Semester 1 Closure Geometer: CPM Chapters 1-6 Period: DEAL Take time to review the notes we have taken in class so far and previous closure packets. Look for concepts you feel very comfortable with and topics you need more help with. Look for connections between ideas as well as connections with material you learned previously. Chapter 1: Shapes and Transformations Chapter 2: Angles and Measurements Chapter 3: Justification and Similarity Chapter 4: Trigonometry and Probability Chapter 5: Completing the Triangle Toolkit Chapter 6: Congruent Triangles For questions 1-6, use the figure to the right. 1) What is different name for plane R. (answers will vary) plane ABC 2) What is different name for 1. _ BAD 3) What is different name for line m. AE or AD or DE 4) Name a pair of congruent segments. BD and DC 5) Name three collinear points. Points A, D, E or Points B, D, C 6) Name a pair of supplementary angles. (answers will vary) BDE or EDC 7) Name the transformation(s) that are not isometric. Justify your answer. A dilation is not isometric because the side lengths are not the same- does not preserve size. (Figures are isometric if the pre-image and images are congruent- same shape AND size.) 8) Using transformations show that a b. Translate 1 down along the line c making 1 and 2 supplementary. Since 1 and 2 were same side interior angles in the original picture. Therefore, if same side interior angles are supplementary, then the lines a and b are parallel. 9) Use the following regular hexagon to answer the following questions. A) Name the image of B after a 240 clockwise rotation about Z. point D B) Name the image of A reflected over FC. Point E
10) Find the measure of the numbered angles. m 1 = 110 m 1 = 112 m 1 = 138 m 1 = 80 m 2 = 80 m 3 = 100 m 4 = 80 11) Given a b cut by a transversal c, which of the following statements is NOT true. A) 1 and 3 are corresponding angles B) 2 and 3 are same-side int. angles C) 3 and 4 are vertical angles D) 1 and 4 are alternate int. angles 12) Refer to the diagram to answer the following questions: Assume the lines are parallel. A) Name a pair of same-side interior angles. Answers will vary (ex: 2 and 3) B) Name a pair of corresponding angles. Answers will vary (ex: 2 and 4) C) Name a pair of alternate interior angles. Answers will vary (ex: 2 and 6) 13) Write the coordinates of the vertices of the image ABCD for each transformation. A) Translation (x, y) (x + 4, y 2) A (7, 2), B (11, 2), C (10, -1), D (6, -1) B) reflection across x = -1 A (-5, 4), B (-9, 4), C (-8, 1), D (-4, 1) C) rotation of 180 o about the origin A(-3, -4), B (-7, -4), C (-6, -1), D (-2, -1) D) Dilation with scale factor of 2 about the origin A(6, 8), B(14, 8), C(12, 2), D(4, 2)
14) Complete the transformations below and then answer the corresponding questions. a. Reflect JKLM across the y-axis to form J K L M. Then rotate J K L M 90 o counterclockwiseabout the origin to form J K L M. b. What single rigid transformation transforms JKLM to J K L M? 180 o rotation about the origin K J L L K M M J 15) How many lines of symmetry does a square have? Hint: Draw a diagram. 4 lines of symmetry 16) Classify the triangle by its sides. a. equilateral b. isosceles 17) Classify the triangle by its angles. obtuse triangle 18) Find the value of x. x = 7
19) Find the value of x if AC = 20. X = 5 units 20) Find the value of x if B is the midpoint of AC. X = 6 units 21) Find m TQM. X = 28 m TQM =44 22) Find the value of x. x = 14 23) Find the value of x. x = 54 24) Find the measure of the sides of equilateral JKL if JK = 5x 7 and JL = 2x + 5. The measure of each side is 13 units.
