Undefined Terms and Intuitive Concepts of Geometry

Undefined terms:In geometry, definitions are formed using known words or terms to describe a new word. There are three words in geometry that are not formally defined. These threeundefinedterms are point, line and plane.POINT(an undefined term)In geometry, a point has no dimension (actual size). Even though we represent a point with a dot, the point has no length, width, or thickness. Our dot can be very tiny or very large and it still represents a point. A point is usually named with a capital letter. In the coordinate plane, a point is named by an ordered pair, (x,y).

LINE(an undefined term)In geometry, a line has no thickness but its length extends in one dimension and goes on forever in both directions. Unless otherwise stated a line is drawn as a straight line with two arrowheads indicating that the line extends without end in both directions. A line is named by a single lowercase letter,, or by any two points on the line,.

PLANE(an undefined term)In geometry, a plane has no thickness but extends indefinitely in all directions. Planes are usually represented by a shape that looks like a tabletop or a parallelogram. Even though the diagram of a plane has edges, you must remember that the plane has no boundaries. A plane is named by a single letter (planem) or by three non-collinear points (plane ABC).

Intuitive Concepts:There are a few basic concepts in geometry that need to be understood, but are seldom used as reasons in a formal proof.Collinear Pointspoints that lie on the same line.

Coplanar pointspoints that lie in the same plane.

Opposite rays2 rays that lie on the same line, with a common endpoint and no other points in common. Opposite rays form a straight line and/or a straight angle (180:).

Parallel linestwo coplanar lines that do not intersect

Skew linestwo non-coplanar lines that do not intersect.

Consider the following theorems relating lines and planes. A diagram is supplied for each theorem that represents one possible depiction of the situation.If a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them.Through a given point there passes one and only one plane perpendicular to a given line.

Through a given point there passes one and only one line perpendicular to a given plane.

Two lines perpendicular to the same plane are coplanar.

Two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane.If a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane.

If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.

If a plane intersects two parallel planes, then the intersection is two parallel lines.

If two planes are perpendicular to the same line, they are parallel.

The angle where two planes meet is called adihedral angle. Woodworkers and construction workers deal with dihedral angles. For example, creating a rafter for a hip roof requires an understanding of dihedral angles.Euclidean Geometry(the high school geometry we all know and love) is the study of geometry based on definitions, undefined terms (point, line and plane) and the assumptions of the mathematician Euclid (330 B.C.)Euclid's textElementswas the first systematic discussion of geometry. While many of Euclid's findings had been previously stated by earlier Greek mathematicians, Euclid is credited with developing the first comprehensive deductive system. Euclid's approach to geometry consisted of proving all theorems from a finite number of postulates (axioms).Euclidean Geometry is the study offlat space. We can easily illustrate these geometrical concepts by drawing on a flat piece of paper or chalkboard. In flat space, we know such concepts as: the shortest distance between two points is one unique straight line.

the sum of the angles in any triangle equals 180 degrees.

the concept of perpendicular to a line can be illustrated as seen in the picture at the right.

In his text, Euclid stated his fifth postulate, the famousparallel postulate, in the following manner:If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

Today, we know theparallel postulateas simply stating:Through a point not on a line, there is no more than one line parallel to the line.

The concepts in Euclid's geometry remained unchallenged until the early 19th century. At that time, other forms of geometry started to emerge, called non-Euclidean geometries. It was no longer assumed that Euclid's geometry could be used to describe all physical space.

non-Euclidean geometries:are any forms of geometry that contain a postulate (axiom) which is equivalent to the negation of the Euclidean parallel postulate.Examples:1. Riemannian Geometry(also calledelliptic geometryorspherical geometry):A non-Euclidean geometry using as itsparallel postulateany statement equivalent to the following:If l is any line and P is any point not on l , then there are no lines through Pthat are parallel to l .

Riemannian Geometry is named for the German mathematician, Bernhard Riemann, who in 1889 rediscovered the work of Girolamo Saccheri (Italian) showing certain flaws in Euclidean Geometry.

