Unit 1 Section 2 Points Lines Rays and

Unit 1 – Section 2 Points, Lines, Rays and Planes

Section 1 -3: Points, Lines, & Planes The student will: - understand basic terms of geometry undefined (general description) defined - name points, lines, and planes - understand basic postulates of geometry including what a postulate is Materials Needed: Definition Sheets, Postulate Sheets, Notes, Textbook, & a writing utensil

In geometry, some words such as point, line, and plane are undefined. In order to define these words you need to use words that need further defining. It is important, however, to have general descriptions of their meaning.

Point • • • A B A location C Has no size (no dimensions) Represented by a dot Named with a capital letter A set of points can form a geometric figure Space – the set of all points (pg. 17) D

Line • A series of points that extends in two opposite directions without end (one-dimensional) • Can be named by any two points on the line or with a single lowercase letter R S T Y t line t (read “line t”) RS (read “line RS”) or SY (read “line SY”) or TR (read “line TR”) & more Collinear Points – points that lie on the same line (pg. 17)

Example 1 n • Identifying Collinear Points a. Are points E, F, and C collinear? name the line on which they lie. Points E, F, and C are collinear. They lie on line m. b. Are points E, F, and D collinear? If so, name the line on which they lie. If so, m C F E P D l Points E, F, and D are not collinear. c. Are points F, P, and C collinear? d. Name line m in three other ways. e. Why are arrowheads used when drawing a line or naming a line such as EF?

Plane • A flat surface that has no thickness • Contains many lines and extends without end in the directions of all its lines (two-dimensional) • Named with a single capital letter or by at least three of its noncollinear points. P A B C Plane P Plane ABC Coplanar – points and/or lines in the same plane (pg. 17)

Example 2 • Naming a Plane How many flat surfaces does the box (icecube) have? front back top bottom right side left side Each one of these flat surfaces lies in a different plane. H G E F D A C B List three different names for the plane represented by the top of the box (ice cube). List three different names for the plane represented by the right side of the box (ice cube. )

Postulate – an accepted statement of fact (pg. 18) Postulate 1 -1 (Pg. 18) Through any two points there is exactly one line. B t A Line t is the only line that passes through points A and B. Postulate 1 -2 (Pg. 18) If two lines intersect, then they intersect in exactly one point. A D C B E AE and BD intersect at C.

Postulate 1 -3 (Pg. 18) If two planes intersect, then they intersect in exactly one line. What points are in the purple plane? R, S, & T Name the purple plane. Plane RST What points are in the blue plane? S, T, & W Name the blue plane. Plane STW What points are in both planes? S&T R S T W Name the intersection of the two planes. Plane RST and plane STW intersect in ST. When you have two points that both lie in two different planes, the line through those two points is the intersection of the planes.

Example 3 Finding the Intersection of two planes. What is the intersection of plane HGFE and plane BCGF? H G E Plane HGFE and plane BCGF intersect in GF. F D A Name two planes that intersect in BF. What is the intersection of plane ABFE and plane AEHD? C B

Postulate 1 -4 (Pg. 19) Through any three noncollinear points there is exactly one plane. Example 4 H G E A G E F D H D C B a. Shade the plane that contains A, B, and C. F A C B b. Shade the plane that contains E, H, & C. c. Name another point that is in the same plane as points A, B, and C. d. Name another point that is coplanar with points E, H, and C.

Section 1 -4: Segments, Rays, Parallel Lines & Planes The student will: - identify and name segments and rays - identify and name parallel lines and skew - identify and name parallel planes Materials Needed: Definition Sheets, Notes, & a Writing Utensil lines

Many geometric figures, such as squares and angles, are formed by parts of lines called segments or rays. Segment – the part of a line consisting of two endpoints and all points between them. (Pg. 23) - a segment is named using its two endpoints - always use segment notation when naming segments B A AB (read “Segment AB”) or BA (read “Segment BA”) Are these two names for the same segment?

Ray – the part of a line consisting of one endpoint and all the points of the line on one side of the endpoint. (Pg. 23) - a ray is named with its endpoint (always listed first) and any other point on the ray - always use ray notation when naming rays Y Y X X YX (read “ray YX”) is the ray that starts at Y and then passes though X and continues on in that same direction without end. XY (read “ray XY”) is the ray that starts at X and then passes though Y and continues on in that same direction without end. Are these two names for the same ray?

Opposite Rays – two collinear rays with the same endpoint. (Pg. 23) - Opposite Rays always form a line. Q R S RQ and RS are opposite rays. Example 1 Q Name the segments and the rays in the figure at the right. LP and PL form a line. Are they opposite rays? Explain. P L

Lines that do not intersect may or may not be coplanar. Parallel Lines – coplanar lines that do not intersect. (Pg. 24) is the symbol for parallel Skew Lines – noncoplanar lines; they are not parallel and they do not intersect (Pg. 24) D C A B E AB & CG are skew. G H F AB || EF Classify AB and HG. Because I can draw a single plane that contains both of these lines, AB || HG.

Segments or rays are parallel if they lie in parallel lines. They are skew if they lie in skew lines. Example 2 A N J Name all labeled segments that are parallel to DC. B D C G H I GH, JI, & AB are parallel to DC. Name all labeled segments that are skew to DC. NJ, JG, & HI are skew to DC. Name all labeled segments that are parallel to GJ. Name all labeled segments that are skew to GJ. Name another pair of parallel segments; of skew segments.

Parallel planes are planes that do not intersect. (Pg. 24) G A H B J D C Plane ABCD || Plane GHIJ What other planes in this figure are parallel? I

Example 3 (second part) S P Q R W T a. Name three pairs of parallel planes. b. Name a line that is parallel to PQ. c. Name a line that is parallel to plane QRUV. U V

Homework p. 19 #9 -13 odd, 19, 23, 27, 31, 35, 39, 43 p. 25 #1 -7 odd, 13 -21 odd, 27 -33 odd Be sure to use proper notation on your answers. Additional Information p. 25 #7 – has 3 answers #17 -21 – if false, be sure to explain!

Reference • Laurie E. Bass, A. J. (2009). Geometry. Upper Saddle River, New Jersey: Pearson Education.