# Collinear Points that lie on the same line.

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1 Point - A point has no dimension. It is represented by a dot. It represents a fixed point in space. Plane - A plane has two dimensions. It is represented by a shape that looks like a floor or a wall, but it extends without end. Through any three points not on the same line, there is exactly one plane. You can use three points that are not all on the same line to name a plane. Coplanar - Points that lie in the same plane. Points S, P, T and V are coplanar. They all lie in plane R. Line - A line has one dimension. It is represented by a line with two arrowheads, but it extends without end. Through any two points, there is EXACTLY one line. You can use any two points on a line to name it. Collinear Points that lie on the same line. Points S, P, and T are collinear. Points P and Q are also collinear. Endpoint A point that is the beginning or end of a segment or ray. Segment A piece of a line which consists of two endpoints. Written as AB or BA Ray A piece of a line which consists of one endpoint and extends indefinitely in the other direction. AB starts a point A and runs through point B BA starts a point B and runs through point A So BA and AB are different rays. 2

2 Opposite Rays Two rays that share a common endpoint and extend in opposite directions. Theorem - A rule that can be proved. CA and CB are opposite rays Postulate A rule that is accepted without proof. (also called axiom) Distance The distance between points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B. Distance of a segment is written as: AB or the length of AB Congruent Segments Line segments that have the same length. Construction A geometric drawing that uses a limited set of tools, usually and compass and a straightedge (a ruler without marks) or AB = CD Angle Two different rays with the same endpoint. This angle can be named BAC CAB A Point A is the vertex of the angle. (Segment) Bisector A point, ray, line, line segment, or plane that intersects the segment at its midpoint. Interior/Exterior of an Angle A point is in the interior of an angle if it is between points that lie on each side of the angle. Vertex The common endpoint of two different rays that form an angle. 3

3 Congruent Angles Two angles are congruent if they have the same measure. Vertical Angles Two angles whose sides form two pairs of opposite rays. Adjacent Angles Two angles that share a common vertex and side, but have no common interior points. Acute/Right/Obtuse/Straight Linear Pair Two adjacent angles are a linear pair if their noncommon sides are opposite rays. The angles in a linear pair are supplementary angles. Euclidian Geometry aka flat or parabolic geometry. Named after the greek mathematician Euclid. Geometry based on the assumption of a flat surface. Non-Euclidean Geometry Geometry that is not based on a flat surface. Some types of non-euclidean geometry: Hyperbolic Geometry Elliptic Geometry Complementary/Supplementary Complementary angles the sum of their measures is 90 Supplementary angles the sum of their measures is 180 4

4 1.1 Vocabulary Grid Write down a definition, example or sketch a picture of each word. After you have filled in as many as possible walk around the room and find a different person for each box to help you fill it in. As you fill out others sheet give an educated guess if you do not know a term. You have 8 minutes to get all the boxes filled. Point Intersection Segment Endpoint Line Coplanar Plane Ray Defined Term Opposite Ray Undefined Term Collinear 5

5 Section 1.1 Points, Lines, and Planes Ordered Pair (x,y) Quadrant II Quadrant I x-coordinate y-coordinate Quadrant III Quadrant IV Collinear Points: points that lie on the same line Name three collinear points: Name three points that are not collinear: Ex 1: Find the coordinates of three points that lie on the graph of y=3x+5 and one point that doesn t lie on the line x 3x + 5 y Ex 2: Do points R (4,8) and S (-3,-7) lie on the line with the equation y=2x-1? Undefined Terms No formal definition, but it is understood. The terms point, line, and plane are undefined terms. Point: An exact position in space; Represented by a dot 6

6 Line: Collection of points along a straight path extending infinitely in opposite directions. Plane: Flat surface extending infinitely in all directions. Coplanar: Points that lie in the same plane. Noncoplanar: Points that do not lie in the same plane. Practice: Defined Terms: Terms that can be described using known words such as point or line. Line Segment: Part of a line consisting of two endpoints and all points on the line between the endpoints. Ray: Part of a line with one endpoint that extends infinitely in the opposite direction. 7

7 Practice: Intersections Two or more geometric figures intersect if they have or more points in common. Perpendicular Lines: Two lines that intersect to form a right angle. (opposite reciprocal slope) Parallel Lines: Two coplanar lines that never intersect. (same slope) 8

8 Section 1.2 Segments & Congruence The length of a segment is the distance between its two endpoints. The length of ST can be written as Postulate/Axiom A rule that is accepted without proof Ruler Postulate: The numbers on a ruler are a real-life example of a number line. The distance between two points on a number line is the absolute value of the difference of the coordinates. Example 1: What is the distance from X to Y? X Y Distance from X to Y can be written as or When three points are collinear, you can say that one point is between the two others. Example 2: Find PQ, QR, and PR if P is located at -3, Q is located at 1, and R is located at 6. Segment Addition Postulate: 9

9 Example 3: Find LM if L is between N and M, NL = 6x-5, LM = 2x+3, and NM = 30 Example 4: Find the length of MN if N is between M and P, MN=3x+2, NP=18, MP=5x 10

10 Congruent Same value Congruent Segments: Line segments that have the same length. Example 5: Find AB if K is between A and B, AK=2x+10, KB=5x+4, and AK is congruent to KB. 11

11 You will need a compass and a ruler to do the following. Use the following segments to connect them on the dotted line provided: A B B C D M R T A D Add segments AB and BC : AB=, BC=, AC= + = Add segments AD and DM : AD=, DM=, AM= + = 12

