Unit 2: Basic Geometric Elements


 Melvyn Thornton
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1 Lesson 2.1: Points, Lines, and Planes Lesson 2.1 Objectives Define and write notation of the following: (G1.1.6) Point Opposite rays Line Collinear Plane Coplanar Ray End point Line segment Initial point Intersection StartUp Give your definition of the following: Unit 2: Basic Geometric Elements Point Line These terms are actually said to be, or have no formal definition. However, it is important to have a general agreement on what each word means. Point A has dimension, it is merely a. Meaning it takes up no space. It is usually represented as a. When labeling we designate a capital letter as a name for that point. We may call it. Line A extends in dimension. Meaning it goes straight in either a vertical, horizontal, or slanted fashion. It extends in two directions. It is represented by a line with an arrow on each end. When labeling, we use lowercase 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 Plane A extends in dimensions. Meaning it stretches in a vertical direction as well as a horizontal direction at the same time. It also extends. 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. M Or the plane can be named by picking three points in the plane and saying. 1
2 Collinear The prefix co means the, or. Linear means. Coplanar are points that lie on the. Line Segment Consider the line AB. It can be broken into smaller pieces by merely chopping the arrows off. This creates a or segment that consists of A and B. This is symbolized as Ray A consists of an point where the figure begins and then continues in one direction forever. It looks like an arrow. This is symbolized by writing its point first and then naming any other point on the ray,. Or we can say ray AB. Betweenness When points lie on a line, we can say that one of them is the other two. This is only true if all three points are. We would say that is between and. Opposite Rays If C is between A and B on a line, then ray and ray are rays. are only opposite if they are. Intersections of Lines and Planes Two or more geometric figures if they have one or more in common. If there is no point or points shown, they the figures do not intersect. The of the figures is the set of points the figures have in common. Two intersect at. Two intersect at. 2
3 Example 2.1 Draw the following 1. AB 4. Plane DEF 2. CD 5. DE intersected by FG at point H. 3. EF 6. If M is between N and L, draw the opposite rays MN and ML. Example 2.2 Answer the following 1. Name 3 points that are collinear. 2. Name 3 points that are not collinear. 3. Name 3 points that are coplanar. 4. Name 4 points that are not coplanar. 5. What are two ways to name the plane? 6. What are two names for the line that passes through points C and B. Homework 2.1 Lesson 2.1 Point, Line, Plane p12 Due: 3
4 Lesson 2.2: Segments 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. Length Finding the between points and is written as Writing is also called the of line segment AB. Postulate 2: Segment Addition Postulate If B is between A and C, then. Also, the opposite is true. If, then is A and. Example Sketch and write the segment addition postulate if point E is between points D and F. 2. Sketch and write the segment addition postulate if point M is between points N and P. Example 2.4 Find 1. GJ 2. KM 3. XY 4. LM 4
5 Distance Formula To find the on a graph between two points. A(, ) B(, ) Congruent Segments Segments that have the same are called segments. This is symbolized by. Example 2.5 Find the distance of each segment and identify if any of the segments are congruent. 1. J(1,1) K(0,5) 2. L(2,1) M(2,0) 3. A(4,3) B(1,6) 4. D(2,3) E(2,0) Midpoint The of a segment is the that divides the segment into two segments. The midpoint the segment, because bisect means to divide into parts. Midpoint Formula Example 2.6 Find the midpoint. 1. R(3,1) S(3,7) 2. T(2,4) S(6,6) Finding the Other End Many may say finding the midpoint is easy! It is simply the of the two. Now imagine knowing the, one, and trying to 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. the coordinates of the. 2. the coordinates of the known. 5
6 Example 2.7 Find the other endpoint given one endpoint, E, and the midpoint, M. 1. E(0,5) 2. E(1,3) M(3,3) M(5,9) Segment Bisector A is a, ray,, or plane that intersects the original segment at its. Example 2.8 Use the diagram to find the given measure if line l is a segment bisector. Homework 2.2 Lesson 2.2 Line Segments p34 Due: 6
