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Name: Geometry (A) Unit 1: Supplemental Packet Supplements 1-1 : Nets and Drawings for Visualizing Geometry A net is a two-dimensional diagram that you can fold to form a three-dimensional figure. A net shows a three-dimensional figure as a folded-out flat surface. 1. If the net at the right folds into the cube beside it, which letters will be on the top and front of the cube? Top _______ Front _______ 2. If the net in problem 1 folds into the cube at the right, which letters will be on the top and right side of the cube. Top _______ Right _______ 3. What is the net for the graham cracker box to the right? Label the net with its dimensions. 4. What is the net for the figure at the right? Label the net with its dimensions. Think: How many surfaces (shapes) does the figure contain? Do all the rectangles have the same dimensions? Is there only one possible net drawing for the figure at the right? If not, draw another. 1 5. What is the net for the figure on the right? Label the net with its dimensions. An isometric drawing shows a corner view of a three-dimensional figure. It allows you to see the top, front, and side of the figure. An orthographic drawing is another way to represent a three-dimensional figure. An orthographic drawing shows three separate views: a top view, a front view and a right-side view. Can you visualize the orthographic drawing for the isometric drawing at the right? Try to draw it. Top view: Front view: Right-side view: 2 How close were you? Additional Practice: 1. The net below folds into a cube. Which letters will be on the top and front of the cube? Top _______ Front _______ 2. What is the net for the cereal box at the right? Label the net with its dimensions. 3. What is the front orthographic drawing for the isometric drawing below? HW 1-1: Practice 1-1 WS 3 Supplements 1-2: Points, Lines, and Planes 1-3: Measuring Segments Important Postulates and Theorems To Remember! Use the diagram below to answer the following. Give two other names for line PQ: Give two other names plane R: Name three points that are collinear: Name four points that are coplanar: Give two other names for line ST: Name a point that is not coplanar with points Q, S, and T: Draw a sketch of the given question then solve. 1. Line RS bisects PQ at point R. Find RQ if PQ = 14 cm. 2. Point T is a midpoint of UV. Find UV if UT = 4 ½ yards. In the diagram, M is the midpoint of the segment. Find the indicated length. 3. Find LN 4. Find MR 4 Supplements 1-4: Measuring Angles 1-5: Exploring Angle Pairs Important Postulates and Theorems To Remember! VERTICAL ANGLES ARE CONGRUENT 1. DEF is a straight angle. What is the mDEC and mCEF? 2. Are BFD and CFD adjacent angles? Explain. 3. Are AFB and EFD are vertical angles? Explain. Additional Examples 1) Name the three angles in the diagram. Why can’t you just call this angle B??????? _______ , or _______ _______ , or _______ _______ , or _______ 5 2) Name the three angles in the diagram. Why can’t you just call this angle G??????? _______ , or _______ _______ , or _______ _______ , or _______ 3) What type of angles do the x and y axis form in the coordinate plane? _________________ 4) Given that m GFJ = 155o, find a) m GFH __________ b) m HFJ __________ 5) Given that VRS is a right angle, find a) m VRT __________ b) m TRS __________ 6) Identify all pairs of congruent angles in the diagram. If m P = 120o, what is m N? _______ _______ _______ _______ m N = ______o 7) WY bisects XWZ, and m XWY = 29o. Find m XWZ _________o 8) Identify all pairs of congruent angles in the diagram. If m B = 135o, what is m A? _______ _______ _______ _______ m A = ______o 6 9) KM bisects LKN and m LKM = 78o. Find m LKN _________o 10) Two angles form a linear pair. The measure of one angle is 4 times the measure of the other. Find the measure of each angle. 11) The exterior sides of two adjacent angles are opposite rays. The measure of one angle is 3 times the measure of the other. Find the measure of each angle. 12) In the diagram: a) All points shown are _________________. b) Points A, B, and C are ________________. c) DBE and EBC are ______________ angles. d) ABC is a ____________ angle. 13) Find m<EGT and m<TGC if EG is perpendicular to CG, and m<EGT = 7x + 2 and m<TGC = 4x. 14) Find the value of each angle. 7 Supplements 1-7: Midpoint and Distance in the Coordinate Plane Find the coordinates of the midpoint of the segment with the given endpoints. 1. L(4, 2) and P(0, 2) 2. G(-2, -8) and H( -3, -12) Using the given point R and the midpoint M, find the other endpoint. 3. R(6, 0), M(0, 2) 4. R( 3, 4), M(3, -2) Horizontal & Vertical Lines Find the length of line segment GF: _________ Find the length of line segment HJ: _________ So, GF _____ HJ Find the length of the line segment. Round to the nearest tenth of a unit. If there are two segments compare the lengths. 5. 6. RS: R(5, 4), S(0, 4) TU: T(-4, -3), U(-1, 1) 8 7. Suppose point P partitions the directed line segment from L to M in the ratio 2 to 1. You can think of LM as being divided into 3 congruent parts so that the length of LP is 2 parts and the length of PM is 1 part. The directed line segment from L to M starts at L and ends at M. 1) Graph L at (-4,1) and M at (5,-5) 2) As you move from L to M, the horizontal distance is __________ (right/left) and the vertical distance is ______(up/down). 