1. PALMER RIDGE HIGH SCHOOL Geometry Mrs. Rothwell Fall Semester 2015

2. WELCOME!  What is Geometry ? The study of the properties of shapes and the space around them – from simple 2 dimensional triangles to complex solids Architects & Engineers (planning) Designers (products/artwork) Geographers & Scientists Computer Programmers (logic)

3. Seating Course Syllabus Supplies: Binder & Graph Paper / Calculator Math Department Policies Daily Notes: PowerPoints (use color!) / Fusion Page Assignment Sheets (keep in binder) Vocabulary Homework (format & timeliness) Getting Support After school homework help (3-4pm in library on T/Th) 9 sections (see schedule of other geometry teachers) Find a friend or form a study group!

4.  Algebra Skills Pre - Assessment Why we are taking it – Critical skills for Geometry (not part of your class grade but will appear in IC – Infinite Campus) Review Assignment (A 4 : 20 points) Next class period 50 problems on test Answer Key Seeking help Retest: Aug. 27/28 20 problems

5. CHAPTER 1  1.1 Points, Lines and Planes  1.2 Linear Measure  1.3 Distance and Midpoints  1.4 Angle Measure QUIZ (16 points)  1.5 Angle Relationships  1.6 Polygons CHAPTER TEST (100 points)

6. Section 1.1: Points, Lines, and Planes Objective: Identifying and modeling the basic elements in Geometry (geometry = earth + measurement). point line plane *These are referred to as undefined terms since they can only be explained using examples and descriptions To model real world objects we must use these basic elements over and over! Other Vocabulary: space - coplaner - collinear - Points located on the same line Points that lie on the same plane A boundless, 3 dimensional set of all points Our Basic Elements: intersection - Location at which elements meet

7. ELEMENT CHART PointLinePlane Description How do we model it? How is it drawn? How do we name it? Facts Using words or symbols simply a location made up of points a flat surface made up of points; has no thickness as a dot line with an arrowhead at each end as a shaded, slanted 4-sided figure P B A n T X Y Z use a capital letter use the letters of two points on the line or a lowercase script letter use three letters of non-collinear points on the plane or an uppercase script letter has neither shape nor size there is exactly one line through any two points there is exactly one plane through any three non-collinear points Point P Plane XYZ Plane XZY Plane YXZ Plane YZX Plane ZXY Plane ZYX Plane T

8. Examples: What is the intersection of planes A and B ? Give at least 2 names for a line containing point D. W Give at least 2 names for a plane containing point C. Y G K Name 3 collinear points. How many planes are shown in the figure? D C B A G P Are points G, A, B, & E coplaner? E t At what point do lines t and r intersect ? Where does line t and plane M intersect ? Which line is contained within plane M ? Section 1.1: Examples

9. #23 page 10: Draw & label a figure for the following relationship: Points Z(4, 2), R(-4, 2), and S are collinear, but points Q, Z, R, and S are not?

10. Section 1.2: Linear Measure & Precision Objective: We will discover how to measure segments and determine the accuracy of our measurement & how to find missing measurements. (Why are measurements important?) Vocabulary: Precision: Line segment: can be measured has 2 endpoints written by naming the endpoints A B or (line AB) (ray AB) When there is no symbol above the endpoints it means the between A and B ! distance Depends on the smallest unit available on the measuring tool. What is AB ? always remember units!

11. “betweeness of points” Section 1.2 Draw & label A C Place point B between points A and C B X Y Points A, B, & C are now ___________. collinear AB + BC = ______ AC When two segment have the same measure. Congruent : (on the same line) A B AB XY ____ = ____ The symbol we use for congruent : _______ RED tick marks indicate the segments are congruent.

12. Examples: 2. Find XY. 1. What is the length of AC ? 3. Find y and PQ if P is between Q and R; PQ = 2y, QR = 3y + 1, and PR = 21. Draw a figure to represent this information. 2.5 cm Section 1.2: Examples A C B 3 cm 6.5 X Z Y 10 5.5 cm 3.5 units 2y Q R P 3y + 1 y = 20 21 Set up an equation (a relationship!). 2y + 21 = 3y + 1 Solve and then go back to the question to reconfirm what you are solving for ! PQ = 40 4. Find x, TU, & SU if T is between S and U. ST = 7x, SU = 10x - 3, and ST = 28. 7x S U T 10x - 3 x = 4SU = 37 TU = 9

13. Section 1.3: Distance & Midpoints Objective: We will discover how to find the distance between two points and how to find the midpoint of a segment. Using a Number Line Pythagorean Theorem -35 QR = _____ QR or 60 The length of a segment can never be ________! negative *actual distance What if the line segment is not completely horizontal or vertical ? 25 0 S 10 SR = _____ QS = _____ 45 15 50 0 0 b a c hypotenuse a 2 + b 2 = c 2 * absolute value is the distance a number is from zero on the number line.

