7.1 Construction Introduction Per Date In pairs, discuss the meanings of the following vocabulary terms. The first two you should attempt to recall from memory, and for the rest you should try to agree on a working definition suggested by the names of each term. You may use words, drawings, and contextual clues to support your definitions. Once you have developed your working definition with evidence, carefully sketch a visual of the vocabulary term. Try to be as exact as possible. Vocabulary Term Definition with Evidence Parallel Lines asic Construction Perpendicular Lines Midpoint of a Line Segment (#VOC) Perpendicular isector (#VOC) ngle isector (#VOC) Reflection: Was it difficult to be accurate with your sketches? What did you use to ensure increased accuracy? What could you use to ensure increased accuracy? Page 1
7.2 Grandma s Hallways Per Date It s time to begin actually building Grandma s house. We re at the point where we need to lay out on the floor the location for the walls. Right now the carpenters are laying out the location for the central hallway and the short hallway opposite its midpoint that leads to the bathroom (as shown on Grandma s house plan). The head carpenter is in charge of this critical step and he s already laid out one wall of the long hallway (see the solid line in the inset below.) ut now he must designate the critical locations for the opposite wall and the connecting short hallway (dotted lines in the inset). elow is a copy of the work order received by the head carpenter from Grandma s Foreman to complete this critical task. Your job is to explain the geometric constructions necessary to completing each task. We must do this to determine what we need to learn to properly lay out Grandma s walls. WORK ORDER #1 1. Goal: Create a short hallway perpendicular to the long hallway, located in such a manner that the line running down the middle of the short hallway bisects the initial wall. Specifications: wall is laid out as indicated by the solid line in the sketch below. Lay out a hallway that will extend perpendicular and opposite the middle of the wall. Note: you can arbitrarily choose the widths of the halls at this point. 2. Goal: Finish the long hallway that uses the original wall as a side. Specifications: Lay out a wall that runs parallel to the original wall (horizontal dotted lines) and perpendicular to the other hallway created. 1. efore proceeding to a step-by-step process for completing this task, explore with a partner how to accurately create this drawing using only patty paper, compass, and straightedge, given one side of the primary hallway as indicated below. You can assume the line below represents the solid line in the inset above. nd that its length is the same as that of the entire main hallway. Note: a straightedge is not a ruler used to measure lengths. Page 2
7.2 Grandma s Hallways Per Date We will now investigate what pieces of this puzzle we can achieve using various tools. 2. Using only patty paper and a straightedge, explore how to perform the following constructions. Determine a way to find the midpoint of this line segment. (Remember, you have a straightedge, but not a ruler.) Determine how to draw a perpendicular line to through point C. C Determine how to draw a line through point C that is parallel to. C Page 3
7.2 Grandma s Hallways Per Date Determine how to draw a perpendicular bisector of. Reflections: I. What construction(s) were easy to accomplish with patty paper and a straightedge? Why? II. What construction(s) were difficult to accomplish with patty paper and a straightedge? Why? III. What tools might make these constructions easier to complete? IV. If you were the Head Carpenter, would patty paper and a straightedge be useful in laying out the lines on Grandma s floor? Why or why not? Page 4
7.2 Grandma s Hallways Per Date 3. We are now going to determine how to find the midpoint and the perpendicular bisector using only a compass and straightedge. Explore with a partner how you might locate the perpendicular bisector with only these two tools. Can you discover a method to construct a perpendicular line through a specific point (not on the line)? What about if that point is on the line? Try to draw some examples of both constructions. Page 5
7.2 Grandma s Hallways Per Date Here is a sketch that suggests a method for constructing the perpendicular bisector using only a compass and a straightedge. Were your ideas close? Note: in this sketch we ve used the length of the line segment KLas our radius. It turns out that you can actually use any radius greater than half the length of the segment, and you don t need to draw the entire circles, just their intersects.!### " Thus, since every point on the perpendicular bisector MN simply corresponds to a different choice of the radius used, this implies that every point on the perpendicular bisector of line segment KL is equidistant from the endpoints K and L. This is the Perpendicular isector Theorem, which we prove in L10 Perpendicular Lines and Triangles. On the next page we show how to construct perpendicular lines that pass through a given point that is located either on or off the line. Time permitting, see if you can discover how to do these on your own, without looking at the next page first. Page 6
