h 3 Worksheets S15 KEY LEVEL 2 Name 3.1 Duplicating Segments and ngles [and Triangles] Warm up: Directions: Draw the following as accurately as possible. Pay attention to any problems you may be having. You draw a figure using measuring tools, such as a protractor and a ruler. The lengths and angle measures need to be fairly precise. Mark the measures in the diagrams. Duplicate will mean to make an exact copy. Draw a duplicate line segment. Draw D. Draw a duplicate angle. Draw m = m. Do the length of the sides matter? Why? No, they are rays. Duplicate the triangle. Draw TRI by measuring only the side lengths. I MUST match up vertices! Draw an Equilateral Triangle. Draw equilateral triangle EQU, with sides all equal to. T R an t do with out an angle measure or a compass! U an t do with out an angle measure or a compass! Need to locate where the 2 sides meet! E Q Did you have any problems with making the drawings accurate? What were the issues? Yes, you can t figure out how the sides meet. You would have to just guess at it, because you don t have angle measures. S. Stirling Page 1 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name EXERISES Lesson 3.1 elow is Page 147-148 #1 3, 7, 8, 17 Use only a compass and a straight edge unless the instructions say to draw or measure! 1. Duplicate the line segments below them. 1) Use compass to measure. 2) With point on, use compass to make an arc, label intersection. Repeat for the other segments. D E F 1) Use compass to measure 2. onstruct line segment XY with length + D.. 2) With point on X, use compass to make an arc. 3) Use compass to measure D. 4) dd on the length, use compass to make an arc, label intersection Y. 3. onstruct line segment XY with length + 2 EF D. 1) Make a segment equal to + 2EF as you did before. 7. Duplicate triangle by copying the three sides, SSS method. 2) With point on the last arc, make a segment equal to D toward the left. Label intersection Y. Follow the Notes page 1. I 8. onstruct an equilateral triangle TRI. Each side should be the length of this segment. Follow the Notes page 1. T R 17. Use your ruler to draw a triangle with side lengths 8 cm, 10 cm, and 11 cm. Explain your method! Labels were provided to make the explanation clearer. 1) Draw a segment on the ray, = 11 cm. 2) Measure 10 cm with your compass. Swing an arc with center. 3) Measure 8 cm with your compass. Swing an arc with center. Label the intersection of the two arcs. 4) onstruct sides and. S. Stirling Page 2 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name 3.2 onstructing Perpendicular isectors Investigation 1: () Using the definitions from your notes, draw the following. Remember that when you draw, you are measuring, so remember to write your measures in your diagrams! Write down your steps too. Draw at least 3 bisectors of. Draw a perpendicular bisector Draw D, the perpendicular bisector of. How many bisectors can you draw through one point? Infinite Is it possible to draw more than one perpendicular bisector? No () In the drawing is the perpendicular bisector of EG. Using your ruler measure the distance of point,, and D from the endpoints of EG. Label them. What do you notice? G The distance from any one point on the perpendicular bisector is the same distance from endpoints E and G. E Is this true for any segment bisector? Draw a counterexample. Must be a perpendicular bisector. is closer to G than E. D S. Stirling Page 3 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name EXERISES Lesson 3.2 elow is Page 151-153 #1 3 and Exercise #12. Use only a compass and a straight edge on this page! Use only a compass and a straight edge on 1 3! 1 & 3. onstruct the perpendicular bisector of then construct the perpendicular bisector of EF at the right. Since EF is too close to the edge, you Follow the Notes page 2. need to make two points, and D, that are equidistant from E and F that were produced from different sized radii. E D 2. onstruct perpendicular bisectors to divide QD into four congruent segments. F 1) With compass set longer than 1 2 QD, construct a Q Y X Z D perpendicular bisector. Label the intersection X. 2) onstruct a perpendicular bisector of QX. Label the intersection Y. 3) onstruct a perpendicular bisector of XD. Label the intersection Z. QY = YX = XZ = ZD 7. onstruct perpendicular bisectors of each side of LI. (Make your lines long!) nything interesting happen? They all intersect in one point. L I Next page please! S. Stirling Page 4 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name EXERISES Lesson 3.2 Review Problems Page 148 Exercises #12. Make sure you can state the conjectures (properties) you are using. 130 50 50 50 130 130 50 50 50 155 90 115 65 a = 50, b = 130, c = 50, d = 130, e = 50, f = 50, g = 130, h = 130, k = 155, m = 115, n = 65 Page 153 #15 20 15. F 16. E 17. 18. 