What is Geometry?

1. What is Geometry? Make 2 lists with your table: What geometry content are you confident about? What geometry content are you nervous about?

2. Agenda • Go over exams • The difference between conjectures (hypotheses, assumptions) and proofs • Some basic assumptions • Some basic terms and ideas • Exploration 8.1 Part 4 • Assign homework

3. Geometry • Points, lines, planes, angles • Curves, Polygons, circles, polyhedra, solids • Congruence, similarity • Reflections, rotations, translations, tessellations • Distance, Perimenter, Area, Surface Area, Volume, Temperature, Time, Mass, Liquid vs. Solid Capacity • Above, below, beside, left, right, upside-down, perception, perspective

4. Geometry • Notice that nowhere on the previous list is the word “proof.” • An example shows that something is true at least one time. • A counter-example shows that something is not true at least one time. • A proof shows that something is true (or not true) all of the time. • This is what all of mathematics is based upon, not just geometry.

5. Geometry • If we believe something to be true… • Assumption/Axiom/Postulate • Conjecture/Hypothesis • Definition • If we can prove something to be true… • Theorem • Property These words are not interchangeable!!

6. Some words are hard to define • Describe the color red to someone. • Can you define the color red? • Can you define or describe the color red to someone who is blind?

7. Some words are hard to define • Point: a dot, a location on the number line or coordinate plane or in space or time, a pixel • Line: straight, never ends, made up of infinite points, has at least 2 points • Plane: a flat surface that has no depth that is made up of at least 3 non-collinear points. • We say these terms are undefined.

8. With undefined terms, we can describe and define… • Segment • Ray • Angle • Collinear points • Coplanar lines • Intersecting lines • Skew lines • Concurrent lines

9. D • • B • C A p Symbols • We use some common notation. • Line, line segment, ray: 2 capital lettersAB AB AB BA or t • Point: 1 capital letter D • Plane: 1 upper or lower case letter Pp • Angle: 3 capital letters with the vertex in the center, or the vertex letter or number  ACD

10. • • • • • D F G E B A C Try these Name 3 rays. Name 4 different angles. Name 2 supplementaryangles. Name a pair of verticalangles. Name a pair of adjacent angles. Name 3 collinear points.

11. • • E H F • D • G • C B • A Try these Name 2 right angles. Name 2 complementaryangles. Name 2 supplementaryangles. Name 2 vertical angles. True or false: AD = DA. If m  EDH = 48˚, find m  GDC.

12. Euclid’s Postulates • 1. A straight line segment can be drawn joining any two points. • 2. Any straight line segment can be extended indefinitely in a straight line. • 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. • 4. All right angles are congruent.

13. A C Euclid’s Fifth Postulate • 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

14. 63˚ u m n t Try these • Assume lines l, m, nare parallel. • Copy this diagram. • Find the value of each angle. l

15. Exploration 8.1 • Exploration 8.1 • Part 4 #1a - e--copy or cut and tape the figures so that the groups are easy to distinguish.

16. Warm Up • Sketch a pair of angles whose intersection is:a. exactly two points.b. exactly three points.c. exactly four points. • If it is not possible to sketch one or more of these figures, explain why.

17. Agenda • Go over warm up. • Go over Exploration 8.1. • Play with protractors. • Exploration 8.6. • More practice problems. • Assign homework.

18. How did you group the polygons? • Compare your answers. Were they all the same or different? Write a few sentences to describe your group’s findings.

19. Use Geoboards • On your geoboard, copy the given segment. • Then, create a parallel line and a perpendicular line if possible. Describe how you know your answer is correct.

20. Use a protractor to check

21. Exploration 8.6 • Do part 1 using the pattern blocks--make sure your justifications make sense. • You may not use a protractor for part 1. • Once your group agrees on the angle measures for each polygon, trace each onto your paper, and measure the angles with a protractor. • List 5 or more reasons for your protractor measures to be slightly “off”.

22. 6 7 3 5 4 2 n 1 m More practice problems • Given m // n. • T or F:  7 and  4are vertical. • T or F:  1   4 • T or F:  2   3 • T or F: m  7 + m  6 = m  1 • T or F: m  7 = m  6 + m  5 • If m  5 = 35˚, find all the angles you can.

23. More practice problems • Think of an analog clock. • A. How many times a day will the minute hand be directly on top of the hour hand? • B. What times could it be when the two hands make a 90˚ angle? • C. What angle do the hands make at 7:00? 3:30? 2:06?

24. More practice problems • Sketch four lines such that three are concurrent with each other and two are parallel to each other.

25. True or False • If 2 distinct lines do not intersect, then they are parallel. • If 2 lines are parallel, then a single plane contains them. • If 2 lines intersect, then a single plane contains them. • If a line is perpendicular to a plane, then it is perpendicular to all lines in that plane. • If 3 lines are concurrent, then they are also coplanar.