12 Section 1.3 Midpoint and Distance Formulas (Day 1) Midpoint (of a segment) The point that divides the segment into two congruent segments. Segment Bisector A point, ray, line, line segment, or plane that intersects the segment at its midpoint. To determine the midpoint on a number line, add the two endpoints and divide by 2. Midpoint Formula: x 1 + x 2 2 Example 1 Midpoint X A Y B What is the coordinate of the midpoint of segment AB? What is the coordinate of the midpoint of segment XY? Example 2 Midpoint: Line RM bisects segment LK at point D. Find DK if LK is

13 Example 3 - Midpoint: Point M is the midpoint of segment VW. Find the length of segment VM. Example 4 - Midpoint: If P is the midpoint of segment CD, PC=x and PD=5x-4, find the value of x and the length of CD. Coordinate Plane Midpoint Formula: Example 5 Coordinate Plane Midpoint: The endpoint of segment RS are R(1,-3) and S(4,2). Find the midpoint. Example 6 Missing Endpoint: The midpoint of segment JK is M(2,1). One endpoint is J(1,4). Find the coordinates of endpoint K. 14

14 Coordinate Plane Distance Formula Example 7 - Distance: Find the length of RS if R(1,-3) and S(4,2). Example 8 - Distance: Determine if the two segments AB and LK are congruent. A(4,6) B(7,2) L(-1,-6) K(4,-6) 15

15 1.3 Using the Midpoint and Distance Formulas (Day 2) Midpoint Formula Distance Formula The coordinates of the midpoint of a segment are the averages of the x-coordinates and of the y-coordinates of the endpoints. Find the midpoint and the distance for the following segments: 1. (2,3) and (2, 7) 2. ( 1,5) and (5,5) 3. (2,4) and (4,8) Find the midpoint and the distance for the following segments: 4. (4,5) and (9,17) 5. ( 2,8) and ( 6, 4) 6. (2,3) and (11,16) 16

16 Given the following endpoint and midpoint, find the missing endpoint and the distance: 7. Endpoint: (4,-4) 8. Endpoint: (-3, 0) 9. Endpoint: (4, 4) Midpoint: (1,-1) Midpoint: (0, 1) Midpoint (-1, -2) Endpoint Endpoint Endpoint Distance Distance Distance 17

17 Section 1.4 Measure and Classify Angles An is formed by two different rays with the same endpoint. Name angles with three letters. The vertex must be in the middle. None of the angles should be named < X, because all three angles have X as their vertex. Angles are measured in. If an angle is 32 degrees, it can be written two different ways: Angles are measured with a protractor. The image to the right demonstrates how a protractor is used. 18

18 Classifying Angles Angle Addition Postulate Example 1 Find Angle Measures: Given that find Example 2 - Find Angle Measures: 19

19 Congruent Angles Angles that have the same measure. Angle Bisector A ray that divides and angle into to congruent angles. Since < XYW and < ZYW are congruent, ray YW is the angle bisector of < XYZ. Example 3 Double an Angle Measure: Example 4 Challenge 20

20 Use the steps and copy the following segment: Use the steps above and copy the following angle right next to it not under: 21

21 Section 1.5 Angle Pair Relationships Adjacent Angles Two angles that share a common vertex and side. Complementary Angles Two angles whose sum is 90 Supplementary Angles Two angles whose sum is 180 Sum is 90 Sum is 180 Example 1: Find the measure of the complement and supplement of 1. m 1 = 22 Example 2: What are the measures of angles 2 and 3? 22

22 Example 3: Linear Pair Vertical Angles Vertical angles are always congruent!! Example 4: FGH and HGJ form a linear pair. Find the measures of the angles if m FHG=(12x+1) and m HGJ=(4x-9). (Draw a picture to help you) Example 5: Find the values of x and y. 23

23 1.5 Class Practice 1) Tell whether the angles are supplementary, complementary, vertical, bisecting. For each identify if adjacent or not. 2) Find m 2=, 3= 3) Solve for x and y: Equation: Equation: Equation: 4) Use one of the learned concepts ( Complementary, supplementary, vertical, angle addition), set up an equation and solve it. 24

24 Section 1.6 Classify Polygons Polygon A closed plane figure with the following properties: 1. It is formed by three or more sides (have to be straight) 2. No lines in the interior of the shape Vertex (plural: vertices) A polygon can be named by listing the vertices in consecutive order. Two possible names for this polygon are ABCDE and Convex and Concave All interior angles are At least one 25

25 Example 1: Naming Polygons A polygon is named by the number of sides. 26

26 Equilateral All sides congruent Equiangular All angles congruent Regular both equilateral and equiangular Example 2: Example 3: Find the value of x in the regular polygon. Section 1.7 Find Perimeter, Circumference and Area Perimeter: Distance around a figure Circumference: Distance around a circle Area: Amount of surface covered by a figure 27

27 Example 1: Find the perimeter and area of the rectangular basketball court shown. Example 2: Find the approximate circumference and area of the patch shown. 28

28 Example 3: The base of a triangle is 28 meters. Its area is 308 square meters. Find the height of the triangle. Example 4: Find the area and perimeter of the triangle. (Use distance formula to find the length of each side) 1.7 Class Practice 1) 29

29 Use the distance (length) formula to find out the length of each one QR, QS, and RS QR= QS= RS= Perimeter= 2) Read this solved problem and solve the next one in a similar way: 30

30 h= 3) a) Circumference rend ring= Circumference yellow ring= b) --This Page is intentionally left blank 31

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