Geometry

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Geometry

Proof

Constructions

Points, lines, and planes

Angles

Parallel lines and planes

Perpendicular lines and planes

Geometry, the study of points, lines, and other figures in space, is a very old branch of mathematics. Its ideas were undoubtedly used, intuitively if not formally, from earliest times. Walking along a straight line toward a particular destination is the shortest way

to get there; lining an arrow up with the target is the way to hit it; sitting in a circle around a fire is the most equitable way to share the warmth. Early humans need not have been students of formal geometry to know and to use these ideas that all relate to geometry.

As early as 2,600 years ago the Greeks had not only discovered a large number of geometric properties, they had begun to see them as abstract ideas to be studied in their own right. By the third century BC, they had created a formal system of geometry. Their system began with the simplest ideas and, with these ideas as a foundation, went well beyond much of what is taught in schools today.

Proof

Typically one learns arithmetic and algebra by experiment or by being told how to do it. Geometry, however, is taught logically. Its ideas are established by means of proof. One starts with definitions, postulates, and primitive terms; then proves his or her way through the course.

The reason for this procedure goes back to the forenamed Greeks, and in particular to Greek mathematician Euclid of Alexandra (c.325–c.265 BC). Twenty-three hundred years ago he wrote a book called the Elements. This book contains no exercises, no experiments, no applications, no questions—just proofs, the proof of one proposition after another.

For centuries the Elements was the basic text in geometry. British mathematician and ancient Greek historian Sir Thomas Little Heath (1861–1940), in his 1925 translation of the Elements, quotes Indian-British mathematician Augustus De Morgan (1806-1871): “There never has been. . .a system of geometry worthy of the name, which has any material departures. . .from the plan laid down by Euclid.” Nowadays the Elements word ’behold’ has been replaced with texts that have exercises, problems, and applications, but the emphasis on proof remains. Even the most obvious fact, such as the fact that the opposite sides of a parallelogram are equal, is supposed to go unnoticed, or at least unused, until it has been proved. Whether or not this makes sense, the reader will have to decide for himself or herself, but sensible or not, proof is and will probably continue to be a dominant component of a course in geometry.

Proofs can vary in formality. They can be as formal as the two-column proofs used in textbooks in which each statement is identified as an assumption, a definition, or the consequence of a previously proved property. They can be informal with much left for the reader to fill in. They can be almost devoid of explanation, as in the ingenious proof of the Pythagorean theorem given by Hindu mathematician and astronomer Bhaskara Acharya (1114–1185) in the twelfth century. His proof consisted of a single word ’behold’ and a drawing.

Constructions

Another lasting influence of Euclid’s Elements is the emphasis which is placed on constructions. Three of the five postulates on which Euclid based his geometry describe simple drawings and the conditions under which they can be made. One such drawing (construction) is the circle. It can be drawn if one knows where its center and one point on it are. Another construction is drawing a line segment between two given points. A third is extending a given line segment. These are the so-called ruler-andcompass constructions upon which Euclidean geometry is based.

Modern courses in geometry are frequently based on other postulates. Some, for example, permit one to use a protractor to draw and measure angles; some allow the use of a scale to measure distances. Even so, the traditional limitations that the Euclidean postulates placed on constructions are often observed. Protractors, scales, and other drawing tools, which would be easier and more accurate to use, are forbidden. Constructions become puzzles, intriguing but separate from the logical structure of the course, and not overly practical.

Points, lines, and planes

Points, lines, and planes are primitive terms; no attempt is made to define them. They do have properties, however, which can be explicitly described. Among the most important of these properties are the following.

Two distinct points determine exactly one line. That line is the shortest path between the two points. Bricklayers use these properties when they stretch a string from corner to corner to guide them in laying bricks.

Two points also determine a ray, a segment, and a distance, symbolized for points A and B by AB (or BA when B is the endpoint), AB, and AB respectively. (Some authors use AB to symbolize all of these, leaving it to the reader to know which is meant.) Three non-collinear points determine one and only one plane.

The photographer’s tripod exploits this to hold the camera steady; while the chair on an uneven floor rocks back and forth between two different planes determined by two different combinations of the four legs.

If two points of a line lie in a plane, the entire line lies in the plane. It is this property that makes the plane flat. Two distinct lines intersect in at most one point; two distinct planes intersect in at most one line. If two coplanar lines do not intersect, they are parallel. Two lines that are not coplanar cannot intersect and are called skew lines. Two planes which do not intersect are parallel.

