Unit 2 Basic Geometric Elements
Lesson 2.1 Points, Lines, and Planes
Lesson 2.1 Objectives • Define and write notation of the following: (G1.1.6) • Point • Line • Plane • Ray • Line segment • Collinear • Coplanar • End point • Initial point • Opposite rays • Intersection
Give your definition of the following: Point Line These terms are actually said to be undefined, or have no formal definition. However, it is important to have a general agreement on what each word means. Start-Up
Point • A point has no dimension, it is merely a location. • Meaning it takes up no space. • It is usually represented as a dot. • When labeling we designate a capital letter as a name for that point. • We may call it Point A. A
B A Line • A line extends in one dimension. • Meaning it goes straight in either a vertical, horizontal, or slanted fashion. • It extends forever in two directions. • It is represented by a line with an arrow on each end. • When labeling, we use lower-case letters to name the line. • Or the line can be named using two points that are on the line. • So we say Line n, or AB n
A C B Plane M • A plane extends in two dimensions. • Meaning it stretches in a vertical direction as well as a horizontal direction at the same time. • It also extends forever. • It is usually represented by a shape like a tabletop or a wall. • When labeling we use a bold face capital letter to name the plane. • Plane M • Or the plane can be named by picking three points in the plane and saying Plane ABC.
A B C Collinear • The prefix co- means the same, or to share. • Linearmeans line. • So collinear means that points lie on the sameline. We say that points A, B, and C are collinear.
A C B Coplanar • Coplanar points are points that lie on the same plane. M So points A, B, and C are said to be coplanar.
B A Line Segment • Consider the line AB. • It can be broken into smaller pieces by merely chopping the arrows off. • This creates a line segment or segment that consists of endpointsA and B. • This is symbolized as AB
B A Ray • A ray consists of an initial point where the figure begins and then continues in one direction forever. • It looks like an arrow. • This is symbolized by writing its initial point first and then naming any other point on the ray, . • Or we can say rayAB. AB
A B C Betweenness • When three points lie on a line, we can say that one of them is between the other two. • This is only true if all three points are collinear. • We would say that B is between A and C.
Opposite Rays • If C is between A and B on a line, then ray CA and ray CB are oppositerays. • Oppositerays are only opposite if they are collinear. C B A
A m n Intersections ofLines and Planes • Two or more geometric figures intersect if they have one or more points in common. • If there is no point or points shown, they the figures do not intersect. • The intersection of the figures is the set of points the figures have in common. • Two lines intersect at onepoint. • Two planes intersect at oneline.
Draw the following F D E B D G A H F E D C E F N M L Example 2.1
Example 2.2 Answer the following • Name 3 points that are collinear. • C, B, D • Name 3 points that are not collinear. • ex: A, B, E or A, B, C • Name 3 points that are coplanar. • ex: A, B, E or B, C, D or B, C, E • Name 4 points that are not coplanar. • ex: A, B, E, C • What are two ways to name the plane? • Plane ABE or Plane F • What are two names for the line that passes through points C and B. • line g or
Homework 2.1 • Lesson 2.1 – Point, Line, Plane • p1-2 • Due Tomorrow
Lesson 2.2 Distance, Midpoint, and Segment Addition
Lesson 2.2 Objectives • Utilize the distance formula. (G1.1.3) • Apply the midpoint formula. (G1.1.5) • Justify the construction of a midpoint. (G1.1.5) • Utilize the segment addition postulate. (G1.1.3) • Identify the symbol and definition of congruent. (G1.1.3) • Define segment bisector. (G1.1.3)
E C D Postulate 1: Ruler Postulate • The points on a line can be matched to real numbers called coordinates. • The distance between the points, say A and B, is the absolute value of the difference of the coordinates. • Distanceis always positive. A B
Length • Finding the distance between points A and B is written as • AB • Writing AB is also called the length of line segment AB.
A B C Postulate 2: Segment Addition Postulate • If B is between A and C, then • AB + BC = AC. • Also, the opposite is true. • If AB + BC = AC, then B is between A and C. BC AB AC
D F E M P N Example 2.3 • Sketch and write the segment addition postulate if point E is between points D and F. DE + EF = DF Sketch and write the segment addition postulate if point M is between points N and P. NM + MP = NP
Example 2.4 Find • GJ • GJ = 16 • KM • KM = 36 • XY • 71-29 XY = 42 • LM • x + 2x = 18 3x = 18 x = 6 LM = 6
Distance Formula To find the distance on a graph between two points A(1,2) B(7,10) We use the Distance Formula AB = (x2 – x1)2 + (y2 – y1)2 Distance can also be found using the Segment Addition Postulate, which simply adds up each segment of a line to find the total length of the line.