19. D 20. Page 158 #13 show your table, and #16. Write your answers below. #13 Rectangle, n 1 2 3 4 5 6 n 35 alc. Value (1)(2) (3)(3) (5)(4) (7)(5) (9)(6) (11)(7) (2 n 1)(n + 1) (69)(36) # of shaded triangles 2 9 20 35 54 77 (2 n 1)(n + 1) 2484 #16 Sketch D and EF D. S. Stirling Page 5 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name 3.3 onstructing Perpendiculars to a Line Warm up: () You are standing at point and need to run to hurch Hill Road as quickly as possible. How would you determine the shortest distance to the road? Try some different measures and use centimeters, cm, for convenience. Draw them in the figure below. What geometric figure will give you the shortest distance from point to the line (road)? X The shortest distance from the point to the line must be measured along the perpendicular segment from the point to the line. So the shortest distance is X = 2.4 cm. ll of the other distances are longer than X. () How could you measure the distance from point to each of the sides of DP? Think about how you measured your distance from you and the road. (Treat each side of the angle as a road.) Find these distances in cm. D X Y P Need to make a perpendicular from the point to each side. is closer to P Or Y < X. than to D. () Hans, H, is a mountain climber and Jose, J, is a cliff diver. (You can see them on their mountains pictured below.) What would you need to measure to determine Hans altitude at the top of his mountain? Show it in the drawing. How far will Jose dive before hitting the water surface? Show it in the drawing, and you may need to draw some water first. lso, label sea level in each drawing. H J 1.5 cm 1.5 cm S. Stirling Page 6 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name EXERISES Lesson 3.3 elow is Page 156-158 #1 3, 8, 10, 12, 18, 20. 1. Draw perpendiculars from the point P to both sides of IG. Which side is closer to point P? To find the distances from P to each side, you must first draw the perpendiculars to each side through point P (see notes page 3). P is closer to side I, but not by much! I P G 2 & 3. Draw altitudes from all three vertices of each triangle below. Observe where the altitudes are located (inside, outside or on). lso identify the type of triangle (acute, right or obtuse). cute triangle. ll altitudes are inside the triangle. R Right triangle. One altitude is inside; the other two are on the triangle, RG and GT. T G T See notes page 4!!! Obtuse triangle. One altitude is inside; the other two are outside the triangle, so you need to extend the sides. O T S. Stirling Page 7 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name 8. Draw an altitude M from the vertex angle of the isosceles right triangle. What do you notice about this segment? Write at least 3 statements! M bisects. M is the midpoint of. M is the perpendicular bisector of. M bisects. M. m = m = 45. m M = m M = 90. M M both are isosceles right triangles. 10. Draw and/or construct a square LE given L as a side. Explain how did it and support your reasoning with properties we ve learned. Need to make right angles at and L and then make L = L = E. Draw E. 3.4 cm L 3.4 cm 3.4 cm E 12. Draw the complement of (without measuring ). Explain how did it and support your reasoning with properties. Use a protractor to draw a 90º angle at So m D + m = 90. Remember to mark 90º angles! S. Stirling Page 8 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name 18. Draw a triangle with a 6 cm side and an 8 cm side and the angle between them measuring 40º. Draw a second triangle with a 6 cm side and an 8 cm side and exactly one 40º angle that is not between the two given sides. re the two triangles congruent? Hint: start each triangle with the 8 cm segment on the rays below. Two triangles DEF are possible, because side DF can intersect EF in two different places! Only one triangle is possible. Your triangle should be congruent to. F F D E 20. Draw two triangles. Each should have one side measuring 5 cm and one side measuring 7 cm, but they should not be congruent. Start with the 7 cm segments. Many different triangles are possible with only two given sides. S. Stirling Page 9 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name EXERISES Lesson 3.4 elow is Page 161-162 #6 8, 12. 6. Draw and/or construct an isosceles right triangle with z the length of each of the two legs. z 1) Make a 90º angle at either endpoint, I made m = 90. 2) Either use a compass or a ruler to measure length = z. Make sides of, the legs of, equal length z. Label the intersections and. 3) Draw. 7. Draw and/or construct RP with angle bisector R and the perpendicular bisector of RP. Place your answer on the ray below. P R R 6.6 cm P 1) With a compass, duplicate RP as on Notes page 1. You cannot locate point without a compass! 2) Measure 26 m RP =, and bisect it. 3) Find the midpoint of RP and draw the perpendicular bisector. RP Note: since is scalene, is not the midpoint of P and the perpendicular bisector does not pass through. 8. Draw and/or construct MSE with angle bisector S and altitude S. Place your answer on the ray below. 