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Unit 2 Basic Geometric Elements Lesson 2.1 Points, Lines, and Planes Lesson 2.1 Objectives Define and write notation of the following: Point Line Plane Ray Line segment Collinear Coplanar End point Initial point Opposite rays Intersection (G1.1.6) Start-Up 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. 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 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 B A Plane M C B 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. Collinear The prefix co- means the same, or to share. Linear means line. So collinear means that points lie on the same line. A B C We say that points A, B, and C are collinear. Coplanar Coplanar points are points that lie on the same plane. A M C B So points A, B, and C are said to be coplanar. 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 endpoints A 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, AB . Or we can say ray AB. B A Betweenness When three points lie on a line, we can say that one of them is between the other two. A This is only true if all three points are collinear. We would say that B is between A and C. B C Opposite Rays If C is between A and B on a line, then ray CA and ray CB are opposite rays. Opposite rays are only opposite if they are collinear. A C B Intersections of Lines 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 one point. Two planes intersect at one line. A m n Example 2.1 4. Plane DEF D Draw the following 1. AB 5. B DE intersected by FG at po int H . G H F CD D C E D A 2. F E 6. If M is between N and L, draw the opposite rays MN and ML. 3. EF E F N M L Example 2.2 Answer the following 1. Name 3 points that are collinear. 1. 2. C, B, D Name 3 points that are not collinear. 2. 3. ex: A, B, E or A, B, C Name 3 points that are coplanar. 3. 4. ex: A, B, E or B, C, D or B, C, E Name 4 points that are not coplanar. 4. 5. ex: A, B, E, C What are two ways to name the plane? 5. 6. Plane ABE or Plane F What are two names for the line that passes through points C and B. 6. line g or BC 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) 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. Distance is always positive. E C A B D Length Finding the distance between points A and B is written as AB Writing AB is also called the length of line segment AB. 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 A B C AC Example 2.3 1. Sketch and write the segment addition postulate if point E is between points D and F. D E F DE + EF = DF 2. Sketch and write the segment addition postulate if point M is between points N and P. N M P NM + MP = NP Example 2.4 Find 1. GJ GJ = 16 1. 2. KM KM = 36 2. 3. XY 71-29 XY = 42 3. 4. LM 4. x + 2x = 18 3x = 18 x=6 LM = 6 Distance Formula To find the distance on a graph between two points A(1,2) AB = B(7,10) We use the Distance Formula (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. LE NT If you want to state two segments are congruent, then you write LE NT 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. 3. A(4,3) 1. J(1,1) 2 2 2 2 ( 1 4) (6 3) (0 1) (5 1) B(-1,6) K(0,5) (5) 2 (3) 2 (1)2 (4) 2 JK LM 2. L(2,1) M(-2,0) 25 9 34 5.83 1 16 17 4.12 (2 2) 2 (0 1) 2 (4) 2 ( 1) 2 16 1 17 4.12 4. D(2,-3) E(-2,0) (2 2) 2 (0 3) 2 (4)2 (3)2 16 9 25 5 Midpoint The midpoint of a segment is the point that divides the segment into two congruent segments. J The midpoint bisects the segment, because bisect means to divide into two equal parts. O We say that O is the midpoint of line segment JY. Y Midpoint Formula We can also find the midpoint of segment AB by using its endpoints in… The Midpoint Formula A(1,2) B(7,10) 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. Example 2.6 Find the midpoint 1. R(3,1) S(3,7) ( (y1 + y2) (x1 + x2) , 2 2. 