Congruent Segments • Segments that have the same length are called congruent segments. • This is symbolized by . Hint: If the symbols are there, the congruent sign should be there. If you want to state two segments are congruent, then you write If you want to state two lengths are equal, then you write
Example 2.5 (x2 – x1)2 + (y2 – y1)2 Find the distance of each segment and identify if any of the segments are congruent. • J(1,1)K(0,5) A(4,3)B(-1,6) D(2,-3)E(-2,0) L(2,1)M(-2,0)
O J Y Midpoint • The midpoint of a segment is the point that divides the segment into two congruent segments. • The midpointbisects the segment, because bisect means to divide into two equal parts. We say that O is the midpoint of line segment JY.
A(1,2) B(7,10) ( ) , Midpoint Formula We can also find the midpoint of segment AB by using its endpoints in… The Midpoint Formula Midpoint of AB = (y1 + y2) (x1 + x2) 2 2 This gives the coordinates of the midpoint, or point that is halfway between A and B.
(y1 + y2) (x1 + x2) 2 2 ( ) , Example 2.6 Find the midpoint • R(3,1)S(3,7) T(2,4)S(6,6)
E M ? Finding the Other End • Many may say finding the midpoint is easy! • It is simply the average of the two endpoints. • Now imagine knowing the midpoint, one endpoint, and trying to find the coordinates of the other endpoint. • Try to remember what the midpoint formula does and work it backwards. • So here is what we are going to do: • Double the coordinates of the midpoint. • Subtract the coordinates of the known endpoint.
Example 2.7 Find the other endpoint given one endpoint, E, and the midpoint, M. • E(0,5)M(3,3) E(-1,-3)M(5,9)
T H O J Y Segment Bisector • A segment bisector is a segment, ray, line, or plane that intersects the original segment at its midpoint.
Example 2.8 Use the diagram to find the given measure if line l is a segment bisector. 109 in ST = ½(109 in) ST = 54.5 in
Homework 2.2 • Lesson 2.2 – Line Segments • p3-4 • Due Tomorrow
Lesson 2.3 Angles and Their Measures
Lesson 2.3 Objectives • Identify more than one name for an angle. (G1.1.6) • Identify angle measures. (G1.1.6) • Classify angles as right, obtuse, acute, or straight. (G1.1.6) • Apply the angle addition postulate. (G1.1.3) • Utilize angle vocabulary to solve problems. (G1.1.6) • Define angle bisector and its uses. (G1.1.3)
What is an Angle? • An angle consists of two different rays that have the same initial point. • The rays form the sides of the angle. • The initial point is called the vertex of the angle. • Vertex can often be thought of as a corner.
Naming an Angle • All angles are named by using three points • First, name a point that lies on one side of the angle. • Second, name the vertex next. • The vertex is always named in the middle. • Finally, name a point that lies on the oppositeside of the angle. So we can call It WON Or NOW W N O
Using a Protractor • To measure an angle with a protractor, do the following: • Place the cross-hairs of the protractor on the vertex of the angle. • Line up one side of the angle with the 0o line near the bottom of the protractor. • Read the protractor for the where the other side of the angle points.
Example 2.9 Protractor Stations
Congruent Angles • Congruent angles are angles that have the same measure. • To show that we are finding the measure of an angle… • Place a “m” before the name of the angle. Congruent Angles Equal Measures
Example 2.10 Give another name for the angle in the diagram above. Then, tell whether the angle appears to be acute, obtuse, right, or straight. • JKN • NKJ, K • right • KMN • NMK • straight • PQM • MQP • acute • JML • LMJ • acute • PLK • KLP • obtuse
Other Parts of an Angle • The interior of an angle is defined as the set of points that lie between the sides of the angle. • The exteriorof an angle is the set of points that lie outside of the sides of the angle. Exterior Interior
Postulate 4: Angle Addition Postulate • The Angle Addition Postulate allows us to add each smaller angle together to find the measure of a larger angle. What is the total? 49o 32o 17o
Example 2.11 Use the given information to find the indicated measure. 3x + 15 + x + 7 = 94 4x + 22 = 94 4x = 72 25o x = 18 3x + 1 + 2x – 6 = 135 5x – 5 = 135 85o 3(28) + 1 5x = 140 84 + 1 x = 28 85
C T R A Adjacent Angles • Two angles are adjacent angles if they share a common vertex and side, but have no common interior points. • Basically they should be touching, but not overlapping. CAT and TAR areadjacent. CAR and TAR are notadjacent.
Angle Bisector • An angle bisector is a ray that divides an angle into two adjacent angles that are congruent. • To show that angles are congruent, we use congruence arcs.
Example 2.12 8x – 16 = 4x + 20 4x = 3x + 6 5x – 11 = 4x + 1 4x – 16 = 20 x = 6 x – 11 = 1 4x = 36 x = 12 x = 9 2(4(9) + 20) 4(6) + 3(6) + 6 5(12) – 11 + 4(12) + 1 2(56) 60 – 11 + 48 + 1 24 + 18 + 6 mABC = 48o mABC = 98o mABC = 112o