Riemannian Geometry is the study ofcurved surfaces. Consider what would happen if instead of working on the Euclidean flat piece of paper, you work on a curved surface, such as a sphere. The study of Riemannian Geometry has a direct connection to our daily existence since we live on a curved surface called planet Earth.

What effect does working on a sphere, or a curved space, have on what we think of as geometrical truths? In curved space, the sum of the angles of any triangle is now always greater than 180. On a sphere, there are no straight lines. As soon as you start to draw a straight line, it curves on the sphere.

In curved space, the shortest distance between any two points (called ageodesic) is not unique. For example, there are many geodesics between the north and south poles of the Earth (lines of longitude) that are not parallel since they intersect at the poles.

In curved space, the concept of perpendicular to a line can be illustrated as seen in the picture at the right.

2.Hyperbolic Geometry(also calledsaddle geometryorLobachevskian geometry):A non-Euclidean geometry using as itsparallel postulateany statement equivalent to the following:Iflis any line andPis any point not onl, then there exists at least two lines throughPthat are parallel tol.

Lobachevskian Geometry is named for the Russian mathematician, Nicholas Lobachevsky, who, like Riemann, furthered the studies of non-Euclidean Geometry.

Hyperbolic Geometry is the study of asaddle shaped space. Consider what would happen if instead of working on the Euclidean flat piece of paper, you work on a curved surface shaped like the outer surface of a saddle or a Pringle's potato chip.

Unlike Riemannian Geometry, it is more difficult to see practical applications of Hyperbolic Geometry.Hyperbolic geometry does, however, have applications to certain areas of science such as the orbit prediction of objects within intense gradational fields, space travel and astronomy. Einstein stated that space is curved and his general theory of relativity uses hyperbolic geometry.

What effect does working on a saddle shaped surface have on what we think of as geometrical truths? In hyperbolic geometry, the sum of the angles of a triangle is less than 180. In hyperbolic geometry, triangles with the same angles have the same areas.

There are no similar triangles in hyperbolic geometry.

In hyperbolic space, the concept of perpendicular to a line can be illustrated as seen in the picture at the right.

Lines can be drawn in hyperbolic space that are parallel (do not intersect). Actually, many lines can be drawn parallel to a given line through a given point.

Graphically speaking, the hyperbolic saddle shape is called ahyperbolic paraboloid, as seen at the right.

It has been said that some of the works of artist M. C. Escher illustrate hyperbolic geometry. In his workCircle Limit III(follow the link below), the effect of a hyperbolic space's negative curve on the sum of the angles in a triangle can be seen. Escher's print illustrates a model devised by French mathematician Henri Poincare for visualizing the theorems of hyperbolic geometry, the orthogonal circle.M. C. Escher web site:http://www.mcescher.comChoose Galleries: Recognition and Success 1955-1972: Circle Limit IIIAnswer the following questions dealing with lines and planes.1.In this rectangular sided box,which set of sides lie in the same plane?

Top of FormChoose one:Bottom of Form

2.Top of FormWhen two planes intersect, two lines are formed.TRUE or FALSE?Choose one:TRUEFALSEBottom of Form

3.Which of the following statements is TRUE?

Top of FormChoose one:Planeslandqare parallel planesPlaneslandqintersect in line.PointPis in planel.Bottom of Form

4.If two lines intersect, only one plane contains both the lines.TRUE or FALSE?

Top of FormChoose one:TRUEFALSEBottom of Form

5.Top of FormFor this rectangular solid, planeGHBandEFCare _____.Choose one:perpendicularparallelneitherBottom of Form

6.Top of FormWhich of the following statementsis true?Choose one:Linelies in planel.The intersection of lineand planelis point A.Planelis perpendicular to line.Bottom of Form

7.If two points lie in a plane, the line joining them also lies in the same plane.TRUE or FALSE?

