Math 112b StudyGuide! (condensed and weekly notes together)
Math 112b StudyGuide! (condensed and weekly notes together) Math 112b Sec 004
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This 12 page Study Guide was uploaded by amber weiss on Friday February 26, 2016. The Study Guide belongs to Math 112b Sec 004 at Southern Illinois University Edwardsville taught by cheryl eames in Spring 2016. Since its upload, it has received 66 views. For similar materials see elementary mathmatics in Mathmatics at Southern Illinois University Edwardsville.
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Date Created: 02/26/16
Chapter 9 Section 3 SNOW DAY PRESENTATION 02/24/2016 Chapter 9 Section 3 Terminology: Space- is an undefined term (like point, line and plane) Dihedral angle- angle formed when 2 planes meet Parallel- if the line doesn’t intersect the plane Perpendicular- if the line if intersects and is perpendicular to EVERY point in plane Two planes are parallel if they do not intersect Two planes are perpendicular if dihedral angle measures 90 degrees Polyhedral- figures in space whose sides are polygonal regions Faces- polygonal regions or sides Edges- 2 faces intersect Vertices- 3 or more faces intersect Tetrahedron- 4 triangles for faces Cube- 6 square faces Octahedron- with 8 triangular faces Dodecahedron- 12 pentagon faces Icosahedron- 20 triangular faces Semiregular polyhedral- polyhedral whose faces are 2 or more regular polygons with the same arrangement of polygons around each vertex Pyramid- polyhedron with a polygon base and triangles as sides Right pyramid- pyramid whose sides are all isosceles triangles Prism- polyhedron with 2 parallel and congruent bases Cone- circular base and a lateral surface that slopes to the vertex Cylinder- 2 parallel circular bases of same size and a lateral surface that joins to bases Net- 2 dimensional pattern for a three dimensional figure Name Faces Vertices edges Triangular prism 5 6 9 Triangular pyramid 4 4 6 Cube 6 8 12 Square pyramid 5 5 8 Hexagonal prism 8 12 18 Hexagonal pyramid 7 7 12 octahedron 8 6 12 Euler’s Formula- FACES + VERTICES = (E+2) add 2 more edges The worked out solutions are already given to us on blackboard in the notes!!! DOWLOAD THEM Chapter 9 Section 4 02/29/2016 Chapter 9 Section 4 Terminology: Reflection symmetry- when it is folded in half on a line of symmetry Line of symmetry- 2 images are congruent and line up on top of each other (mirror/reflection) Rotation symmetry- if it can be turned about a point less than 360 degrees and coincide with itself Math 112b Review Chapter 9 Sections 1-4 Chapter 9 Section 1 Terminology: Geometry- lines, angles, surfaces and solids Undefined terms- point, line, plane Defined terms- might use undefined terms to define other things Axioms- assume to be true without proving them Theorems- things we prove using undefined terms, definitions, axioms and other theorems that have been proved. Collinear- points are called if same if share same line Line segment- consists of 2 points (endpoints) and all points between are collinear Segment- endpoint and all collinear points between them Line- goes on forever in 2 directions Ray- point on line with all points that lie on 1 side of point Acute angle- less than 90 degrees Right angle- exactly 90 degrees Obtuse angle- more than 90 degrees Straight angle- 180 degrees exactly Reflex angle- greater than 180 degrees Vertical angles- opposite angles formed by 2 intersecting lines Adjacent angles- 2 angles vertex and common sides do NOT overlap Complementary- 2 angles sum of 90 degrees Supplementary- 2 angles sum is 180 degrees Perpendicular lines- intersect create right angles Parallel- do not intersect coplanar Corresponding angles- occupy SAME position relative to transversal and original lines (are congruent) Alternate exterior angles- occupy opposite sides of transversal on exterior of original lines (congruent) Alternate interior angles- occupy opposite sides of transversal on interior of original lines (congruent) Curve- set of points which can be connected by a single smooth, continuous motion (drawn without lifting pen) Simple curve- curve can be drawn without lifting pencil and WITHOUT intersecting itself (maybe start and end point) Closed curve- curve but has to touch bach to start (start stop at same point) but doesn’t