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Geometry: Section 1.2 _______________________________________________ Start Thinking: How would you describe a point, line and plane. Remember, with only your description I should be able to draw the figure. Undefined Terms: Name Description Diagram and Notation Point Line Plane Collinear Coplanar PRACTICE: 1a.) Rename Line M in 2 ways. 2a.) Name Plane N in 2 ways. 3a.) Name a point that is coplanar with A B C. 4a.) Name a point that is collinear with E and A 1b.)What point is coplanarr with C, B and G? 2b.) How many points are coplanar with C? 3b.) What points are coplanar with B, D and G? Defined Terms: Name Description Diagram and Notation Segment Ray Opposite rays PRACTICE: 1.) Name the Largest Segment in two different ways. 2.) Name in two different ways. 3.) What two segments create 4.)What is an opposite ray with 5.)What is the overlap of and ? ? 6.) Do rays with a common end point always create a line? Draw a diagram. Definition: Postulate/ Axiom: _______________________________________________________________________________ ______________________________________________________________________________________________. Postulate 1.1: Though any two points there exists exactly _______________ line(s). Postulate 1.2: If two lines intersect, they intersect at _______________________________. Postulate 1.3: If two planes intersect, then they intersect at __________________________. Postulate 1.4: Through three noncollinear points there is exactly one ________________________. PRACTICE: 1.) and intersect at ____________. 2.) Plane HEF and Plane DCE intersect at ______________. 3.) Plane HGB and intersect at _________________. 4.) Plane FEC intersects Plane ADB at ____________________. Sometimes, Always, Never 1.) Three points are ________________________ coplanar. 2.) Three points are _____________________ collinear. 3.) Two intersecting lines are ________________________ coplanar. Determine the location where the following two lines intersect. 1.) Y = -3x + 8 and -2x + y = -2 2.)Y = 3 and y = -2/3x + 7 Reasoning: Though there are not actually any truly real world examples of the geometric figures we have discussed, give examples of figures that represent the following. Explain why they can’t actually be classified as the figure. 1.)Point: 2.) Line 3.)Plane 3.) Ray Geometry: Section 1.3 ***_______________________________________________*** Vocabulary: Definition Example and Notation Line Segment: Length of a Line Segment: Postulate 1.6: If three points A, B and C are collinear and C is between A and B then __________________________. Practice: 1.) contains point J. If MN= 59 and MJ=8x-14 and JN= 4x +1 then what does x equal? 2.) contains point G. If KG= 3x+2 and GM=2x-3 and KM= 4x +10 then what is KM? Definition Examples and Notation Congruent: Equal: Midpoint: Segment Bisector: 3.) contains point J such that 4.) P is the midpoint of . If MJ= 6x-7 and JN=5x+1 then what does x equal? . If M is at -12 and N is at 2 where is P? 4.) has M as its midpoint. If T is located at -4 on the number line and U is to the right of T, and TM = 8x+11 and MU=12x – 1, then where is U located? 6.) intersects at its midpoint K. If AK= 3x+10 and KB=46 then what does x equal? intersects . They both bisect on another and intersect at a 90 degree angle at P. If KP=3 and MP = 3x+1 and PN=4x then what is the distance between M and K? 7.) Relate two objects to one another using the word “congruent” and then change the sentence to describe an aspect of them using the word equal. Geometry: Section 1.4 _______________________________________________ Vocabulary: Definition Example and Notation Angle: Measure of an Angle: Ex1: Name ALL the angles in the diagrams. Term Example Acute angle: An angle that measure between _________________________ Right angle: An angle that measures ________________________ Obtuse angle: An angle that measures _____________________________ Straight angle: An angle that measures ____________________ Congruent angles: Ex2: Identify congruent angles Ex3: In the diagram below, bisects LKN and mLKM = 78°. Find mLKN. If mP = 120°, what is mN? ______ If mL = 20°, what is mM?____ Ex3: Identify all pairs of congruent angles. FINDING ANGLE MEASURES Ex5: Use the diagram to find the measure of the indicated angle. Then classify the angle. ANGLE ADDITION POSTULATE: If P is in the interior of RST, Ex6: Given that mGFJ = 155°, find mGFH a right angle, and mHFJ. mRST = ________ + ________. EX 7: Practice B: Given that VRS is find mVRT . More examples EX: <ABC is a straight angle. Point D is in the interior of the angle. If m<ABD is 3x -7 and m<DBC = 2x+2 then what is m<ABD in degrees? EX: is bisected by . If = 8x -2 and = 6x+22 then what is EX: Easier: John has a weird clock where the hour hand only moves at the top of the hour and does not move as the hour is passing. What is the measure of the angle between the hour hand and the minute hand at 12:20? Harder: The hour hand of