Transformations and Congruence

Name Date Class UNIT 1 Transformations and Congruence Unit Test: C 1. Draw ST. Construct a segment bisector and label the intersection of segments Y. If SY = a + b, what is ST? Explain your reasoning. Use the statement for 5 6. If two figures are congruent, then their corresponding angles are congruent. 5. State the inverse of the statement and determine whether it is true. Explain your reasoning. Use the graph for 2 4. 6. State the converse and determine whether it is true. Explain your reasoning. 7. Draw the preimage and image of a triangle with coordinates T(2, 1), U(0, 1), and V(3, 3) under a translation along < 4, 2>. 2. Suppose you reflect figure FGHI across the line y = x. What will be the coordinates of F G H I? 3. What will be the midpoints of FF, GG, HH, and II? 4. What will be the midpoints of FF and HH, if FGHI is translated using the vector 2, 3? 8. On the grid above, draw a reflection of T U V across the line y = x. 9. Which of these transformed figures, TUV, T U V, or T U V, if any, are congruent? Explain. 1

Name Date Class UNIT 1 Transformations and Congruence Unit Test: C Use the graph for 10. 13. State the angle of rotational symmetry and the number of lines of symmetry if either exists. Use the graph for 14 16. 10. What is the equation for the line of reflection for ABCD to A B C D on the graph above? Explain your reasoning. Use the graph for 11 13. 11. Dilate the figure shown by a factor of 1 2 on the graph shown. Be sure to note the location of P. 14. What translation was used to map the solid image to the dashed image? 15. What reflection was used to map the solid image to the dashed image? 16. Mitch believes the dashed image could also be mapped using rotation about the origin. Do you agree? Explain your reasoning. 12. If the dilated figure is rotated 90 counterclockwise about the origin, what will be the coordinates of P? 17. For two triangles, ABC PQR, m P = 45, m B = (5x + 2), and m Q = (6x 16). Can m C be determined? If so, state the measure. If not, explain why not. Justify all reasons. 2

Name Date Class UNIT 2 Lines, Angles, and Triangles Unit Test: C Use the figure for 1 2. 5. Write an equation for the line that passes through (10, 0) and is parallel to 2x y = 7. 1. If ( x ) m 1= + 4, what is m 2? 2. For what measure will m 3 = m 4? 3. Write an equation for the line that passes through (2, 5) and is perpendicular to 3x + 4y = 8. 6. The sum of the interior angles of a convex polygon is 3240. How many sides must the polygon have? Show your work. 7. Shane believes that the sum of the interior angles of an octagon is 1440. What mistake did Shane make? How would you explain the correct solution to Shane? For 8 9, use the figure. Two triangles are shown. 8. If ( x ) m 1= 3 + 15, what is m 3 in terms of x? 9. For what value of x is DEF an isosceles triangle? Explain your reasoning. 4. What congruency statement would not result in BCD QRS? Explain. 10. Can a triangle have side lengths 5, 8, and 13? Explain why or why not. 1

Name Date Class UNIT 3 Quadrilaterals and Coordinate Proof Unit Test: A 1. Parallelogram EFGH is shown. A. Name a congruency involving the sides. B. Name a congruency involving the angles. C. Name a congruency involving the diagonals. ABCD is a quadrilateral with BE ED. 2. Given that EC = 8.4 cm, what is the measure of AE so that ABCD is a parallelogram? 3. What are the coordinates of the fourth vertex C so that rectangle ABCD is formed? 4. STUV is a parallelogram. What must be true for SU and TV in order for STUV to be a square? 5. Quadrilateral ABCD has diagonal AC represented by y = 2x + 5 and BD 1 represented by y = x 4. Based on 2 the slopes of these lines, what is true about the quadrilateral? 8. State whether each quadrilateral must have congruent diagonals. A parallelogram Yes No B rhombus Yes No C rectangle Yes No D isosceles trapezoid Yes No E kite Yes No 11. A parallelogram has vertices D(2, 1), E(2, 6), F(5, 6), G(5, 1). Determine whether DEFG is a rhombus, rectangle, both, or neither. Explain your reasoning. Quadrilateral ABCD is shown. 13. What is m B? 125

