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## Geometry

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**Geometry**Chapter 1**Lesson (1-1)POINTS, LINES, & PLANES**What are the 3 BASIC UNDEFINED TERMS IN GEOMETRY? Answer: Point, line, & plane They do not have any shape or size. They are generally defined using examples.**Lesson (1-1)POINTS, LINES, & PLANES**A point • is a location. • has no shape or size. • is named using one capital letter. EX: A**Lesson (1-1)POINTS, LINES, & PLANES**A line: • is made up of infinitely many points. • has no thickness.**Lesson (1-1)POINTS, LINES, & PLANES**A line is named 2 ways. 1. Using 2 capital printed letters representing 2 points on the line E D**Lesson (1-1)POINTS, LINES, & PLANES**2. Using a lowercase, cursive letter. line m**Lesson (1-1)POINTS, LINES, & PLANES**• A plane: • Is a flat surface made up of points that extends infinitely in all directions. • has no thickness.**Lesson (1-1)POINTS, LINES, & PLANES**A plane is named 2 ways. 1.) using 3 capital printed letters representing 3 points that are not all on the same line EX. plane BCD or plane DBC or plane CBD, etc**Lesson (1-1)POINTS, LINES, & PLANES**• 2. using 1 capital, cursive letter plane N**Lesson (1-1)POINTS, LINES, & PLANES**COLLINEAR POINTS: Points that lie on the same line COPLANAR POINTS: Points that lie on the same plane**Lesson (1-1)POINTS, LINES, & PLANES**INTERSECTION: The set of points that 2 or more geometric figures have in common 1.) 2 lines intersect in a ________. In the diagram, line aand line b intersect at point R. point**Lesson (1-1)POINTS, LINES, & PLANES**2.) A line and a plane intersect in _________________________________ ___________________ EX In the first diagram, In the second diagram, line k and plane Bline p lies completely in intersect at point P.plane Q , so their intersection is line p. a point OR the line if the line lies in the plane.**Lesson (1-1)POINTS, LINES, & PLANES**line 3.) 2 planes intersect in a. EX**Lesson (1-1)POINTS, LINES, & PLANES**POSTULATE OR AXIOM: An accepted statement of fact POSTULATE Through any 2 points there is . EX exactly one line**Lesson (1-1)POINTS, LINES, & PLANES**POSTULATE If 2 lines intersect, then they intersect in . EX exactly one point**Lesson (1-1)POINTS, LINES, & PLANES**POSTULATE If 2 planes intersect, then they intersect in . exactly one line**Lesson (1-1)POINTS, LINES, & PLANES**POSTULATE Through any 3 noncollinear points there is . exactly one plane**Lesson (1-1)POINTS, LINES, & PLANES**Assignment: Worksheet Front: #1, 7, 10 – 13, 15, 19 Back: #1, 4, 5, 8, 24, 33, 35, 40, 43**Segments, Rays, Parallel Lines, and Planes**Segment: The part of a line consisting of 2 endpoints and all points between them Q P consists of points P and Q and all of the points between them To name a line segment use 2 capital letters and a segment above them.**Segments, Rays, Parallel Lines, and Planes**What is the difference between and ? SEGMENT AB is at the right in blue A B LINE AB is at the right in red A B**Segments, Rays, Parallel Lines, and Planes**IN CLASS: Front of worksheet #1,7,10-13,15,19 Assignment: Back of worksheet #1,4,5,8,24,33,35**Segments, Rays, Parallel Lines, and Planes**Ray: The part of a line consisting of one endpoint and all the points of the line on one side of the endpoint**Segments, Rays, Parallel Lines, and Planes**Naming a RAY: FIRST: Name the endpoint. SECOND: Name another point on the line closer to the arrow. THIRD: Write over the letters.