Danny’s clock does move as the minute hand moves. What would be the angle between the hour hand and the minute hand at 12:30? Geometry Lecture 1.5 ______________________________ Word Bank: Segment Point Congruent Ray Vertex Equal Line Coplanar Opposite Rays Plane Collinear Right Angle DIAGRAM Adjacent angles ________________________________________________________________ _____________________________________________________________________________ Vertical angles _________________________________________________________________ _____________________________________________________________________________ Complementary angles __________________________________________________________ ______________________________________________________________________________ Supplementary angles___________________________________________________________ ______________________________________________________________________________ Linear pair_____________________________________________________________________ ______________________________________________________________________________ Angle Bisector __________________________________________________________________ _______________________________________________________________________________ Ex 1: Identify an example of the following types of angles/ objects. a.) Adjacent Angles b.)Vertical Angles c.)Supplementary Angles d.)Complementary Angles e.)Angle Bisector f.) Linear Pair Ex 2: Draw a figure with the following characteristics. 1.)∠ADB and ∠BDC are adjacent bisects ∠ADC ∠EDF and ∠BDC are vertical angles. EX 3: ∠BMC and ∠CMD are complementary. If m∠BMC = 6x +1 and m∠CMD = 4x+11. Solve for x. EX 4: ∠ABC and ∠CBD are Supplementary. If m∠ABC = 20 and m∠CBD = 15x+10 . Solve for x. EX 5: bisects ∠ADC. If m∠ADB = 8x – 2 and ∠BDC = 6x + 38 then what is m∠ADC Vertical Angles Vertical Angles are ______________________________. Example: 1.) 2.) Challenge Practice: 1.) : bisects ∠AFC . Solve for x. 2.) Find the values of x and y shown in the diagram. Geometry: Section 1.7 _____________________________________________________ Vocabulary: Length of a Line Segment: _________________________________________________________________________ THE DISTANCE FORMULA. The distance d between two points A(x1, y1)and B(x2, y2) is: ______________________________ ***Developing the Distance formula*** Example: A(2, 9) B(-3, 10) AB= ? Ex 2: Given points J(6,6) and K(2,-4), find JK. Ex 1: Luisa takes the subway from Oak Station to Jackson Station each morning. Oak Station is 1mi west and 2mi south of City Plaza. Jackson Station is 2mi east and 4mi north of City Plaza. City Plaza is at (0,0). Using the distance formula, determine how far Luisa travels by subway. Vocabulary: Midpoint of a segment: ____________________________________________________________________________ THE MIDPOINT FORMULA: the formula for the location of the midpoint is Ex 2: Given points J(6,6) and K(2,-4), find midpoint of Ex 3: Use the midpoint formula to find the endpoint C. If midpoint of is M(3,4) and endpoint is A(1,6). Find the coordinate of endpoint C. : Ex 4: The Line Segment AB has J as its midpoint. If A (1, 5) and J (7, -7) what is the location of B? Ex5: has its midpoint at R. If we have the following locations, G(1,3) and R (4, 7) and RH = 2x+1 then what does x equal? EX 6: Ex 7: has a midpoint M. has its midpoint at A. Determine the location of A if J (2,1) and K(12,7). has a midpoint at L. If M is located at (0,0) and N is at (a,b) then determine the expression that describes the distance between MN as well as the location of L. EX 8: Flagstaff has decided to put a tightrope over the buildings in downtown. If they plan to have one end at the corner of RT 66 and Humphrys and the other end is at san Francisco and Dale and a city block is 1000ft, then how long will the tight rope be? If the tightrope walker walks at a rate of one mile per hour, then how long will it take to walk the whole length to the nearest second? Geometry Section 1.8 _________________________________________________ START THINKING: How would you describe area and perimeter? When might you need each? Formulas: Figure Perimeter Square Rectangle Triangle Circle Parallelogram Let’s think: Do you need to know area or perimeter to know how much to buy of: a. b. c. d. Wallpaper for a bedroom Fencing for a backyard Crown molding for a ceiling Paint for a basement floor Area . Practice: Calculate the area and perimeter of the following. 1.) 3.) 2.) 4.) Working Backwards 1.) What is the length of x given the figure below and that the area is 528 cm. 2.) Given an area, find the value of x. A = 399 in.2 5.) Determine the area of the following composite figures. a.) b.) Not to Scale KM More Challenging Problems: 1.) The base of a triangle is four times its height. The area of the triangle is 50 square inches. Find the base and height. 2.) The length of a rectangle is 4 less than twice the width. If the perimeter is 48 in then what are dimensions of the rectangle?