Name Date Class Benchmark Test Modules 1 6 For 1 2, use the graph. 5. ABC maps to ABC as follows. Preimage Image A( 3, 4) A ( 1, 3) B( 5, 1) B ( 3, 2) C(2, 3) C (4, 2) Use coordinate notation to write the rule that maps the preimage to the image. 1. Determine the measure of each segment. Then indicate whether the statements are true or false. A AB B CD C EF CD True False EF True False AB True False 2. What is the midpoint of AB? 6. Write the transformation in words for the rule ( xy, ) ( x, y). 7. Use the graph. Use the following information for 3 4. In the figure, m PQS = 38. Specify the component form of the vector that maps ABC to A B C. 3. What is the value of x? 4. What is m PQR? 8. Line segment AB with endpoints A(7, 2) and B( 1, 2) is reflected over the x-axis. What are the coordinates of the midpoint of AB? 179

Name Date Class Benchmark Test Modules 1 6 9. Use the graph. What are the coordinates of the image of triangle JKL after a reflection over line m? 10. Parallelogram QRST has vertices Q( 4, 2), R( 2, 4), S(0, 1), and T( 2, 1). What are the coordinates of its image after a counterclockwise rotation of 270 about the origin? 13. A point is located at (3, 2). Which transformation will result in an image located in Quadrant III? A Reflection over x-axis Yes No B Reflection over y-axis Yes No C Translation 4 units left and 3 units up Yes No D Rotation 90 clockwise about the origin Yes No E Reflection over the line y = x Yes No Use the figures for 14 15. Given RST XYZ Use the figure for 11 12. 14. What is m T? 15. What is YX? 16. Line segment GH is reflected across the x-axis and rotated 90 clockwise about the origin. 11. How many lines of symmetry does the figure have? 12. What are the angles of rotation less than 360 for the figure? What are the coordinates of points G' and H'? 180

Name Date Class Benchmark Test Modules 1 6 Use the graph for 17 18. Use the figure for 22 23. 22. Name all angles congruent to 4. 17. What transformations can you use to show that quadrilaterals ABCD and A B C D are congruent? 18. Express the transformations as a single mapping rule in the form of (x, y) (?,?). 19. Triangle PQR has vertices P(0, 2), Q(3, 4), and R(4, 2). If triangle PQR is rotated 270 clockwise about the origin, what is the length of side P R? Use the following information for 20 21. GHJ PQR and PQR STU Complete the following using a side or angle of STU. 20. H 23. Name all angles supplementary to 2. 24. Write an equation in slope-intercept form for the line that passes through ( 2, 2) and is parallel to 2x + 4y = 16. 25. Write an equation in slope-intercept form for the line that passes through (4, 1) and is perpendicular to 4x y = 3. 26. Look at the figure below. 21. JG In the figure, m 1 = m 5. What theorem can be used to prove that line a and line b are parallel? 181

Name Date Class Benchmark Test Modules 1 6 Use the following information for 27 28. In the figure below, EB is the perpendicular bisector of AC and FC is the perpendicular bisector of BD. 32. Use the figures. 27. If AE = 6 cm and FD = 9 cm, what is FB? 28. If AC = 8 cm, what is CD? Determine the value of x that makes the triangles congruent. For 33 34, state the additional congruency statement or statements needed to prove BCD QRS for the given theorem. 29. The measures of two vertical angles are represented by the expressions (2x + 7) and (4x 5) Find the value of x. Use the following information for 30 31. In the figure, m 4 = 154. 33. SAS Theorem 34. ASA Theorem 30. What is m 3? 31. What is m 1? 182

Name Date Class Mid-Year Test Modules 7 12 For 1 2, right triangles ABC and TUV are shown in the graph. 5. Use the figure. What is sum of the measures of the interior angles of the polygon? 1. Show that the triangles are congruent using the AAS Theorem. 6. The sum of the measures of the interior angles of a polygon is 2160. How many sides does the polygon have? For 7 8, use the figure. 2. Show that the triangles are congruent using the HL Theorem. For 3 4, use the figure. 7. If m 1 = (2x + 10), what is m 3 in terms of x? 8. For what value of x is ABC an isosceles triangle? 3. What value of x makes KM the angle bisector of LKN? 4. If KM is the angle bisector of LKN, what is m LKN? 9. Triangle DEF is an equilateral triangle. If DE = 6x + 4y, what is the perimeter of the triangle in terms of x and y? 183