Top of FormChoose one:TRUEFALSEBottom of Form

8.Top of FormFor this rectangular solid, which plane(s) containDand are parallel to planeFEG?Choose one:planesDABandGAD.planesDCBandFCB.only planeDAB.Bottom of Form

Beach Ball Investigation forNon-Euclidean GeometryTopic Index|Geometry Index|Regents Exam Prep Center

This activity can be accomplished by groups of students or the activity can be performed as a demonstration in front of the class by the teacher and/or student volunteers.

Materials:beach balls (or other larger balls) (could be done with balloons with some care) string, protractors, rulers or yardsticksNote:The ball can be marked by the teacher with a permanent marker before the activity begins. This will allow the ball to be used with several different groups/classes of students. Mark the poles. Mark two other points on the ball allowing for adequate distance between the points. Mark the vertices of several different sized triangles on the ball and

draw the triangles using great circles to form the sides. Label all of your points for easy reference for students' answers.

The shortest distance between two points on a sphere is along the arc of the great circle joining the points. The shortest distance between points on any surface is called ageodesic. In a plane, a straight line is a geodesic. On a sphere, a great circle is a geodesic.Student Tasks:1. Using the string, determine the length of thegreat circleof the spherical ball. Pull the string tight to the ball between the two poles, to approximate a geodesic. Record this length.2. Find the distance between the two designated, but non-connected, points on the ball. (These points will not be the poles.) Record this length. Is the geodesic you used for this length unique, or are other geodesics possible for this measurement?3. Locate the vertices of each of the triangles on the ball. Using a great circle as a geodesic, find the lengths of the sides of the triangles. Record the lengths for all of the triangles.4. To the best of your ability, use the protractor to measure the angles in each of the triangles. Record the measurements for each triangle.5. Make a concluding statement about the relationship between the angels of a triangle on a sphere.6. A discussion of Euclidean geometry versus non-Euclidean geometry would follow.

Basic ConstructionsTopic Index|Geometry Index|Regents Exam Prep Center

In geometry, constructions utilize only two tools- thestraightedge(an unmarked ruler) and thecompass.Never draw freehand when doing a construction!

The Compass:Compasses come in a variety of styles. Become familiar with the compass you will be using before beginning your constructions.

Suggestions for working with a compass: Place several pieces of paper under your worksheet to allow the compass point to remain stable. Hold the compass lightly and allow the wrist to remain flexible. If you cannot manage to move your wrist when drawing circles, try rotating the paper under the compass.The Straightedge:A straightedge is generally a clear plastic tool devoid of markings. It most often appears in the shape of a triangle. A portion of a straightedge is visible in the lower left corner of the picture on the left below.If you do not have a straightedge, a ruler may be used. Just remember to completely ignore the markings on the ruler.

Basic Geometrical Constructions:The basic constructions used in geometry include: copy a line segment copy an angle bisect a line segment bisect an angle construct perpendicular lines construct parallel lines construct isosceles triangle construct equilateral triangleEach construction is developed separately in the following web pages.Several videos showing the actual constructions are included.

While the constructions listed above are considered the basic geometrical constructions, you should also be able to construct situations that require the use of these constructions.Copy a Line Segmentand an AngleTopic Index|Geometry Index|Regents Exam Prep Center

ATTENTIONVideo Users:These videos require that you have available a means of displaying video such as Windows Media Player, Real Player, QuickTime, etc.Video files are lengthy and may take some time to load depending upon your connection. Please be patient.When the video is loading for the first time, you may experience some choppy sound and movement. Allow the video to finish loading and then play again for a smooth delivery.

Remember -- use your compass and straight edge only!

Areference lineis a line upon which you produce copies of existing figures.Copy a line segment

Video ofCopy Line Segment

Given:(Line segment)Task:To construct a line segment congruent to (line segment).

Directions:1.If a reference line does not already exist, draw a reference line with your straightedge upon which you will make your construction. Place a starting point on the reference line.