14. Length of one side a = ____ Section 1.3: Distance & Midpoints b = ____ Monument: Starbucks to my house (I 25: y-axis) a 2 + b 2 = c 2 6 3 6 2 + 3 2 = c 2 36 + 9 = c 2 45 = c 2 The distance between two points with coordinates (x 1, y 1 ) and (x 2, y 2 ) is given by: (4, 5) b a c (-2, 2) c 2 = a 2 + b 2 Length of other side subscript tells which point North Baptist Rd. as the crow flies (using a drone !) in your car = 9 miles

15. Find the distance between: Section 1.3: Distance & Midpoints a 2 + b 2 = c 2 3 2 + 4 2 = c 2 9 + 16 = c 2 25 = c 2 (-2, -1) b a c (-5, 3) Try these 5 = c d = 5 Pythag. Thm. Distance Formula Doesn’t matter which point is #1 or #2; just be consistent! L(3, 5) and R(1, -1) Recap: what is the benefit of using the Distance Formula? You don’t need to graph! L(3, 5) and M(7, 9)

16. Objective: How do we locate the middle of a segment? Midpoint Theorem Midpoint: = What is the midpoint of QS ? example: Midpoint of SR: Find the midpoint of (-1, 1) and (5, -3) Coordinates of the midpoint of a segment whose endpoints are (x 1, y 1 ) and (x 2, y 2 ) are: Section 1.3: Distance & Midpoints Remember, this is a location, not a distance! -7 Q R 7 0 S 3 =5= (5, 4) (1, 2) avg. y avg. x (3, 3) (-1, 1) (5, -3) (3, 3) (2, -1) Recap: To find the midpoint coordinates we are essentially taking the ___________ of the coordinates! The midpoint __________ the segment (cuts it in half)! average bisects

17. Example: Separate the coordinates: Section 1.3: Distance & Midpoints E(-6, 4) F(-5, -3) S(-1, 5) (-7, 11) (2, 7) D ( ?, ? ) Multiply both sides by 2 to eliminate the fraction T(-4, 3) R(2, 7)

18. Section 1.4: Angle Measure Objective: We will discover how to measure segments and classify angles Also, we will be able to identify and use congruent angles as well as the bisector of an angle. Vocabulary: Naming an angle: order matters (the vertex letter must be in the center) name 4 possibilities How big is a degree ? extends indefinitely in one direction has 1 endpoint written by naming the endpoint first A B Ray not opposite ray to 360 degree s in a circle Angle formed by 2 non-collinear rays X Y Z Vertex (and is collinear) common endpoint ______ of angle _______ of angle A B C interior exterior 4 Sides: and

19. Section 1.4: Angle Measure What are the measures of angles 6 and 8 ? * watch the order of the points (the vertex is in the center) How do we write this with the points in this problem ? ___________________ using a protractor (place center at vertex and align side with the 0 line): Angles with the same measure are called: M R N Angle Bisector: _________________ _________________ __________ A B C congruent angles a ray that divides example: Name Measure Model right angle acute angle obtuse angle A B C Q P 6 8 angles. D a. Name all angles that have point W as the vertex. 1 3 2 5 4 X WV Z Y

20. Section 1.5: Angle Relationships Angle Pairs Adjacent Angles (next to) Vertical Angles (across) Linear Pair (form a line) Descriptions Example(s) Two angles that lie in the same plane B Have a common vertex Have a common side Two non adjacent angles A C D B A C D Formed by two intersecting lines Pair of adjacent angles Their non-common sides are opposite rays B A C D E B C D E Example: X V W Y Z Name 2 obtuse vertical angles T Name 2 acute vertical angles Name 2 acute adjacent angles What do all the angles sum up to ? ______ No common interior points

21. Section 1.5: Angle Relationships Angle Relationships Complementary Angles Supplementary Angles DescriptionsExample(s) Two angles whose measures have a sum of 90 Perpendicular lines intersect to form 4 right angles. Two angles whose measures have a sum of 180 2 Z X Y W H G E F 1 P R Q X Y Z N M

22. N K O I J H #6; page 41: The measure of the supplement of an angle is 60 less than three times the measure of the complement of the angle. Find the measure of the angle, its complement, and its supplement. the angle = x the complement =90 - x the supplement =180 - x = - 60 3(90 – x) x = 15 x = 84 x + (x + 12) = 180 y = 96 Unless the diagram is marked, or the information is given, never ________ anything. assume M L x = 7 Section 1.5: Angle Relationships

23. Polygon _______________________________ A closed figure whose sides are all segments Non-examples: examples: quadrilateral no curves Convex polygon Concave polygon Section 1.6: Polygons None of the lines containing a side pass through the polygon Lines containing the sides pass through the interior Objective: We will discover how to identify and classify polygons no crossed lines not “open” shapes triangle pentagon hexagon heptagon octagon nonagon n - gon 6 sides 7 sides 8 sides 9 sides decagon 10 hendecagon 11 dodecagon 12

24. Regular Polygon P = a + b + c Perimeter Section 1.6: Polygons The sum of the lengths of all sides Example: square equilateral triangle regular pentagon P = w + l + w + l P = s + s + s + s b a c s s s s ww l l P = 4s P = 2w + 2 l Lets find the perimeter of a triangle Choose 3 pts. Write the coords. Draw the triangle Find the lengths (formula?) Determine the perimeter