If the length of the diameter from pole to

pole is greater than the length of the diameter connect-

ing two opposite points on the equator, then the

spheroid is said to be oblate. If, on the other hand, the

polar diameter is less than the equatorial diameter, then

the spheroid is called prolate. The Earth, for example,

is not a perfect sphere but is an oblate spheroid.

See also E

ARTH

;

ELLIPSOID

;

SPHERE

.

oblique This term is used in a number of geometric

settings to mean either “not at right angles” or “does

not contain a right angle.” For example, two intersect-

ing lines drawn in the plane are oblique if they meet at

an

ANGLE

different from 90°, or three lines meeting at a

point in three-dimensional space are oblique if they are

not mutually perpendicular. A single line drawn in the

plane is called an oblique line if it is neither horizontal

nor vertical, and an oblique coordinate system has axes

that are not at right angles.

Any angle that is not a multiple of 90°is called an

oblique angle. An oblique triangle is one that does not

contain a right angle.

An oblique cone is a cone with its vertex not

directly above the center of its base (and so the line

connecting the vertex to the center of the base is not at

right angles to the base). An oblique prism has lateral

edges that are not perpendicular to the base.

See also C

ARTESIAN COORDINATES

;

RIGHT ANGLE

;

SKEW LINES

.

obtuse angle An

ANGLE

between 90°and 180°is

called an obtuse angle. A

TRIANGLE

in which one of its

angles is obtuse is called an obtuse triangle. According

to the

LAW OF COSINES

, a triangle with side-lengths a, b,

and c, and corresponding angles A, B, C opposite those

sides, satisfies:

For the angle Cto be obtuse, it must be the case that

cosC< 0, that is, a2+ b2< c2. Thus a triangle a, b, c is

obtuse if one of the following inequalities holds: a2+ b2

< c2, c2+ a2< b2or b2+ c2< a2.

An angle between 180°and 360°is called a reflex

angle. An angle of 180°degrees is a straight angle, and

one of 360°is called a full turn or a

PERIGON

.

See also

ACUTE ANGLE

; P

YTHAGORAS

’

S THEOREM

;

TRIANGLE

.

obverse Changing the predicate Bof a statement of

the form “all Aare B” from positive to negative, or

vice verse, and negating the statement as a whole pro-

duces the obverse of the statement: “no Ais not B.”

For example, the obverse of the statement “all men are

mortal” is “no man is immortal.” Euler diagrams show

that any statement of the this type is logically equiva-

lent to its obverse.

See also

ARGUMENT

.