Name Date Class Mid-Year Test Modules 7 12 10. State whether the given side lengths can form a triangle. A 5, 8, 12 Yes No B 20, 18, 2 Yes No C 15, 26, 9 Yes No D 4.75, 12.25, 16.25 Yes No 17. Vertices of a quadrilateral are at ( 2, 1), (2, 4), and (4, 2). What is the location of the 4th vertex so that a parallelogram is formed? 18. State whether the following are true or false for all parallelograms. A Opposite sides are congruent. True False B Consecutive angles are congruent. True False C Diagonals bisect each other. True False D Diagonals are congruent. True False 20. If m DAB = (4y 12) and m ABC = 64, for what value of y is ABCD a parallelogram? 21. If ABCD is a parallelogram, BD = 3x 15, AC = x + 9, and AED = 2y + 18, for what value of x and y is the figure a square? Explain your reasoning. 22. GIJL is a trapezoid with midsegment HK. If IJ = 7 cm and GL = 15 cm, what is HK? 23. Quadrilateral ABCD is shown. Use the following information for 19 21. ABCD is a quadrilateral with BE ED and BCD DAB. 19. If EC = 20 cm and AE = 3x 4, for what value of x is ABCD a parallelogram? What is m D? 185

Name Date Class End-of-Year Test Modules 1 23 For 1 2, use the graph. 6. State the converse of the statement. Then determine whether the converse is true. Explain. If two angles are vertical angles, then they are congruent. 1. Which segment is congruent to EF? 7. Use the graph. 2. What is the midpoint of GH? Use the following information for 3 4. In the figure, m KJL = 32. Write the vector that maps RST to R S T. Use the figure for 8 9. 3. What is the value of x? 4. What is m KJM? 5. Line segment PQ with endpoints P(4, 2) and Q( 2, 0) is rotated 90 clockwise around the origin. What are the coordinates of the midpoint of PQ? 8. How many lines of symmetry does the figure have? 9. What are the angles of rotation less than 360 for the figure? 191

Name Date Class End-of-Year Test Modules 1 23 Use the following information for 10 11. In the figures below, ABC LNM. 14. In the figure, m 2 = 75. 10. What is the value of x? 11. What is the value of y? Use the graph for 12 13. What is m 7? 15. The measures of two complementary angles are represented by the expressions (3x + 16) and (5x + 18). Find the value of x. 16. Write an equation for the line that passes through (1, 3) and is perpendicular to 1 y = x + 5. 2 12. What transformations can you use to show that quadrilaterals DEFG and D'E'F'G' are congruent? 17. Write an equation for the line that passes through (3, 2) and is parallel to 2x + 3y = 3. 18. In the figure, the measure of 2 is 55. 13. Express the transformations as a single mapping rule in the form of (x, y) (?,?). What is the measure of 4? 192

Name Date Class End-of-Year Test Modules 1 23 19. Use the figures. 23. In the figure, PQ PS. Determine the value of x that ensures that the triangles are congruent. For 20 21, state the additional congruency statement or statements needed to prove ABC XYZ for the given theorem. Explain why PQR PSR. 24. In the figure, m BAC = 9x + 4 and m BAD = 3x + 8. 20. ASA Theorem 21. AAS Theorem 22. Look at the figure below. What value of x indicates that AD is the angle bisector of BAC? 25. Use the figure. What is the value of x? Are triangles DEF and FGH congruent? Explain why or why not. If the triangles are congruent, write a congruence statement. 26. The sum of the measures of the interior angles of a regular polygon is 900. How many sides does the polygon have? 193

Name Date Class End-of-Year Test Modules 1 23 27. Triangle RST is an isosceles triangle with m R = 120. What is m S? Explain your reasoning. 36. GIJL is a trapezoid with midsegment HK. 28. The sides of a triangle measure 5 meters and 8 meters. What are the possible side lengths for the third side? Show your work. If IJ = 18 cm and GL = 42 cm, what is HK? 37. Triangle PQR is shown in the graph. 34. ABCD is a quadrilateral with BE ED and BCD DAB. If EC = 16 cm, m ABC = 64, AE = 3x 5, and m DAB = (4y 12), for what values of x and y is ABCD a parallelogram? 35. State whether each quadrilateral has congruent diagonals. A parallelogram Yes No B rhombus Yes No C rectangle Yes No D isosceles trapezoid Yes No E kite Yes No Classify the triangle. Explain your reasoning. 38. A parallelogram has vertices D( 4, 1), E(2, 5), F(4, 3), and G( 2, 3). Determine whether DEFG is a rhombus, rectangle, or neither. Explain your reasoning. 195