3) Divide each of these distances by 3: horizontal / 3 = _______ 4) Multiply each of these distances by 2: horizontal * 2 = _______ vertical / 3 =________ vertical * 2 = _______ 5) Move L(-4, 1) _____ to the(right/left) and _______ (up/down) … where does the point P go? ( , ). REASONING How did you know where to start? _________________________________________________________________ Why did you divide the segment into 3 parts? ____________________________________________________ Why do you make this move 2 times? _____________________________________________________________ How did you know where to put P? ________________________________________________________________ 8. RS is the directed line segment from R(-2,-3) to S(8,2). What are the coordinates of point Q that partitions the segment in the ratio 2 to 3? The directed line segment from R to S starts at ______ and ends at ______. 1) Graph R at (-2,-3) and S at (8, 2) Optional – On Your Own Graph Paper! 2) As you move from R to S, the horizontal distance is __________ (right/left) and the vertical distance is ______(up/down). 3) Divide each of these distances by 5: horizontal / 5 = _______ vertical / 5 =________ 4) Multiply each of these distances by 2: horizontal * 2 = _______ vertical * 2 = _______ 5) Move R(-2, -3) _____ to the(right/left) and _______ (up/down) … where does the point Q go? ( , ). 9 9. Point C lies on the directed line segment from A(5, 16) to B(-1,2) and partitions the segment into a ratio of 1 to 2. What are the coordinates of C? The directed line segment from A to B starts at ______ and ends at ______. 1) Graph A at (5,16) and B at (-1,2) Optional – On Your Own Graph Paper! 2) As you move from A to B, the horizontal distance is __________(right/left) and the vertical distance is ______(up/down). 3) Divide each of these distances by 3: horizontal / 3 = _______ 4) Multiply each of these distances by 1: horizontal * 1 = _______ vertical / 3 =________ vertical * 1 = _______ 5) Move A(5, 16) _____ to the(right/left) and _______ (up/down) … where does the point C go? ( , ). 10. The endpoints of directed line segment XY are X(2,-6) and Y(-6,2). What are the coordinates of point P on XY such that XP is ¾ of the distance from X to Y? The directed line segment from X to Y starts at ______ and ends at ______. 1) Graph X at (2,-6) and Y at (-6, 2) Optional – On Your Own Graph Paper! 2) As you move from X to Y, the horizontal distance is __________ (right/left) and the vertical distance is ______ (up/down). 3) Multiply each of these distances by 3/4: horizontal * 3/4 = _______ vertical * 3/4 = _______ 4) Move X (2, -6) _____ to the(right/left) and _______ (up/down) … where does the point P go? ( , ). Optional Formula to partition any line segment with endpoints (x1, y1) , (x2, y2) into a ratio of m:n: ( (mx2 + nx1)/(m+n) , (my2 + ny1)/(m+n) ) Try this on the problem #9 above: m = ________ n = ________ x1 = ________ x2 = ________ y1 = ________ y2= ________ 10 Supplements 1-8: Perimeter, Circumference and Area (With Review of Polygons) Classifying Polygons Polygon _________________________________________________________________________________________________________ _________________________________________________________________________________________________________ _________________________________________________________________________________________________________ Convex Polygon _________________________________________________________________________________________________ Concave Polygon _________________________________________________________________________________________________ n-gon _________________________________________________________________________________________________ Equilateral Polygon ________________________________________________________________________________ Equiangular Polygon _______________________________________________________________________________ Regular Polygon______________________________________________________________________________________ Names of Polygons Name Sides Name Sides 3 11 4 12 5 n 6 7 8 9 10 11 1. Tell whether the figure is a polygon and whether it is convex or concave. a) b) c) 2. The head of a bolt is shaped like a regular hexagon. The expressions shown represent side lengths of the hexagonal bolt. Find the length of a side. 3. The expressions (4x + 8)o and (5x – 5)o represent the measures of two congruent angles . Find the measure of an angle. 12 1. Find the dimensions of the garden, including the path. 2. What is the circumference and area of the circle in terms of ? What is the circumference and area of the circle to the nearest tenth? 3. What is the area and perimeter of EFG? 4. Graph quadrilateral JKLM with vertices J(-3, -3), K(1, -3), L(1, 4) and M(-3,1). What is the perimeter of JKLM? 13 5. You are designing a poster that will be 3 yds wide and 8 ft high. How much paper do you need to make the poster? Leave your answer in terms of feet. 6. What is the area of the figure at the right? All angles are right angles. 7. Triangle JKL has vertices J(1, 6), K(6, 6) and L(3, 2). Find the approximate perimeter and area of triangle JKL. Round all answers to the nearest tenth. 8. The base of a triangle is 24 feet. Its area is 216 square feet. Find the height of the triangle. HW 1-8: Practice 1-8 WS 14