A line which does not lie in a plane either intersects that plane in a single point, or is parallel to the plane.

Angles

An angle in geometry is the union of two rays with a common endpoint. The common endpoint is called the vertex and the rays are called the sides. Angle ABC is the union of BA and BC. When there is no danger of confusion, an angle can be named by its vertex alone. It is also handy from time to time to name an angle with a letter or number written in the interior of the angle near the vertex. Thus angles ABC, B, and x are all the same angle.

When the two sides of an angle form a line, the angle is called a straight angle. Straight angles have a measure of 180°. Angles that are not straight angles have a measure between 180° and0°. The reflex angles, whose measures exceed 180°, encountered in other branches of mathematics are not ordinarily used in geometry. If the sum of the measures of two angles is 180°, the angles are said to be supplementary. Right angles have a measure of 90°. Lines that form right angles are also said to be perpendicular. If the sum of the measures of two angles is 90°, the angles are called complementary. Angles that are smaller than a right angle are called acute. Those angles that are larger than a right angle but smaller than a straight angle are called obtuse. When two lines intersect, they form two pairs of opposite or vertical angles. Vertical angles are equal.

A ray that divides an angle into two equal angles is called an angle bisector. Points on an angle bisector are equidistant from the sides of the angle.

Parallel lines and planes

Given a line and a point not on the line, there is exactly one line through the point parallel to the line.

Two coplanar (lines on the same plane) lines l1 and l2 in Figure 1, cut by a transversal t, are parallel if and only if:

1) Alternate interior angles (e.g., d and e) are equal.

2) Corresponding angles (e.g., b and f) are equal.

3) Interior angles on the same side of the transversal are supplementary (see Figure 1).

These principles are used in a variety of ways. A draftsperson uses 2) to rule a set of parallel lines. Number 1) is used to show that the sum of the angles of a triangle is equal to a straight angle.

If a set of parallel lines cuts off equal segments on one transversal, it cuts off equal segments on any other transversal (see Figure 2) a draftsperson finds this idea useful when he or she needs to subdivide a segment into parts that are not readily measured, such as thirds. If transversal AC in Figure 2 is slanted so that AC is three units, then the parallel lines through the unit points will divide AB into thirds as well.

If a set of parallel planes is cut by a plane, the lines of intersection are parallel. This property and its converse are used when one builds a bookcase. The set of shelves are, one hopes, parallel, and they are supported by parallel grooves routed into the sides.

Perpendicular lines and planes

If A is a given point and CD a given line, then there is exactly one line running through A that is perpendicular to CD. If B is the point on line CD that also resides on the line running perpendicular to

CD, then that line, AB, is the shortest distance from point A to line CD.

In a plane, if CD is a line and B a point on CD, then there is exactly one line through B perpendicular to CD. If B happens to be the midpoint of CD, then AB is called the perpendicular bisector of CD. Every point on AB is equidistant from C and D.

If a line QP is perpendicular to a plane at a point P, then it is perpendicular to every line in the plane that passes through P. Carpenters use this property when they make sure that a doorframe is perpendicular to the floor. Otherwise the door will rub on the floor, as someone who lives in an old house is likely to know.

A line will be perpendicular to a plane if it is perpendicular to two lines in the plane. The carpenter, in setting up the door frame, need not check every line with his or her square; two lines will do.

If perpendiculars are not confined to a single plane, there will be an infinitude of lines through B perpendicular to CD, all lying in the plane that is perpendicular to CD. If B is a midpoint, this plane will be the perpendicular-bisector plane of CD, and every point on this plane will be equidistant from C and D.

Two planes are perpendicular if one of the planes contains a line that is perpendicular to the other plane. The panels of folding screens, for example, stay perpendicular to the floor because the hinge lines are perpendicular to the floor.

Triangles

Triangles are plane figures determined by three non-collinear points called vertices. They are made up of the segments, called sides, which join them. Although the sides are segments rather than rays, each pair of them makes up one of the triangle’s angles.

Triangles may be classified by the size of their angles or by the lengths of their sides. Triangles whose angles are all less than right angles are called acute. Those with one right angle are right triangles. Those with one angle larger than a right angle are obtuse. (In a right triangle, the side opposite the right angle is called the hypotenuse and the other two sides legs.) Triangles with no equal sides are scalene triangles. Those with two equal sides are isosceles. Those with three equal sides are equilateral. There is no direct connection between the size of the angles of a triangle and the lengths of its sides. The longest side, however, will be opposite the largest angle; and the shortest side, opposite the smallest angle. Equal sides will be opposite equal angles.