7.2 Grandma s Hallways Per Date Constructing a Perpendicular Line through a point not on the line. Constructing a Perpendicular Line through a point on the line Page 7
7.2 Grandma s Hallways Per Date 4. Complete the hallway markings using the compass and straightedge techniques you ve learned. Follow the Hints below. athroom *Hints: Main Hallway Wall i. First draw the center of the side hallway by drawing the perpendicular bisector that passes through the midpoint of the main hallway. ii. iii. iv. Choose a width, using your compass, which is half the width you would like to use for your hallways and mark two points on the main hallway, equidistant from the midpoint of the main hallway. This represents the width of the short hallway. Draw one side of the short hallway by creating the line perpendicular to the main hallway and passing through one of the points you just located. Draw the other side of the short hallway the same way. v. Draw the other wall of the main hallway using the same width and perpendicular lines. Page 8
7.2 Grandma s Hallways Per Date Grandma s Hallway Constructions Rubric (circle box you feel you deserve for each category) Line Quality 8 Clear, consistent lines meeting all cross points Neatness 8 No unnecessary or incorrect marks ccuracy of ngle Measures 12 Perfect angle measurements 6 few wavy lines, line wavers off cross points slightly 6 One or two unnecessary or incorrect marks 9 One or two angles with 1-2 measurement errors 4 Inconsistent lines, lines are wavy 4 Three or four unnecessary or incorrect marks 6 Three or four angles with 1-2 measurement errors 2 Lines are not straight, did not use a ruler or straight edge 2 More than four unnecessary or incorrect marks 3 More than four angles with 1-2 measurement errors Student Comments to Teacher: Grandma s Hallway Constructions Rubric (teacher-graded rubric) Line Quality 8 Clear, consistent lines meeting all cross points Neatness 8 No unnecessary or incorrect marks ccuracy of ngle Measures 12 Perfect angle measurements 6 few wavy lines, line wavers off cross points slightly 6 One or two unnecessary or incorrect marks 9 One or two angles with 1-2 measurement errors 4 Inconsistent lines, lines are wavy 4 Three or four unnecessary or incorrect marks 6 Three or four angles with 1-2 measurement errors 2 Lines are not straight, did not use a ruler or straight edge 2 More than four unnecessary or incorrect marks 3 More than four angles with 1-2 measurement errors Teacher Comments to Student: Page 9
7.3 Construction Homework Per Date Practice: For each segment, construct a perpendicular line through the given point using only a compass and straightedge. Leave your tick marks to show your work. 1) 2) 3) C 4) Using only a compass and straightedge, construct 2 different size rectangles in the space below. Page 10
7.4 Grandma s Hallway Continued Per Date uilding houses always seems rather simple to young apprentices; after all, it mostly consists of measuring and cutting wood, and pounding nails. fter only a few days on the job an apprentice often thinks s/he can do the work as well as anyone. In the middle of laying out Grandma s hallways the Head Carpenter took his lunch break. He had only located the midpoint of the first wall when he took the break. Scotty, a young apprentice on the job for less than a week decided to stay behind and impress the boss with his ability to continue laying out the main hallway. His results are shown below (dotted line that appears to be parallel to the solid line). Scotty doesn t have any tools, being only an apprentice, and the Head Carpenter took all the tools except his protractor. There is also another faint dotted line on the floor that was left by an abandoned layout. How can Scotty use only the protractor to check the accuracy of his work? Hint: recall what you ve learned about parallel lines cut by a transversal. Reflection: How did Scotty do? Explain what information is important in this situation. Given two intersecting lines, say l and m, what could you check to see if the lines are parallel? Note: In reality a Carpenter will simply measure the desired width of the hallway at either end and connect the dots. Nevertheless, the fact presented herein, and many others as well, are well known and often used by the best master builders if not the apprentices. Page 11
7.5 Grandma s Garden Plot Per Date Now that Grandma s hallway is under construction, she has more time to work in her garden. Grandma would like to grow some rose bushes and hibiscus plants in an obtuse corner of her lot. WORK ORDER #2 1. Goal: Design the layout for a flower garden with equal area for roses and hibiscus flowers. Specifications: Divide the non-perpendicular corner of the lot into two congruent sections for roses and hibiscus flowers by drawing an angle bisector as indicated in the sketch. This is a blown up version of the flower garden located in the lower left-hand corner of Granma s Plot Plan. 2. Goal: Enclose the areas for the Roses and Hibiscus in triangular plots that have equal area. Specifications: Pick a point on the dotted line that represents a point on your desired boundary for your growing plot. Draw equal length lines from this point to the two solid rays. Explore with a partner how you might bisect using only a compass and straightedge. Hint: think about how you used the compass and straightedge in earlier constructions. Page 12