2. Place the point of the compass on pointA.3.Stretch the compass so that the pencil is exactly onB.4.Without changing the span of the compass, place the compass point on the starting point on the reference line and swing the pencil so that it crosses the reference line. Label your copy.Your copy and (line segment)are congruent. Congruent means equal in length.Explanation of construction:The two line segments are the same length, therefore they are congruent.Copy an angle

Video ofCopy an Angle

Given:Task:To construct an angle congruent to.

Directions:1.If a reference line does not already exist, draw a reference line with your straightedge upon which you will make your construction. Place a starting point on the reference line.2. Place the point of the compass on the vertex of(pointA).3. Stretch the compass to any length so long as it stays ON the angle.4. Swing an arc with the pencil that crosses both sides of.5. Without changing the span of the compass, place the compass point on the starting point of the reference line and swing an arc that will intersect the reference line and go above the reference line.

6. Go back toand measure the width (span) of the arc from where it crosses one side of the angle to where it crosses the other side of the angle.7. With this width, place the compass point on the reference line where your new arc crosses the reference line and mark off this width on your new arc.8. Connect this new intersection point to the starting point on the reference line.Your new angle is congruent to.Explanation of construction:When this construction is finished, draw a line segment connecting where the arcs cross the sides of the angles. You now have two triangles that have 3 sets of congruent (equal) sides. SSS is sufficient to prove triangles congruent. Since the triangles are congruent, any leftover corresponding parts are also congruent - thus, the angle on the reference line andare congruent.Bisect a Line Segmentand an AngleTopic Index|Geometry Index|Regents Exam Prep Center

ATTENTIONVideo Users:These videos require that you have available a means of displaying video such as Windows Media Player, Real Player, QuickTime, etc.Video files are lengthy and may take some time to load depending upon your connection. Please be patient.When the video is loading for the first time, you may experience some choppy sound and movement. Allow the video to finish loading and then play again for a smooth delivery.

Remember -- use your compass and straight edge only!

Bisect - cut into two congruent (equal) pieces.Bisect a line segment(Also know as Construct a Perpendicular Bisector of a segment)

Video ofBisect a Segment

Given: (Line segment)Task: Bisect.

Directions:1.Place your compass point onAand stretch the compass MORE THAN half way to pointB, but not beyondB.2. With this length, swing a large arc that will go BOTH above and below.(If you do not wish to make one large continuous arc, you may simply place one small arc aboveand one small arc below.)3.Without changing the span on the compass, place the compass point onBand swing the arc again. The two arcs you have created should intersect.

4.With your straightedge, connect the two points of intersection.5.This new straight line bisects. Label the point where the new line andcross asC.has now been bisected andAC = CB. (It could also be said that the segments are congruent,.)

(It may be advantageous to instruct students in the use of the "large arc method" because it creates a "crayfish" looking creature which students easily remember and which reinforces the circle concept needed in the explanation of the construction.)Explanation of construction:To understand the explanation you will need to label the point of intersection of the arcs above segmentasDand below segmentasE. Draw segments,,and. All four of these segments are of the same length since they are radii of two congruent circles. More specifically,DA = DBandEA = EB. Now, remember a locus theorem: The locus of points equidistant from two points, is the perpendicular bisector of the line segment determined by the two points. Hence,is theperpendicularbisector of.The fact that the bisector is also perpendicular to the segment is actually MORE than we needed for a simple "bisect" construction. Isn't this great! Free stuff!!!

Bisect an angle

Video ofBisect an Angle

Given:Task:Bisect.

Directions:1.Place the point of the compass on the vertex of(pointA).2. Stretch the compass to any length so long as it stays ON the angle.3.Swing an arc so the pencil crosses both sides of. This will create two intersection points with the sides (rays) of the angle.4.Place the compass point on one of these new intersection points on the sides of.