In comparing triangles it is useful to set up a correspondence between them and to name corresponding vertices in the same order. If CXY and PST are two such triangles, then angles C and P correspond; sides CY and PT correspond; and so on.

Two triangles are congruent when their six corresponding parts are equal. Congruent triangles have the same size and shape, although one may be the mirror image of the other. Triangles ABC and FDE are congruent provided that the sides and angles which appear to be equal are in fact equal.

One can show that two triangles are congruent without establishing the equality of all six parts. Two triangles will be congruent whenever:

1) Two sides and the included angle of one are equal to two sides and the included angle of the other (SAS congruence).

2) Two angles and the included side of one are equal to two angles and the included side of the other (ASA congruence).

3) Three sides of one are equal to three sides of the other (SSS congruence).

Triangle congruence applies not only to two different triangles. It also applies to one triangle at two different times or to one triangle looked at in two different ways. For example, when the girders of a bridge are strengthened with triangular braces, each triangle stays congruent to itself over a period of time, and does so by virtue of SSS congruence.

Two triangles can also be similar. Similar triangles have the same shape, but not necessarily the same size. They are alike in the way that a snapshot and an enlargement of it are alike. When two triangles are similar, corresponding angles are equal and corresponding sides are proportional.

One can show that two triangles are similar without showing that all the angles are equal and all the sides proportional. Two triangles will be similar when:

1) Two sides of one triangle are proportional to two sides of another triangle and the included angles are equal (SAS similarity).

2) Two angles of one triangle are equal to two angles of another triangle (AA similarity).

3) Three sides of one triangle are proportional to three sides of another triangle (SSS similarity).

The properties of similar triangles are widely used. Artists, for example, use them in making smaller or larger versions of a picture. Map makers use them in drawing maps; and users, in reading them.

Figure 3 shows a right triangle in which an altitude BD has been drawn to the hypotenuse AC by AA similarity, the triangles ABC, ADB, and BDC are similar to one another.

By virtue of these similarities, one can write AC/BC = BC/DC and AC/AB = AB/AD. Then, using AD + DC = AC and a little algebra, one ends up with (AB)2 + (BC)2 = (AC)2, or the Pythagorean theorem: “In a right triangle the sum of the squares on the legs is equal to the square on the hypotenuse.” This neat proof was discovered by Bhaskara, as mentioned earlier.

The altitude BD in Figure 3 is also the mean proportional between AD and DC. That is, AD/BD = BD/DC.

In triangle ABC, if DE is a line drawn parallel to AC, it creates a triangle similar to ABC. It therefore divides AB and BC proportionally. Conversely, a line that divides two sides of a triangle proportionally is parallel to the third side. A special case of this is a

segment joining the midpoints of two sides of a triangle. It is parallel to the third side and half its length.

Each triangle has four sets of lines associated with it: medians, altitudes, angle bisectors, and perpendicular bisectors of the sides. In each set, the three lines are, remarkably, concurrent; that is, they all pass through a single point. In the case of the medians, which are lines from a vertex to the midpoint of the opposite side, the point of concurrency is the centroid, the center of gravity. The angle bisectors are concurrent at the incenter, the center of a circle tangent to the three sides. The perpendicular bisectors of the sides are concurrent at the circumcenter, the center of a circle passing through all three vertices. The altitudes, which are lines from a vertex perpendicular to the opposite side, are concurrent at the orthocenter.

Quadrilaterals

Quadrilaterals are four-sided plane figures. Various special quadrilaterals are defined in various ways. The following are typical:

Trapezoid: A quadrilateral with one pair of parallel sides.

Parallelogram: A quadrilateral with two pairs of parallel sides.

Rhombus: A parallelogram with four equal sides.

Kite: A quadrilateral with two pairs of equal adjacent sides.

Rectangle: A parallelogram with four right angles.

Square: A rectangle with four equal sides. It is a special kind of rhombus.

Cyclic quadrilateral: A quadrilateral whose four vertices lie on one circle.

In any quadrilateral, the sum of the angles is 360°. In a cyclic quadrilateral opposite angles are supplementary.