2 T(2,4) S(6,6) 3 3 1 7 , 2 2 6 8 , 2 2 26 46 , 2 2 8 10 , 2 2 3, 4 4,5 ) Finding the Other End M 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: 1. 2. Double the coordinates of the midpoint. Subtract the coordinates of the known endpoint. F (7,13) ? E M (5, 7) 2 E (3,1) (10,14) (3,1) (7,13) Example 2.7 Find the other endpoint given one endpoint, E, and the midpoint, M. 1. E(0,5) 2. M(3,3) (3,3) 2 (6, 6) (0,5) (6,1) E(-1,-3) M(5,9) (5,9) 2 (10,18) (1, 3) (11, 21) Segment Bisector A segment bisector is a segment, ray, line, or plane that intersects the original segment at its midpoint. T J O H So HT is a segment bisector of JY . Y 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 opposite side of the angle. So we can call It Or NOW O WON W N Using a Protractor 1. 2. 3. 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. 54o 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. mWON mNOW WON NOW Equal Measures Congruent Angles Types of Angles Acute Right Obtuse Straight Less than 90 (<90) Equal to 90 (=90) Greater than 90 (>90) Equal to 180 (=180) Looks like Measure 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. 1. JKN NKJ, K 1. 1. 2. KMN 2. NMK 2. 3. MQP 3. acute JML 4. LMJ 4. 5. straight PQM 3. 4. right acute PLK 5. KLP 5. 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 exterior of 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 1. 4x + 22 = 94 4x = 72 25o 2. x = 18 3x + 1 + 2x – 6 = 135 3(28) + 1 84 + 1 85 85o 5x – 5 = 135 5x = 140 x = 28 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. C CAT and TAR are adjacent. CAR and TAR are not adjacent. T A R 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 In the diagram, BD bisects ABC. Find mABC. 2. 1. 4x = 3x + 6 x=6 3. 5x – 11 = 4x + 1 x – 11 = 1 x = 12 8x – 16 = 4x + 20 4x – 16 = 20 4x = 36 x=9 4(6) + 3(6) + 6 24 + 18 + 6 mABC = 48o 5(12) – 11 + 4(12) + 1 60 – 11 + 48 + 1 mABC = 98o 2(4(9) + 20) 2(56) mABC = 112o Homework 2.3 Lesson 2.3 - Angles and Their Measures p5-7 Due Tomorrow Lesson 2.4 Angle Pair Relationships Lesson 2.4 Objectives Identify vertical angle pairs. (G1.1.1) Identify linear pairs. (G1.1.1) Differentiate between complementary and supplementary angles. (G1.1.1) Vertical Angles Two angles are vertical angles if their sides form two pairs of opposite rays. Basically the two lines that form the angles are straight. To identify the vertical angles, simply look straight across the intersection to find the angle pair. Hint: The angle pairs do not have to be vertical in position. Vertical Angle pairs are always congruent! 4 1 3 2 4 1 3 2 Linear Pair Two adjacent angles form a linear pair if their noncommon sides are opposite rays. Simply put, these are two angles that share a straight line. Just like neighbors share a fence line, but they must live on the same side of the road. Since they share a straight line, their sum is… 180o 1 2 Example 2.13 Find the measure of all unknown angles, when m1 = 57o. 1. m2 1. 2. 123o m3 2. 3. 57o m4 3. 123o 16y - 27 = 13y Example 2.14 -27 = -3y y=9 Solve for x and y. 1. 2. 3x + 48 = 180 3x = 132 x = 44 3. 9x + 7 = 5x + 67 7x = 5x + 18 4x + 7 = 67 2x = 18 4x = 60 x = 15 x=9 Complementary v Supplementary Complementary angles are two angles whose sum is 90o. Complementary angles can be adjacent or nonadjacent. Supplementary angles are two angles whose sum is 180o. Supplementary angles can be adjacent or nonadjacent. Example 2.15 Find the measure of all unknown angles, given that m and n form a right angle and the m1 = 22o and 1 4. 1. m2 1. 68o 2. m5 m 2. 22o 3. m6 3. 