1) With a protractor, duplicate m M = 29. M M M 3.7 cm 29º 5.7 cm E S 2) With a compass or a ruler, duplicate MS and ME. 3) Measure m MSE = 38, and bisect it. 4) With a protractor, draw altitude S. You will need to extend side ME first because MES is obtuse! S. Stirling Page 10 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name 12. onstruct a linear pair of angles (that are not congruent). arefully bisect each angle in the linear pair. What do you notice about the two angle bisectors? an you make a conjecture? an you prove that it is always true? The angle bisectors of a linear pair of angles will be perpendicular (or will form D X y y x x Y EXERISES Lesson 3.4 Review Problems Do page 162 #14 16 Put the info. into the drawings! Must use algebra to solve!. lso do #19 & 20. 14. Given that the lines are parallel. Find y. 15. If E bisects R and m R = 84, find m R. a 90 angle). Prove: Label equal angles x and y. y + y + x + x = 180 form a straight angle. 2 y + 2x = 180 simplify y + x = 90 divide both sides by 2. So X Y. 68 x 55º y 110 5x 10 55º 55º 70º 42º E If parallel, alternate interior angles =. 68 x = 5x 10 78 = 6x x = 13 If parallel, alternate interior angles =. ( ) 5 13 10 = 55 55i 2 = 110 = y Or linear pairs supp. 180 2 55 = 70 and If parallel, Same side interior angles supp. 180 70 110 y = = 4x + 18 42º 7x + 4 46º R ngle bisector ( x ) 2 4 + 18 = 84 4x + 18 = 42 4x = 24 x = 6 ( ) m R = 7 6 + 4 = 46 S. Stirling Page 11 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name 16. Given X bisects largest,, or? 66º 6x + 36. Which angle is 7x 3 32º 57 5x 32º 8x + 10 50º X ngle bisector (congruent angles) 7x 3 = 57 5x 12x = 60 x = 5 m = 6 5 + 36 = 66 ( ) ( ) ( ) m = 2 7 5 3 = 64 m = 8 5 + 10 = 50 Largest. 19. ccurate size! 20. ccurate size! S. Stirling Page 12 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name EXERISES Lesson 3.5 & 3.6 Page 164-166 #1, 2, 4, 5, 17; Page 172 #6 On a separate sheet of paper: Page 165 #14, 15; Page 173 Review #15, 16 1. Draw a line parallel to n through P using alternate interior angles. Label what you measured and state the property you used. Various answers, but the alternate interior angles need to be labeled as equal. 2. Draw a line parallel to n through P using corresponding angles. Label what you measured and state the property you used. Various answers, but the corresponding angles need to be labeled as equal. n P n P 4. Draw and/or construct a rhombus with x as the length of each side and as one of the acute angles. Place your answer on the ray below. x 1) Use a protractor to draw 35 2) Either use a compass or a ruler to measure length = x. Make sides of equal length x. m =. 3) With a compass set to length x, locate the intersection of the other two congruent sides. 5. Draw and/or construct trapezoid TRP with TR and P as the two parallel sides and with P as the distance between them. (There are many solutions.) Place your answer on the ray below. T R Y P 1) Either use a compass or a ruler to measure length of TR and duplicate it. 2) Draw a perpendicular line anywhere on TR. Make XY = 4.3 and draw a perpendicular line at Y. 3) Draw P anywhere on this line and draw the two remaining sides of TRP. X S. Stirling Page 13 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name 3.5 Page 165 Review Exercise #17 k = 90 72 = 18 90 72 90 72 108 72 108 108 108 72 108 Vertical angles m = 108 2 = 54 62 62 118 Vertical angles q = 118 2 = 59 a = 72, b = 108, c = 108, d = 108, e = 72, f = 108, g = 108, h = 72, j = 90, k = 18, l = 90, m = 54, n = 62, p = 62, q = 59, r = 118 Page 172 # 6. Draw and/or construct isosceles triangle T with perimeter y and length of the base equal to x. Place your answer on the ray below. y X Y T x 1) Use a compass or a ruler to measure length x = and subtract it from Y. lso construct on the given ray. 2) Since y represents the perimeter of the isosceles triangle, use a compass or a ruler to bisect Y. The resulting segments, =, are the X XY lengths of the remaining sides of the triangle, T and T. 3) Use a compass, set to X or XY to construct the two sides T and T. S. Stirling Page 14 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name Review Problems Page 165 #14, 15; Page 173 Review #15, 16 Page 165 #14 Sketch trapezoid ZOID with ZO ID, point T the midpoint of OI and point R the midpoint of ZD. Sketch TR. Page 165 #15 Draw rhombus ROM with m R = 60 and diagonal O. Page 173 #15 If a polygon has 500 diagonals from each vertex, how many sides does it have? 500 = n 3 503 = n Page 173 #16. Must use actual measures! Draw parallelogram RE so that = 5.5 cm, E = 3.2 cm and m = 110. S. Stirling Page 15 of 16
h 3 Worksheets S15 KEY LEVEL 2 Name 3.8 Page 190 Review Exercise #14 h = 90 52 = 38 52 38 142 = 180 38 52 128 128 128 52 52 52 128 52 52 38 38 71 142 n = 142 2 = 71 38 a = 128, b = 52, c = 128, d = 128, e = 52, f = 128, g = 52, h = 38, k = 52, m = 38, n = 71, p = 38 3.R Page 197-198 Review Exercise #62 & 64 38 38 142 38 106 74 a = 38, b = 38, c = 142, d = 38, e = 50, f = 65, g = 106, h = 74 65 65 50 f = 130 2 = 65 30 30 One possible explanation: Since linear pairs are supplementary m FD = 30. m D = 30 because D and alternate interior angles =. ut its vertical angle has a measure of 26º. This is a contradiction! S. Stirling Page 16 of 16