1.1: Identify Points, Lines, & Planes 1.2: Use Segments & Congruence 02 Building Blocks of Geometry 1.1 and 1.202 Building Blocks of Geometry 1.1: Identify Points, Lines, & Planes 1.2: Use Segments & Congruence Objectives: To learn the terminology and notation of the basic building blocks of geometry To use the Ruler and Segment Addition Postulates To construct congruent segments with compass and straightedge 1

Vocabulary Point Line Segment Line Ray Plane In your notes, define each of these without your book. Draw a picture for each word and leave a bit of space for additions and revisions. Point Line Segment Line Ray Plane

Undefined Terms? In geometry, we always try to define things in simpler terms. Point, line, and plane are considered undefined terms, however, and cannot be made any simpler, so we just describe them. What the Ancient Greeks said: “A point is that which has no part. A line is breadthless length.”

Undefined Terms? In geometry, we always try to define things in simpler terms. Point, line, and plane are considered undefined terms, however, and cannot be made any simpler, so we just describe them. What the Ancient Chinese said: “The line is divided into parts, and that part which has no remaining part is a point.”

Mathematical model of a point Points Basic unit of geometry No size, only location ZERO dimensions Represented by a dot and named by a capital letter Mathematical model of a point

Points Mathematical model of a point A star is a physical model of a point

Mathematical model of a line Lines Straight arrangement of points No width, only length Extends forever in 2 directions ONE dimension Named by two points on the line: line AB or BA or or It can be named with a lower-case script letter: l l Mathematical model of a line

Lines Mathematical model of a line Spaghetti is a physical model of a line

Mathematical model of a line Lines How many lines can you draw through any two points? ONE Mathematical model of a line

Collinear Points Collinear points are points that --?--. Points A, B, and C are collinear Lie on a line What do you supposed NON-collinear means?

Mathematical model of a plane Planes Flat surface that extends forever Length and width but no height (2-D) Represented by a 4-sided figure and named by a capital script letter or 3 (non-collinear) letters on the same plane. Mathematical model of a plane

Planes Mathematical model of a plane Flattened dough is a physical model of a plane

Mathematical model of a plane Planes How many points does it take to define a plane? (tell where the plane is in space?) THREE Mathematical model of a plane

Coplanar Points Coplanar points are points that --?--. Points A, B, and C are coplanar Lie on the same plane What do you suppose NON-coplanar means?

Hierarchy of Building Blocks Space 3-D Planes 2-D Lines 1-D Points Space is the set of all points 0-D

A Romance of Many Dimensions Are there more than three spatial dimensions? ? Point Segment Square Cube 0-D 1-D (Length) 2-D (Area) 3-D (Volume) 1 point 2 points 4 points 8 points 0 “sides” 2 “sides” 4 “sides” 6 “sides”

A Romance of Many Dimensions Are there more than three spatial dimensions? Point Segment Square Cube Hypercube 0-D 1-D (Length) 2-D (Area) 3-D (Volume) 1 point 2 points 4 points 8 points 0 “sides” 2 “sides” 4 “sides” 6 “sides”

A Romance of Many Dimensions Are there more than three spatial dimensions? Point Segment Square Cube Hypercube 0-D 1-D (Length) 2-D (Area) 3-D (Volume) 4-D (Hypervolume) 1 point 2 points 4 points 8 points 0 “sides” 2 “sides” 4 “sides” 6 “sides”

A Romance of Many Dimensions Are there more than three spatial dimensions? Point Segment Square Cube Hypercube 0-D 1-D (Length) 2-D (Area) 3-D (Volume) 4-D (Hypervolume) 1 point 2 points 4 points 8 points 16 points 0 “sides” 2 “sides” 4 “sides” 6 “sides”

A Romance of Many Dimensions Are there more than three spatial dimensions? Point Segment Square Cube Hypercube 0-D 1-D (Length) 2-D (Area) 3-D (Volume) 4-D (Hypervolume) 1 point 2 points 4 points 8 points 16 points 0 “sides” 2 “sides” 4 “sides” 6 “sides” 8 “sides”

WATCH the following Computer animation of hypercube Explanation of 4th dimension

Example 1 Give two other names for and plane R. Name three points that are collinear. Name four points that are coplanar.

Line Segment A line segment consists of two endpoints and all the collinear points between them. Line segment AB or Endpoints

Congruent Segments Congruent segments are line segments that have the same length. Symbol for congruent

Copying a Segment We’re going to try making two congruent segments using only a compass and a straightedge. Here, we’re not using a ruler to measure the length of the segment!

Copying a Segment Draw segment AB.

Copying a Segment Draw a line with point A’ on one end.

Copying a Segment Put point of compass on A and the pencil on B. Make a small arc.

Copying a Segment Put point of compass on A’ and use the compass setting from Step 3 to make an arc that intersects the line. This is B’.

Copying a Segment Click on the image to watch a video of the construction. GSP

A laser is a physical model of a ray A ray consists of an endpoint and all of the collinear points to one side of that endpoint. Ray AB or A laser is a physical model of a ray

Example 2 Ray BA and ray BC are considered opposite rays. Use the picture to explain why. At what time would the hands of a clock form opposite rays? 3:15, 6:00, 1:35,…

Example 3 Give another name for . Name all rays with endpoint J. Which of these rays are opposite rays? HG JG JH JF JE

Intersection Two or more geometric figures intersect if they have one or more points in common. The intersection of the figures is the set of points the figures have in common. The intersection of two planes is a line. The intersection of two lines is a point.

Example 4 What is the length of segment AB? B A

Example 4 You basically used the Ruler Postulate to find the length of the segment, where A corresponds to 0 and B corresponds to 6.5. So AB = |6.5 – 0| = 6.5 cm B A

Example 5 Now what is the length of ? 8.5 – 2 = 6.5 A B

Ruler Postulate The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is its coordinate. The distance between points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B.

Example 6 When asked to measure the segment below, Kenny gave the answer 2.7 inches. Explain what is wrong with Kenny’s measurement. Inches are not divided into tenths

Give Them an Inch… A Standard English ruler has 12 inches. Each inch is divided into parts. Cut an inch in half, and you’ve got 1/2 an inch. Cut that in half, and you’ve got 1/4 an inch. Cut that in half, and you’ve got 1/8 inch. Cut that in half, and you’ve got 1/16 inch. Click the ruler and practice measuring both inches and centimeters

Example 7 Let’s say you found the length of a segment to be 6’ 7” using your dad’s tape measure. Convert this measurement to the nearest tenth of a centimeter (1” ≈ 2.54 cm). 6 7/12 = 6.58 1 = 2.54 6.58 X Cross-multiply X= 16.71

Example 8 Use the diagram to find GH. 15

Example 8 Use the diagram to find GH. Could you as easily find GH if G was not collinear with F and H? Why or why not? No, the parts wouldn’t equal the whole. FG + GH = FH

Segment Addition Postulate If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. BETWEEN indicates collinear points

Example 9 Point A is between S and M. Find x if SA = 2x – 5, AM = 7x + 3, and SM = 25. A M S 25 2x – 5 7x + 3 2x – 5 + 7x+3 = 25 9x = 27 X=3

Example 10 Point E is between J and R. Find JE given that JE = x2, ER = 2x, and JR = 8. WORK IT OUT WITH A PICTURE!! X = 2, why does it not equal -4?

Example 11: SAT Points E, F, and G all lie on line m, with E to the left of F. EF = 10, FG = 8, and EG > FG. What is EG? Work it out with a picture. 18