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Transcript
Objectives: I can Identify and model points, lines, planes, and angles. I can Identify intersecting lines and planes. Basic Geometry Review Unit 3 Lesson 1 We will be identifying basic geometric terms. In geometry, the terms point, line, and plane are considered undefined terms. They are explained through examples and descriptions, not definitions. A point is simply a location. P Drawn as a dot Named by a capital letter Has neither shape nor size called “point P” A line is made up of an infinite set of points extending in two directions. A line has no thickness or width, but it does have length. A B A A B n line has an arrowhead at each end A line can be named by a lowercase letter OR if two points are known, then the line can be named by those letters. Called line n, line AB or AB, line BA or BA We can also represent a line like: AB Points on the same line are said to be collinear. A B Noncollinear points are points NOT on the same line. A B C A plane is a flat surface made up of an infinite set of points that creates a flat surface that extends without ending. X Z Y T A plane has an infinite length and width, but no depth. X Y Z T Drawn as a shaded, slanted 4-sided figure Named as a capital letter or by using three non collinear points on that plane. What could we name this plane? Plane T, plane XYZ, plane XZY, plane YXZ, plane YZX, plane ZXY, plane ZYX. 10 Different planes in a figure: A D B C E H Plane EFGH F G Plane ABCD Plane BCGF Plane ADHE Plane ABFE Plane CDHG Etc. Lesson 1-1 Point, Line, Plane Points that lie on the same plane are said to be coplanar. A B D C M Noncoplanar plane. points are points not in the same K B A F Modeling Points, Lines, and Planes Let’s take a look at a piece of paper Naming Lines and Planes 1. Name a line containing point A. 2. Name a plane containing point C 3. Name three points that A are collinear. l E D C B N 4. Are points E, A, B, and D collinear or non collinear? FACTS It takes at least two points to make a line. It takes at least three points to make a plane. Space is the set of all points. 1. Are points A, B, and C collinear or noncollinear? 2. Are points B, C, and E collinear or noncollinear? 3. What are some ways to name this line? A E B C Intersection is the set of points in both figures. Lines intersect at a point. k j P Example Draw and label a figure for the following situation. Plane R contains lines AB and DE, which intersect at point P. Add point C on plane R so that it is not collinear with AB or DE. A E P C D B Example Choose the best diagram for the given relationship. Plane D contains line a, line m, and line t, with all three lines intersecting at point Z. Also point F is on plane D and is not collinear with any of the three given lines. A. B. C. D. A line and a plane intersect at a point. K P Planes intersect at a line. K B A F True or False 1. Line PF ends at P. false 2. Point S is on an infinite number of lines. true 3. The edge of a plane is a line. false Interpret Drawings Example 1 A. How many planes appear in this figure? Answer: There are two planes: plane S and plane ABC. Interpret Drawings Example 1 B. Name three points that are collinear. Answer: Points A, B, and D are collinear. Interpret Drawings Example 1 C. Are points A, B, C, and D coplanar? Explain. Answer: Points A, B, C, and D all lie in plane ABC, so they are coplanar. Interpret Drawings Example 1 Answer: The two lines intersect at point A. 1) A. point X B. point N A. C. point R B. C. D. point A D. A B C D 2) Draw a surface to represent plane R and label it. ANSWER 2) Draw a surface to represent plane R and label it. Congruent refers to objects that have the same shape or size. *Congruent segments are segments that have equal length! When writing and signifying congruence, we use the ≅ symbol. When drawing a picture of figures that are congruent, we use slashes or ticks. 31 Congruent Segments Definition: Segments with equal lengths. (congruent symbol: ) B Congruent segments can be marked with dashes. A If numbers are equal the objects are congruent. C AB: the segment AB ( an object ) AB: the distance from A to B ( a number ) Correct notation: AB = CD AB CD Incorrect notation: AB CD AB = CD D Congruent Segments Segment Bisector Definition: Any segment, line or plane that divides a segment into two congruent parts is called segment bisector. A F A B E AB bisects DF. E AB bisects DF. B D F A E D D F Plane M bisects DF. B AB bisects DF. Segment Bisector Any segment line or plane that intersects a segment at it’s midpoint. n 4 A X is the midpt of AB 4 X B AX XB n bisects AB If X is between A and B and X is the midpoint of AB, what is the measure of AX if AB = 16x – 6 and XB = 4x +9? Ray Definition: RA : RA and all points Y such that A is between R and Y. A How to sketch: R A Y R How to name: RA ( not AR ) RA or RY ( not RAY ) ( the symbol RA is read as “ray RA” ) Opposite Rays Definition: If A is between X and Y, then ray AX and ray AY are opposite rays. ( Opposite rays must have the same “endpoint” ) X A Y opposite rays D not opposite rays E DE and ED are not opposite rays. Angles An Angle is a figure formed by two rays with a common endpoint, called the vertex. ray vertex ray Angles can have points in the interior, in the exterior or on the angle. E A D B C Points A, B and C are on the angle. D is in the interior and E is in the exterior. B is the vertex. Naming an angle (1) Using 3 points (2) Using 1 point (3) Using a number – next slide Using 3 points: vertex must be the middle letter This angle can be named as ABC or CBA Using 1 point: using only vertex letter * Use this method is permitted when the vertex point is the vertex of one and only one angle. Since B is the vertex of only this angle, this can also be called B . B A C Naming an Angle - continued Using a number: A number (without a degree symbol) may be used as the label or name of the angle. This A number is placed in the interior of the angle near its vertex. The angle to the left can be named B 2 C as 2 . * The “1 letter” name is unacceptable when … more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present. Example K is the vertex of more than one angle. Therefore, there is NO K in this diagram. There is LKM , PKM , and LKP There is also 2 and 3 but there is no 5!!! L M 2 K 3 P 4 Types of Angles Acute Angle:an angle whose measure is less than 90. Right Angle: an angle whose measure is exactly 90 . Obtuse Angle: an angle whose measure is between 90 and 180. Straight Angle:an angle that is exactly 180 . Measuring Angles l Just as we can measure segments, we can also measure angles. l We use units called degrees to measure angles. A circle measures _____ 360º ? A (semi) half-circle measures _____ ? ? A quarter-circle measures _____ 90º One degree is the angle measure of 1/360th of a circle. Adding Angles l l When you want to add angles, use the notation m1, meaning the measure of 1. If you add m1 + m2, what is your result? m1 + m2 = 58. A B 36° m1 + m2 = mADC also. 22° Therefore, mADC = 58. 1 D 2 C Angle Bisector An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles. Example: Since 4 6, UK is an angle bisector. 41° K j 4 U 6 41° 5 3 Example Draw your own diagram and answer this question: If ML is the angle bisector of PMY and mPML = 87, then find: mPMY = _______ mLMY = _______ Congruent Angles Definition: If two angles have the same measure, then they are congruent. Congruent angles are marked with the same number of “arcs”. The symbol for congruence is ≅ Example: 3 5. 3 5 Example Draw your own diagram and answer this question: If ML is the angle bisector of PMY and mPML = 87, then find: mPMY = _______ mLMY = _______ Practice Name all angles that have B as a vertex. A G B D E C F