Geometry Points, Lines, Planes, & Angles
Table of Contents Points, Lines, & Planes Line Segments Pythagorean Theorem Distance between points Area of Figures in Coordinate Plane Midpoint formula Locus & Constructions Angles & Angle Addition Postulate Angle Pair Relationships Angle Bisectors & Constructions
Table of Contents for Videos Demonstrating Constructions Congruent Segments Midpoints Circle Equilateral Triangles Congruent Angles Angle Bisectors
Definitions An "undefined term" is a word or term that does not require further explanation. There are three undefined terms in geometry: Points - A point has no dimensions (length, width, height), it is usually represent by a capital letter and a dot on a page. It shows position only. Lines - composed of an unlimited number of points along a straight path. A line has no width or height and extends infinitely in opposite directions. Planes - a flat surface that extends indefinitely in two- dimensions. A plane has no thickness.
Points & Lines A television picture is composed of many dots placed closely together. But if you look very closely, you will see the spaces. However, in geometry, a line is composed of an unlimited (infinite) number of points. There are no spaces between the points that make a line. .................................... B A You can always find a point between any two other The line above would be called line or line
Points & Lines Line , or , Points are labeled with letters. all refer to the same line Line a Lines are named by using any two points OR by using a single lower-case letter. Arrowheads show that the line continues without end in opposite directions.
Collinear Points Line , or , all refer to the same line Points A, B, and C are NOT collinear points since they do not lie on the same (one) line. Line a Points D, E, and F are called collinear points, meaning they all lie on the same line. Postulate: Any two points are always collinear.
Example: Give six different names for the line that contains points U, V, and W. Answer [This object is a pull tab]
Intersecting Lines If two non-parallel lines intersect in a plane they do so at only one point. and intersect at K. Postulate: two lines intersect at exactly one point.
Example a. Name three points that are collinear b. Name three sets of points that are noncollinear c. What is the intersection of the two lines? a. A, D, C b. A,B,D / A,C,B / C,D,B (others) c. Point D Answer [This object is a pull tab]
Rays or Rays are also portions of a line. Rays start at an initial point, here endpoint A, and continues infinitely in one direction. is read "rayAB"
Naming Rays or Rays and are NOT the same. They have different initial points and extend in different directions.
Opposite Rays Suppose point C is between points A and B Rays and are opposite rays. Opposite rays are two rays with a common endpoint that point in opposite directions and form a straight line.
Collinear Rays Recall: Since A, B, and C all lie on the same line, we know they are collinear points. Similarly, segments and rays are called collinear, if they lie on the same line. Segments, rays, and lines are also called coplanar if they all lie on the same plane.
Example Name a point that is collinear with the given points. a. R and P b. M and Q c. S and N d. O and P
Example Name two opposite rays on the given line e. f. g. h.
1 is the same as . True False Answer False [This object is a pull tab] Hint Read the notation carefully. Are they asking about lines, line segments, or rays? click
2 is the same as . True False Answer True [This object is a pull tab]
3 Line p contains just three points. True False Answer False Hint [This object is a pull tab] click to reveal Remember that even though only three points are marked, a line is composed of an infinite number of points. You can always find another point in between two other points.
4 Points D, H, and E are collinear. True False Answer False [This object is a pull tab]
5 Points G, D, and H are collinear. True False Answer True [This object is a pull tab]
6 Ray LJ and ray JL are opposite rays. Explain your answer. Yes No No, Opposite Rays have same endpoint but point in opposite directions Answer [This object is a pull tab]
7 Which of the following are opposite rays? A and B and C and Answer D D and [This object is a pull tab]
8 Name the initial point of A J B K Answer A C L [This object is a pull tab]
9 Name the initial point of A J B K Answer C L B [This object is a pull tab]
Example Are the three points collinear? If they are, name the line they lie on. a. L, K, J b. N, I, M c. M, N, K d. P, M, I
Planes Planes can be named by any three noncollinear points: Plane KMN, plane LKM, or plane KNL or, by a single letter such as Plane R. (These all name the same plane) Points K, M, and L are coplanar. Points O, K, and L are non-coplanar in the diagram above. However, you could draw a plane to contain any three points Coplanar points are points that lie on the same plane:
Collinear points are points that are on the same line. J,G, and K are three collinear points. F,G, and H are three collinear points. J,G, and H are three non-collinear points. Any three non-collinear points can name a plane. Coplanar points are points that lie on the same plane. F, G, H, and I are coplanar. F, G, H, and J are also coplanar, but the plane is not drawn. F,G, and H are coplanar in addition to being collinear. G, I, and K are non-coplanar and non-collinear.
Intersecting Planes The intersection of these two planes is shown by line A B If two planes intersect, they intersect along exactly one line. As another example, picture the intersections of the four walls in a room with the ceiling or the floor. You can imagine a line laying along the intersections of these planes.
Example Name the following points: A point not in plane HIE A point not in plane GIE Two points in both planes Two points not on
10 Line BC does not contain point R. Are points R, B, and C collinear? Yes No Answer No [This object is a pull tab]
11 Plane LMN does not contain point P. Are points P, M, and N coplanar? Yes No Yes on Plane MNP Answer [This object is a pull tab] Hint: What do we know about any three points? click to reveal
12 Plane QRS contains . Are points Q, R, S, and V coplanar? (Draw a picture) Yes No Answer Yes [This object is a pull tab]
13 Plane JKL does not contain . Are points J, K, L, and N coplanar? Yes No Answer No [This object is a pull tab]
14 and intersect at A Point A B Point B C Point C Answer B D Point D [This object is a pull tab]
15 Which group of points are noncoplanar with points A, B, and F on the cube below. A E, F, B, A B A, C, G, E Answer C C D, H, G, C D F, E, G, H [This object is a pull tab]
16 Are lines and coplanar on the cube below? Yes No Answer Yes [This object is a pull tab]
17 Plane ABC and plane DCG intersect at _____? A C B line DC C Line CG Answer B D they don't intersect [This object is a pull tab]
18 Planes ABC, GCD, and EGC intersect at _____? A line B point C C point A D line Answer B [This object is a pull tab]
19 Name another point that is in the same plane as points E, G, and H A B B C C D Answer D D F [This object is a pull tab]
20 Name a point that is coplanar with points E, F, and C A H B B C D D A Answer C [This object is a pull tab]
21 Intersecting lines are __________ coplanar. A Always B Sometimes C Never Answer A [This object is a pull tab]
22 Two planes ____________ intersect at exactly one point. A Always B Sometimes Answer C C Never [This object is a pull tab]
23 A plane can __________ be drawn so that any three points are coplaner A Always B Sometimes C Never Answer A [This object is a pull tab]
24 A plane containing two points of a line __________ contains the entire line. A Always B Sometimes C Never Answer A [This object is a pull tab]
25 Four points are ____________ noncoplanar. A Always B Sometimes C Never Answer B [This object is a pull tab]
26 Two lines ________________ meet at more than one point. A Always B Sometimes C Never B Answer Look what happens if I place line y directly on top of line x. click to reveal HINT [This object is a pull tab]