**Segments, Rays, Parallel Lines, and Planes**Name the rays. EX D N M F is not the same as .**Segments, Rays, Parallel Lines, and Planes**Opposite Rays: 2 collinear rays with the same endpoint Opposite rays ALWAYS form a line! C B A**Segments, Rays, Parallel Lines, and Planes**and are OPPOSITE RAYS!!! R Q P Together, they make .**Segments, Rays, Parallel Lines, and Planes**Let’s do #2 – 6, 14, 17, & 18 front of the worksheet together for practice!**Segments, Rays, Parallel Lines, and Planes**Parallel lines: Coplanar lines that do not intersect Give some real life examples of parallel lines.**Segments, Rays, Parallel Lines, and Planes**Skew lines: Non-coplanar lines that do not intersect Give some real life examples of skew lines.**Segments, Rays, Parallel Lines, and Planes**Parallel Planes: Planes that do not intersect Give some real life examples of parallel planes.**Segments, Rays, Parallel Lines, and Planes**Let’s do #8,9, & 21 on the front of the worksheet.**Segments, Rays, Parallel Lines, and Planes**Assignment: Do the rest of the back of the worksheet. #2,3,6,7,9-23,25-32,34, 37-39,41,42,44-47 When you have finished this, you should have the front and back of the worksheet completed.**Measuring Segments**A B C D E F ‒2 ‒1 ‒4 ‒3 4 3 2 1 0 To find the length of a segment, think RIGHT minus LEFT.**Measuring Segments**A B C D E F ‒2 ‒1 ‒3 4 3 2 1 0 ‒4 The length of is F - E or 4 – 2 = 2.**Measuring Segments**A B C D E F ‒2 ‒1 ‒4 ‒3 4 3 2 1 0 1. 2. 3. 4.**Measuring Segments**A B C D E F ‒2 ‒1 ‒3 4 3 2 1 0 ‒4 Remember, RIGHT MINUS LEFT. 1. DF= 4 – 1 = 3 2. CE= 2 – 0 = 2 3. BC= 0 – (‒1) = 1 4. AE= 2 – (-3) = 5**Measuring Segments**When we found the lengths of the segments on the previous slide, notice how we wrote the lengths. 1. DF= 4 – 1 = 3 2. CE= 2 – 0 = 2 3. BC= 0 – (‒1) = 1 4. AE= 2 – (-3) = 5 We wrote = 3 and not = 3. DF**Measuring Segments**When we write the length of a segment, we do NOT write “ ― “ over the letters. Find AD & AF. AD = 4 & AF = 7**Measuring Segments**Congruent segments Segments with the same length**Measuring Segments**A B C D E F ‒2 ‒1 ‒4 ‒3 4 3 2 1 0 is congruent to what other segments on the number line?**Measuring Segments**Since all have length 2, they are congruent. The symbol for congruent is ≅ .**Measuring Segments**SEGMENT ADDITION POSTULATE If 3 points A, B, and C are collinear, and B is between A and C, then AB + BC = AC A B C**Measuring Segments**If 3 points A, B, and C are collinear and B is between A and C, then AB + BC = AC A B C If AB = 5, and BC = 8, then we write…. 5 + 8 = 13… AC = 13**Measuring Segments**X, Y, AND Z are collinear and Y lies between X and Z. Sketch a diagram representing this information, and use it to write an equation using the letters X, Y, and Z. Then do the problems on the following slides. X Y Z XY + YZ = XZ**Measuring Segments**1. XY = 5, YZ = 3; Find XZ. X Y Z XY + YZ = XZ 5 + 3 = XZ 8 = XZ**Measuring Segments**2. XY = 4, XZ = 11; Find YZ. XY + YZ = XZ 4 + YZ = 11 YZ = 7**Measuring Segments**3. If XZ = 70, XY = 3a - 2, and YZ = 5a, find the value of a, XY, and YZ. XY + YZ = XZ (3a – 2) + (5a) = 70 8a – 2 = 70 8a = 72 a = 9 XY = 3a - 2 = 3(9) – 2 = 25 YZ = 5a= 5(9) = 45