68o 4. m4 n o 4. 22 5. m3 5. 68o 6. m7 6. 68o 7. m8 7. 22o Example 2.16 A and B are complementary. Find mA and mB. 1. mA = 2x + 12 mB = 9x – 10 1. 2x + 12 + 9x – 10 = 90 11x + 2 = 90 11x = 88 x=8 A and B are supplementary. Find mA and mB. 2. mA = 12x + 32 mB = 4x – 12 2. 12x + 32 + 4x – 12 = 180 16x + 20 = 180 16x = 160 x = 10 mA = 2(8) + 12 mB = 9(8) - 10 mA = 12(10) + 32 mB = 4(10) - 12 mA = 16 + 12 mB = 72 - 10 mA = 120 + 32 mB = 40 - 12 mA = 28o mB = 62o mA = 152o mB = 28o Perpendicular Lines When two lines intersect to form a right angle, they are said to be perpendicular lines. So we can say that a b. a b Homework 2.4 Lesson 2.4 - Angle Pair Relationships p7-8 Due Tomorrow Lesson 2.5 Introduction to Parallel Lines and Transversals Lesson 2.5 Objectives Identify angle pairs formed by a transversal. (G1.1.2) Compare parallel and skew lines. (G1.1.2) Lines and Angle Pairs Alternate Exterior Angles – because they lie outside the two lines and on opposite sides Transversal 1 2 4 3 5 6 8 7 Corresponding Angles – because they lie in corresponding positions of each intersection. of the transversal. Alternate Interior Angles – because they lie inside the two lines and on opposite sides of the transversal. Consecutive Interior Angles – because they lie inside the two lines and on the same side of the transversal. Example 2.17 Determine the relationship between the given angles 1 3 and 9 1) 2 13 and 5 2) 3 Alternate Interior Angles 5 and 15 4) 5 Corresponding Angles 4 and 10 3) 4 Alternate Interior Angles Alternate Exterior Angles 7 and 14 5) Consecutive Interior Angles Postulate 15: Corresponding Angles Postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent. You must know the lines are parallel in order to assume the angles are congruent. 1 2 4 3 5 6 8 7 Theorem 3.4: Alternate Interior Angles If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Again, you must know that the lines are parallel. If you know the two lines are parallel, then identify where the alternate interior angles are. Once you identify them, they should look congruent and they are. 1 2 4 3 5 6 8 7 Theorem 3.5: Consecutive Interior Angles If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. Again be sure that the lines are parallel. They don’t look to be congruent, so they MUST be supplementary. 1 2 4 3 180o = + + = 180o 5 6 8 7 Theorem 3.6: Alternate Exterior Angles If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. Again be sure that the lines are parallel. 1 2 4 3 5 6 8 7 Example 2.18 Find the missing angles for the following: 120o 60o 120o 120o 140o 140o 105o 110o 70o 105o 110o Example 2.19 Solve for x 1. 2. 3x + 15 = 60 3. 2x – 4 = 92 3x = 45 2x = 96 x = 15 x = 48 x – 10 = 100 x = 110 4. 5x – 10 + 75 = 180 5x + 65 = 180 5x = 115 x = 23 Parallel versus Skew Two lines are parallel if they are coplanar and do not intersect. The short-hand symbol for being parallel is //. Lines that are not coplanar and do not intersect are called skew lines. These are lines that look like they intersect but do not lie on the same piece of paper. Skew lines go in different directions while parallel lines go in the same direction. Example 2.20 Complete the following statements using the words parallel, skew, perpendicular. 1) Line WZ and line XY are _________. 1) 2) Line WZ and line QW are ________. 2) 3) perpendicular Line TS and line ZY are __________. 6) 7) parallel Plane RQT and plane WQR are _________. 5) 6) skew Plane WQR and plane SYT are _________. 4) 5) perpendicular Line SY and line WX are _________. 3) 4) parallel skew Line WX and plane SYZ are __________. 7) parallel. Homework 2.5 Lesson 2.5 p11-12 Due Tomorrow Unit 2 Test Monday, October 11th