have to be simple Simple closed curve- starts and stops same point and cannot intersect itself Plane origin- union of simple closed curve and its interior Convex- 2 points in a region Nonconvex (concave)- 2 points outside of region Circle- set of all points in a plane that are equidistant from a fixed point Tangent- touch circle at one point (line) Secant- cut through the circle (line) Chord- stretch across circle (segment) Diameter- line within circle Radius- half of the diameter For more of this, download the chapter 9 section 1 notes! This is just vocab and the main parts! NO EXAMPLES Chapter 9 Section 1 Math 112 B 02/17/2016 Geometry: - Lines - Surfaces - Angles - Solids Based on 4 things: 1. Undefined terms: point, line, plane 2. Defined terms: definitions; might use undefined terms to define other things 3. Axioms: assume to be true without proving them 4. Theorems: things we prove using undefined terms, definitions, axioms, and other theorems that have been proved 1) What are some ral-world items that could use models for points, lines, planes? When using these items, what would you need to clarify or emphasize for students? - Points: a dot; in math, points have NO size - Lines: chalkboard rail, number lone; in math, lines extend forever in both directions - Planes: desk top, floor; in math, goes on forever in all directions 2) What does it take to fix a line? We need two points - - Collinear: if points share the same line - Example: use 3 points, - this would be collinear this would NOT be collinear, because the points do not fall on the same line 3) Are every 2 points collinear? yes 4) Are 3 points collinear? No, sometime yes if fall on same line A line segment consists of 2 points (endpoints) and all points between are collinear. A B C - B bisects line segments into 2 congruent parts 4.Write congruence statement for 2 of above. AB is congruent BC 5. What is another segment pictures above. Line AC 6. Is line AB sane as line BA? Yes, same - Segment: endpoint and all collinear points between them - Line AB = Line BA, order doesn’t matter because direction doesn’t matter - Line: goes on forever in 2 directions - Ray: point on line with all points that lie on ONE side of point A B C 7. Give 2 names for the line above, BA, CB, AC, CA, AB, BC (these have an arrow on top representing an aray) 8. Give another name for Ray AB. RAY AC (again has an arrow on top of AC) 9. Is ray AB same as Ray BA? No, they are not the same because they are going opposite directions from each other An angle is a union between 2 arrays (sides of angle) with common endpoint (vertex) A name with 1 letter: vertex name with 3 letters: vertex MUST BE in B middle C 10. Give 2 names for the angle above. <B, <ABC, <CBA - Acute angle: less than 90 degrees - Right Angle: exactly 90 degrees - Obtuse angle: more than 90 degrees - Straight angle: exactly 180 degrees - Reflex angle: greater than 180 degrees - Vertical angle: opposite angles formed by 2 intersecting lines - Complementary: 2 angles sum of 90 degrees A way to remember, think of it as weight, if you lose (less) weight, than it is a compliment someone gives you. - Supplementary: 2 angles sum is 180 degrees People take supplements for MORE of something, more than 90. - Adjacent: 2 angles vertex and common side do not overlap For the pictures of the angles, the answers are: a) Add up to 90, complementary b) Add up to 180, supplementary c) 3 1 4 2 <1 and <2 are vertical <3 and < 4 are vertical <3 and <1 are adjacent <2 and <4 adjacent - Parallel Lines: DO NOT INTERSECT coplanar - Perpendicular lines: intersect to create RIGHT angles - Corresponding angles: occupy same position relative to transversal and original lines )are congruent) - Transversal= line n below - Alternate exterior angles: occupy opposite sides of the transversal on exterior of original lines (they are congruent) - Alternate interior angles: occupy opposite sides of the transversal on interior of original lines (they are congruent) - Corresponding angles: a. m<1 congruent m<5 b. M<2 congruent m<6 c. M<3 congruent m<7 d. M<4 congruent m<8 - Our parallel lines are line l and line m - M<1 and m<4 = same angles - M<3 and m<2 = same angle - Alternate exterior: m<1 and m<8 because they occupy the opposite sides of transversal - Alternate interior: m<3 and m<6 ; m<4 and m<5 n 1 2 l 3 4 