If needed, stretch your compass to a sufficient length to place your pencil well into the interior of the angle. Stay between the sides (rays) of the angle. Place an arc in this interior - you do not need to cross the sides of the angle.5. Without changing the width of the compass, place the point of the compass on the other intersection point on the side of the angle and make the same arc. Your two small arcs in the interior of the angle should be crossing.6.Connect the point where the two small arcs cross to the vertexAof the angle.You have now created two new angles that are of equal measure (and are each 1/2 the measure of.)Explanation of construction:To understand the explanation, some additional labeling will be needed. Label the point where the arc crosses sideasD. Label the point where the arc crosses sideasE. And label the intersection of the two small arcs in the interior asF. Draw segmentsand. By the construction,AD = AE(radii of same circle) and DF = EF (arcs of equal length). Of courseAF = AF. All of these sets of equal length segments are also congruent. We have congruent triangles by SSS. Since the triangles are congruent, any of their leftover corresponding parts are congruent which makesequal (or congruent) to.Parallel- through a pointTopic Index|Geometry Index|Regents Exam Prep Center

ATTENTIONVideo Users:These videos require that you have available a means of displaying video such as Windows Media Player, Real Player, QuickTime, etc.Video files are lengthy and may take some time to load depending upon your connection. Please be patient.When the video is loading for the first time, you may experience some choppy sound and movement. Allow the video to finish loading and then play again for a smooth delivery.

Remember -- use your compass and straight edge only!

Parallel-through a point

Video ofParallel through a point

Given:PointPis off a given lineTask: Construct a line through P parallel to the given line.

Directions:1. With your straightedge, draw a transversal through pointP. This is simply a straight line which runs throughPand intersects the given line.2. Using your knowledge of the construction COPY AN ANGLE, construct a copy of the angle formed by the transversal and the given line such that the copy is located UP at pointP. The vertex of your copied angle will be point P.

3. When the copy of the angle is complete, you will have two parallel lines.This new line is parallel to the given line.Explanation of construction:Since we used the construction to copy an angle, we now have two angles of equal measure in our diagram. In relation to parallel lines, these two equal angles are positioned in such a manner that they are called corresponding angles. A theorem relating to parallel lines tells us that if two lines are cut by a transversal and the corresponding angles are congruent (equal), then the lines are parallel.Perpendiculars- from a pointonthe line- from a pointoffthe lineTopic Index|Geometry Index|Regents Exam Prep Center

ATTENTIONVideo Users:These videos require that you have available a means of displaying video such as Windows Media Player, Real Player, QuickTime, etc.Video files are lengthy and may take some time to load depending upon your connection. Please be patient.When the video is loading for the first time, you may experience some choppy sound and movement. Allow the video to finish loading and then play again for a smooth delivery.

Remember -- use your compass and straight edge only!

Perpendicular - lines (or segments) which meet to form right angles.Perpendicularfrom a pointONa line

Video ofPerpendicular On Line

Given:PointPis on a given lineTask:Construct a line throughPperpendicular to the given line.

Directions:1. Place your compass point onPand sweep an arc of any size that crosses the line twice (below the line). You will be creating (at least) a semicircle. (Actually, you may draw this arc above OR below the line.)2.STRETCH THE COMPASS LARGER!!3. Place the compass point where the arc crossed the line on one side and make a small arc below the line. (The small arc could be above the line if you prefer.)

4. Without changing the span on the compass, place the compass point where the arc crossed the line on the OTHER side and make another arc. Your two small arcs should be crossing.5.With your straightedge, connect the intersection of the two small arcs to pointP.This new line is perpendicular to the given line.Explanation of construction:Remember the construction for bisect an angle? In this construction, you have bisected the straight angleP. Since a straight angle contains 180 degrees, you have just created two angles of 90 degrees each. Since two right angles have been formed, a perpendicular exists.Perpendicularfrom a pointoffa line.

Video ofPerpendicular Off Line

Given: PointPis off a given lineTask:Construct a line throughPperpendicular to the given line.

Directions:1.Place your compass point onPand sweep an arc of any size that crosses the line twice.2.Place the compass point where the arc crossed the line on one side and make an arc ON THE OPPOSITE SIDE OF THE LINE.3. Without changing the span on the compass, place the compass point where the arc crossed the line on the OTHER side and make another arc. Your two new arcs should be crossing on the opposite side of the line.