The diagonals of any parallelograms bisect each other. The diagonals of kites and any rhombus are perpendicular to each other.

Opposite sides of parallelograms are equal.

Circles

A circle is a set of points in a plane which are a fixed distance from a point called the center, C (see Figure 4) a chord is a segment, DE, joining two points on the circle; a radius is segment, CA, joining the center and a point on the circle; a diameter is a chord, DB, through the center. A tangent, DF, is a line touching the circle in a single point. The words radius and diameter can also refer to the lengths of these segments.

An arc is the portion of the circle between two points on the circle, including the points. A major arc is the longer of the two arcs so determined; a minor arc, the shorter. When an arc is named it is usually the minor arc that is meant, but when there is danger of confusion, a third letter can be used, e.g., arc DAB.

All circles are similar, and because of this fact the ratio of the circumference to the diameter is the same for all circles. This ratio, called pi or p, was shown to be smaller than 22/7 and larger than 223/71 by ancient Greek mathematician Archimedes (287-212 BC) about 240 BC.

An arc can be measured by its length or by the central angle that it subtends. A central angle is one whose vertex is the center of the circle.

An inscribed angle is one whose vertex is on the circle and whose sides are two chords or one chord and a tangent. Angles EDB and BDF are inscribed angles. The measure of an inscribed angle is one half that of its intercepted arc. Any inscribed angle that intercepts a semicircle is a right angle; so is the angle between a tangent and a radius drawn to the point of tangency.

If X is a point inside a circle and AB and CD any two chords through X, then X divides the chords into segments whose products are equal. That is, (AX)(XB) = (CX)(XD).

Area

Areas are expressed in terms of squares such as square inches, meters, miles, etc. Formulas for the areas of various plane figures are based upon the formula for the area of a rectangle, lw, where l is the length and w the width. The area of a parallelogram is bh, where b is the base and h the height (altitude), measured along a line perpendicular to the base. The area of a triangle is half that of a parallelogram with the same base and height, bh/2. When the triangle is equilateral, h = \/3 b/2 so the area is \/3b 2 /4. A trapezoid whose parallel sides are b1 and b2 and whose height is h can be divided into two triangles with those bases and altitudes. Its area is (b1 + b2)h/2.

The area of a quadrilateral with sides a, b, c, and d depends not only on the lengths of the sides but on the size of its angles. When the quadrilateral is cyclic (all four endpoints are on in a circle), its area is given by a remarkable formula discovered by Hindu mathematician and astronomer Brahmagupta (598-668) in the seventh century:

where s is the semi-perimeter (a + b + c + d)/2. This formula includes Heron’s formula, discovered in the first century, for the area of a triangle,

as a special case. By letting d = 0, the quadrilateral becomes a triangle, which is always cyclic.

The area of a circle can be approximated by the area of an inscribed regular polygon. As the number of sides of this polygon increases without limit, its area approaches cr/2, where c is the circumference of the circle and r the radius. Since c = 2πr, the area of the circle is π².

The surface area of a sphere of radius r is four times the area of a circle of the same radius, 4πr².

The lateral surface of a right circular cone can be unrolled to form a sector of a circle (see Figure 5) its

area is πrs, where r the radius of its base and s is the slant height of the cone.

Volumes

The volumes of geometric solids are expressed in terms of cubes that are one unit on a side, such as cubic centimeters or cubic yards. The volume of a rectangular solid (box) whose length, width, and height are l, w, and h is lwh. The volume of a prism or a cylinder is Bh, where B is the area of its base and h its height measured along a line perpendicular to the base. The volume of a pyramid or cone is one-third that of a prism or cylinder with the same base and height, that is Bh/3. The volume of a sphere of radius r is 4πr³/3.

It is interesting to note that the volumes of a cylinder, a hemisphere, and a cone having the same base and height are in the simple ratio 3:2:1.

Other geometries

The foregoing is a summary of Euclidean geometry, based on Euclid’s postulates. Euclid’s fifth postulate is equivalent to assuming that through a given point not on a given line, there is exactly one line parallel to the given line. When one assumes that there is no such line, elliptical geometry emerges. When one assumes that there is more than one such line, the result is hyperbolic geometry. These geometries are called non-Euclidean. Non-Euclidean geometries are as correct and consistent as Euclidean, but describe special spaces. Geometry can also be extended to more than three dimensions. Other special geometries include projective geometry, affine geometry, and topology.