5 6 m 7 8 11. list angles congruent to angle 1 ______4,5,8________ 12. what type of angle pair is <2 and <3 _______vertical________ 13. “ ‘’ <6 and <3______alternate interior_________ 14. “ ‘’ <6 and <2_____corresponding____________ 15. “ ‘’ <1 and <8________alternate exterior_________ 16. if measure of <3=45 degrees, what is the measure of angle 5? M<3 and M<5 are 180 degrees because of substitution M<7 is congruent to m<3 corresponding angles M<7+m<5= 180 supplementary M<3+m<5 = 180 180-45= 135 degrees - Curve: set of points which can be connected by a single smooth, continuos motion (draw without lifting pencil) - Simple curve: curve can be drawn without lifting pencil and witout intersection itself (except maybe start/stopping point) - Closed curve: curve but has to touch back to start doesn’t have to be simple (starts and stops at same point) - Simple closed curve: starts and stops same point cannot intersect itself Classify each following curves appropriately curve may belong to more than 1 category. 17. simple: H, A, C, D, F, G 18. closed: H, E,A,C,D,F 19. simple closed: H,A,C,D,F, THE SHAPES ARE IN OUR 9.1 PACKET - Plane region: union of simple closed curve and its interior - Convex: two points in region - Nonconvex (concave): 2 points outside region - The first shape is convex cause it is connected on the incse - The 2 ndshape is a cncave because the connection outside - Circle: set of all points in a plane that are equidistant from a fixed point (center in plane) - Tangent: touch the circle at ONE point (line) - Secant: cut throught the circle (line) - Chord: stretch across the circle (segment) QUESTION: what is the longest possible chord? Diameter Math 112b Chapter 9 Section 2 02/22/2016 Upcoming: a) Quiz #3 on Wednesday February 24, 2016 b) Comp Quiz #2 on Monday February 29, 2016 c) Test #2 on Wednesday March 2, 2016 1. Describe how you would help students discover the sum of measures of the 3 angles in a triangle. - Measure the angles and add them up (need to know how to use a tool) - Cut, tear the corners of the triangle, stack and see which one is larger ; make another shape, in this case a line. - Rule: a straight line = 180 degrees 2. What must students know before completing number 1? - How to use the tool (protractor) - That a straight line is 180 degrees 3. How can we expand problem 1 to find the sum of measures of the 4 angles in a quadrilateral? - First, we draw a quadrilateral (4 sided figure) - Then, we cut a diagonal from one of the vertexes. - We should be able to see 2 triangles - The black dots represent vertex - the black line represents the diagonal to see the 2 triangles 6 5 2 4 1 3 - M<1+m<2+m<3 = 180 - M<4+ m<5+ m<6 = 180 - M<all = 180+180 = 360 degrees - CONNECT NONADJACENT VERTICES WITH A DIAGANOL - Sum of angles = vertex angle 4. Name # of sides # of triangles Sum of angles Triangle 3 180 1 Quadrilateral 4 360 2 Pentagon 5 540 3 Hexagon 6 720 4 Heptagon 7 900 5 Octagon 8 1080 6 Nonagon 9 7 1260 Decagon 10 8 1440 Dodecagon 12 10 1800 n-gon n (n-2) (n-2) * 180 5. WE SKIPPED NUMBER 5 2 plane figures are congruent if… They are same size and shape Place 1 on other so they coincide Segments are congruent if… Same length Angles are congruent if… Same measure Figures are congruent if… Corresponding sides are congruent Corresponding angles are congruent 6. In each of the following diagrams, explain why they are not regular polygons. a) not congruent sides or angles b) sides are not congruent c) not even a polygon, doesn’t have sides or angles Central angle 7. verte Exterior angle x Name Sum of Vertex angle Central Angle Exterior angle angles Equilateral 180 180/3 = 60 360/3 = 120 180-60=120 Triangle Square 360 360/4 = 90 360/4= 90 180-90=90 Regular Pentagon 540 540/5 = 108 360/5= 72 180-108=72 Regular Hexagon 720 720/6= 120 360/6= 60 180-120=60 Regular Heptagon 900 900/7= 128.6 360/7= 51 180-128.6=51.4 Regular Octagon 1080 1080/8= 135 360/8= 45 180-135=45 Regular Nonagon 1260 1260/9= 140 360/9= 40 180-140=40 Regular Decagon 1440 1440/ 10= 360/10= 36 180-144=36 144 Regular 1620 1620/12= 135 360/12= 30 180-135=45 Dodecagon Regular n-gon (n-2) * 180 Total degrees/ A circle= 360/ 180- vertex number of number of angle straight sides line=180 sides 8. The graph above is #8 9. HOMEWORK 10.HOMEWORK
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