4. With your straightedge, connect the intersection of the two new arcs to pointP.This new line is perpendicular to the given line.Explanation of construction:To understand the explanation, some additional labeling will be needed. Label the point where the arc crosses the line as pointsCandD. Label the intersection of the new arcs on the opposite side as pointE. Draw segments,,, and. By the construction,PC = PDandEC = ED.Now, remember a locus theorem: The locus of points equidistant from two points (CandD), is the perpendicular bisector of the line segment determined by the two points. Hence,is the perpendicular bisector of.The fact that we created a bisector, as well as a perpendicular, is actually MORE than we needed - we only needed to create a perpendicular. Yea, free stuff!!!Constructions: Isosceles and Equilateral TrianglesTopic Index|Geometry Index|Regents Exam Prep Center

Directions for constructing isosceles and equilateral triangles:

Construct an Isosceles Triangle Using Given Segment Lengths:

When constructing an isosceles triangle, you may be given pre-determined segment lengths to use for the triangle (such as in this example), or you may be allowed to determine your own segment lengths. Either way, the construction process will be the same.Construct an isosceles triangle whose legs and base are of the pre-determined lengths given. Construct the new triangle on the reference line.

Using your compass, measure the length of the given "base".

Do not change the size of the compass. Place your compass point on the reference line point and scribe a small arc which will cross the line.

Using your compass, measure the length of the given "leg". Place the compass point where the previous arc crosses the reference line and scribe another arc above the reference line

Without changing the size of the compass, move the compass point to the point on the reference line. Scribe an arc above the line such that it intersects with the previous arc.

You now have three points which will define the isosceles triangle.

Construct an Equilateral Triangle Using a Given Segment Length:When constructing an equilateral triangle, you may be given a pre-determined segment length to use for the triangle (such as in this example), or you may be allowed to determine your own segment length. Either way, the construction process will be the same.

Construct an equilateral triangle whose sides are of given length "a". Construct the new triangle on the reference line.

Using your compass, measure the length of the given segment, "a".

Do not change the size of the compass. Place your compass point on the reference line point and scribe an arc which will cross the line and will rise above the line.

Do not change the size of the compass. Place the compass point where the arc crosses the reference line and scribe another arc which crosses the previous arc.

You now have three points which will define the equilateral triangle.

Alternate Method for Constructing an Equilateral Triangle:

An equilateral triangle can be easily constructed from a circle. The secret to this method is to remember to keep the compass set at the same length as the radius of the original circle.Draw a circle and place a point on the circle. Do not change the size of the compass.

With the compass still set at the same size as the radius of the circle, place the compass point on the point on the circle and mark off a small arc on the circle. Now, move the compass point to this new arc and mark off another arc. Continue around the circle.You now have a circle with six equally divided sections on its circumference.Connect every other point on the circle to form the equilateral triangle.

Construction ActivitiesTopic Index|Geometry Index|Regents Exam Prep Center

1.Draw two quadrilaterals of about the same shape as the ones shown below. Using your straightedge and compass, bisect each side of each figure. Join the midpoints of the four sides of each figure in order, so that two new quadrilaterals are formed. What do you notice?

2.Draw a line segment that is several inches long. Using your straightedge and compass, divide the segment into four equal parts. Into what other number of equal parts, less than ten, can a line segment be easily divided?

3.Draw two triangles of the same shape as shown below. Using your straightedge and compass, bisect each of the three angles of each figure. What do you notice?

4.Using what you know about constructions, can you figure out a way to construct the following items?the altitudes of a scalene triangle

the altitudes of an obtuse triangle

the medians of a scalene triangle

the medians of an obtuse triangle

a square

5.Construct an angle of 30. Justify your construction.

6.Given segments of lengthaandb, construct a rectangle that has a vertex atA on the given reference line. Justify why your construction is correct.

7.Given the following figure, construct a parallelogram having sidesandand