7.5 Grandma s Garden Plot Per Date efore proceeding to using a compass, try answering the following using only patty paper and a straightedge: 1. Determine a way to copy C. C Determine a way to bisect C. C Reflection: i. Which construction was easiest to accomplish with patty paper and a straightedge? Why? ii. If you were asked to map out Grandma s Garden Plot in a real situation, would patty paper and a straightedge be useful? Why or why not? Now let s try it with a compass and straightedge: Page 13
7.5 Grandma s Garden Plot Per Date 2. Determine a way to copy C using only a straightedge and compass. Hint: think about your earlier compass and straightedge constructions. compass can be used to capture distances and locate points, and a straightedge can be used to connect two points. The base of the angle is provided to the right, along with a single arbitrarily chosen point. C Determine a way to bisect C using only a straightedge and compass. Hint: think about your earlier compass and straightedge constructions. C Page 14
7.5 Grandma s Garden Plot Per Date Practice: efore you complete your task to help Grandma and her Garden Plot, practice copying and bisecting angles. 3. isect the following angles on the left, then copy the original angle to the right. C D E F G H I Page 15
7.5 Grandma s Garden Plot Per Date 4. isect the following angles and the left, then copy the bisected angle on the right. J K L M O N 5. Kimo is practicing to help Grandma with her Garden Plot. He believes he has correctly copied and bisected the angle on the left below, with his results to the right. How did Kimo do? Use a straightedge and compass to check Kimo s work. Page 16
7.5 Grandma s Garden Plot Per Date Here is the obtuse corner in Grandma s property lot. Use the information that you are learning (ahead and after this page) to complete Work Order #2. Page 17
7.6 Grandma s Garden Plot Homework Per Date Homework: Complete the following geometric construction tasks with a compass and straightedge. Check your work with a protractor. Remember to leave your marks to demonstrate your accurate construction actions. 1. isect C with a line segment. Name this line segment D. Extend line segment D 4 inches from point (use a ruler for this). 2. Draw a perpendicular bisector from D towards the top-left of the paper. Extend this segment 2 inches from D. Name this line segment EF with point E on D. 3. Draw a parallel line segment to D, named GH and passing through point F. This drawing will be graded for accuracy with respect to following directions and measurements. C Page 18
7.7 Parallel Lines Proofs Per Date Let s now prove some of our previous conjectures. 1. While construction ensues on Grandma s house, her grandson Koa teaches her about what he is learning in class this year. He shows Grandma this image and makes the following conclusion: If j k, then. Grandma looks over the image and adds another line. She asks Koa, ased on your conclusions, does this mean that 2 is congruent to 3? Koa looks carefully at the image and responds. What should Koa s response to Grandma be? What evidence could he have used to support his conclusion? Page 19
7.7 Parallel Lines Proofs Per Date Explore with a partner the following, using patty paper and/or rigid transformations: Which pairs of angles in the above diagram do you need to know are congruent before you can!##"!##" conclude that the lines and DE are parallel? Each answer should be a single pair of angles. Why do you think each single pair implies the desired result? e as specific as possible. The following theorem is the Converse to the Parallel Postulate. Rather than prove this theorem at this time, which is essentially based on the fact that the three angles in a triangle sum to180, we will assume it to be true. Converse of the Parallel Postulate (#THM): If two lines are cut by a transversal such that the interior angles sum to 180 then the two lines are parallel. This is equivalent to saying that if two lines are not parallel then the interior angles will not sum to 180, which is what Koa could have replied to his Grandma in the previous problem. Note: This postulate is very useful when trying to prove two lines are parallel. Page 20
7.7 Parallel Lines Proofs Per Date efore proceeding to the next proofs, identify each pair of angles in the diagram below that have the property: if the pair of angles are congruent then the two lines and DE must be parallel. Which transformations and other facts seem to justify your conjectures? We will prove these conjectures below, although we will use non-transformation results to do so. Pairs of ngles that you believe must be Congruent Transformation or Facts Used to Justify your Result Page 21
7.7 Parallel Lines Proofs Per Date 2. Given: C DE (#THM Corresponding ngles Converse) suur suur Prove: DF. Statement Reason Page 22
7.7 Parallel Lines Proofs Per Date C G D H E F 3. Given: GE DE (#THM lternate Interior ngles Converse) suur suur Prove: DF. Statement Reason Page 23
7.7 Parallel Lines Proofs Per Date C G D H E F 4. Given: C HEF (#THM lternate Exterior ngles Converse)!##" Prove:!##" DF. Statement Reason Page 24
7.8 Parallel Lines Proofs Homework Per Date C G D H E F!##" Given:!##" DF Prove: C + DEH = 180. Statement Reason Page 25