7 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 consists of two different that have the same. The form the of the angle. The initial is called the of the angle. can often be thought of as a. Naming an Angle All are named by using points First, name a that lies on one of the angle. Second, name the next. The is always named in the. Finally, name a that lies on the side of the angle. Using a Protractor To measure an angle with a, do the following: 1. Place the of the protractor on the of the angle. 2. Line up one of the angle with the 0 o line near the of the protractor. 3. Read the protractor for the where the of the angle points. 7
8 Congruent Angles are angles that have the. To show that we are finding the measure of an angle Place a before the name of the angle. Types of Angles Right Looks Like Measure (<90) (=90) (>90) (=180) 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 2. KMN 3. PQM 4. JML 5. PLK Other Parts of an Angle The of an angle is defined as the set of points that lie the sides of the angle. The of an angle is the set of points that lie of the sides of the angle. Postulate 4: Angle Addition Postulate The Postulate allows us to add each smaller angle together to find the measure of a larger angle. 8
9 Example 2.11 Use the given information to find the indicated measure Adjacent Angles Two angles are angles if they a and, but have common interior points. Basically they should be, but not. Angle Bisector An is a that an angle into adjacent angles that are. To show that angles are congruent, we use. Example 2.12 In the diagram, BD bisects ABC. Find m ABC Homework 2.3 Lesson Angles and Their Measures p56 Due: 9
10 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 if their form pairs of opposite rays. Basically the two lines that form the angles are. To identify the angles, simply look straight the to find the angle pair. Hint: The angle pairs do not have to be vertical in position. pairs are always! Linear Pair Two angles form a linear pair if their noncommon sides are opposite rays. Simply put, these are two that a. Just like neighbors share a fence, but they must live on the side of the road Since they share a straight line, their sum is Example 2.13 Find the measure of all unknown angles, when m 1 = 57 o. 1. m 2 = 2. m 3 = 3. m 4 = Example 2.14 Solve for x and y
11 Complementary v Supplementary angles are two angles whose is. angles can be adjacent or nonadjacent. angles are two angles whose is. 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 m 2 = 22 o and m 2 = 2. m 5 = 3. m 6 = 4. m 4 = 5. m 3 = 6. m 7 = 7. m 8 = Example 2.16 A and B are complementary. Find m A and m B. 1. m A = 2x + 12 m B = 9x 10 A and B are supplementary. Find m A and m B. 2. m A = 12x + 32 m B = 4x 12 Perpendicular Lines When two lines intersect to form a angle, they are said to be lines. Homework 2.4 Lesson Angle Pair Relationships p78 Due: 11
12 Lesson 2.5: Lines Cut by a Transversal 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 Example 2.17 Determine the relationship between the given angles 1. 3 and and and and and 14 Postulate 15: Corresponding Angles Postulate If two lines are cut by a, then angles are. You must know the lines are parallel in order to assume the angles are. 12
13 Theorem 3.4: Alternate Interior Angles If two lines are cut by a, then angles are. 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. Theorem 3.5: Consecutive Interior Angles If two lines are cut by a, then angles are. Again be sure that the lines are parallel. They don t look to be congruent, so they MUST be supplementary. Theorem 3.6: Alternate Exterior Angles If two lines are cut by a, then angles are. Again be sure that the lines are parallel. Example 2.18 Find the missing angles for the following: 13
14 Example 2.19 Solve for x Parallel versus Skew Two lines are if they are and intersect. The shorthand for being is. Lines that are and do not intersect are called lines. These are lines that look like they intersect but do not lie on the same piece of paper. lines go in directions while 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. 2. Line WZ and line QW are. 3. Line SY and line WX are. 4. Plane WQR and plane SYT are. 5. Plane RQT and plane WQR are. 6. Line TS and line ZY are. 7. Line WX and plane SYZ are. Homework 2.5 Lesson 2.5 Lines Cut by a Transversal p1112 Due: Unit 2 Test: 14
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