25) QX bisects PQR. If m 1 = 4x -12 and m 2 = 2x + 6, find m XQR. m XQR = 24 26) Find the value of x. x = 16 27) Find the value of x and y. State geometric relationships used in solving. y = 79 triangle angle sum theorem, supplementary angle pair x = 47 alternate interior angles of parallel lines are congruent 28) Find the value of x and y. y = 3 units x = 3 2 units 45-45-90 ratios Multiply each expression below. 29) (2d 3)(d + 4) 30) 2x(x + 3) 31) (r 5) 2 2d 2 + 5d 12 2x 2 + 6x r 2 10r + 25 32) Find the intersection of MN and LO. Point R
33) Find the slope of a line perpendicular to 3y + 2x = 4. m = 3 2 34) What is the slope of the line that contains (6, -4) and (-4, 1)? m = 1 2 35) Which pair of lines is perpendicular? a. y = 3x + 5 b. y = 4x 5 c. y = 5x 3 d. y = 2 x 11 3 y = -3x 8 y = 4x + 2 y = 1 5 x + 3 y = 3 2 x 5 36) Which pair of lines is parallel? a. y = 5x + 3 b. y = -2x 1 c. y = 3x + 8 d. x = 3 5x + y = 10 2x + y = 4 3x + y = 6 y = 3 37) a. Graph a line parallel to y = 3x 1 that contains (2, 3). b.write the equation of the new line in point slope form. y 3 = 3(x 2) 38) a. Graph a line perpendicular to y = 1 x 5 that contains (-2, 2). 2 b. Write the equation of the new line in slope-intercept form. y = 2x 2
39) What is the distance between (-1, 6) and (5, -2)? 10 units 40) The two rectangles are similar. Which is a correct proportion for corresponding sides? 4 m 12 m 8 m a. b. c. d. x 41) The Sears Tower in Chicago is 1450 feet high. A model of the tower is 24 inches tall. What is the ratio of the height of the model to the height of the actual Sears Tower? 1 725 42) Solve the proportion. m = 25 43) The figures are similar. Give the ratio of the perimeters and the ratio of the areas of the first figure to the second. The figures are not drawn to scale. Ratios of the perimeters = 5 6 Ratios of the Areas = 25 36 15 18 44) Find the ratio of the perimeter of the larger rectangle to the perimeter of the smaller rectangle. 5 ft 3 ft 9 ft 11 ft 5 3
45) You want to produce a scale drawing of your living room, which is 24 ft by 15 ft. If you use a scale of 4 in. = 6 ft, what will be the dimensions of your scale drawing? 10 in x 16 in 46) Figure. Name a pair of corresponding sides? TQ and BC 47) ABCD ~ WXYZ. AD = 6, DC = 3, and WZ = 59. Find YZ. The figures are not drawn to scale. B X A C W Zy D Z 29.5 units 48) Triangles ABC and DEF are similar. Find the lengths of AB and EF. A D 5x 5 E x F B 4 C AB = 10 units and EF = 2units 49) State whether the triangles are similar. If so, write a similarity statement and the postulate or theorem you used. If not similar, write not similar. a. ABD~ CBD by SAS~ b. CAB~ ZXY by AA~
50) Campsites F and G are on opposite sides of a lake. A survey crew made the measurements shown on the diagram. What is the distance between the two campsites? The diagram is not to scale. 42.3 m 51) Find the length of the missing sides. Then determine the perimeter of the triangle. 5 52) Find the length of the missing side. The triangle is not drawn to scale. 5 3 25 24 7 units 53) You traveled 10 miles east and then 8 miles south. Exactly how far are you from your starting point? x = 2 41 miles 54) Two students were asked to find the value of x in the figure. The equations they used are below. Decide whether each student is correct or incorrect. If the student is incorrect, explain what they did wrong and then show the correct equation. If the student was correct, write correct. Lee s equation: sin 57 = x 15 Jamila s equation: cos 33 = 15 x Incorrect. The hypotenuse is x and the opposite side is 15. Therefore the equation should be sin 57 = 15 x. Correct
55) Find the length of the leg. If your answer is not an integer, leave it in simplest radical form. Leg = 8 2 16 45 Not drawn to scale 56) Find the value of the variable(s). If your answer is not an integer, leave it in simplest radical form. x y 30 20 x = 30 units, y = 10 3 units 57) Write the tangent ratios for and. P Tan P = 20 21 Tan Q = 21 20 21 29 tangent ratios of complementary angles are reciprocals! R 20 Q Not drawn to scale 58) A large totem pole in the state of Washington is 100 feet tall. At a particular time of day, the totem pole casts a 249-foot-long shadow. Find the angle of elevation to the nearest degree. Draw a diagram to support your work! x = 22 59) Find the value of x. Round to the nearest tenth. 10 x = 8.1 units x Not drawn to scale
60) A spotlight is mounted on a wall 17.4 feet above the ground in an office building. It is used to light an entrance door 9.3 feet away. To the nearest degree, what is the angle of depression from the spotlight to the entrance door? θ =62 o 61) State whether the triangles are congruent. If so, write a congruence statement and the postulate or theorem you used. If not congruent, write not congruent. a. ADB CDB by SAS b. triangles are not congruent (no AA for congruent triangles) 62) If DLQ EMR, then which of the following are NOT true? a. LDQ MRE b. DLQ EMR c. RME QLD d. MRE LQD 63) XYZ ABC. What angle is congruent to Y? Y B 64) Write the converse of the conditional statement: If it is raining, then the ground is wet. If the ground is wet, then it is raining. 65) Is the following a statement a good definition? Explain. A square has four right angles. No, a rectangle also has 4 right angles. A better defition would include the fact that a square also has 4 congruent sides. 66) Find a counterexample for the statement: If two shapes are similar, then they are congruent. An equilateral triangle with side lengths 4 units and an equilateral triangle with side lengths 8 units.
67) State the postulate or theorem you can use to prove the triangles congruent. If the triangles cannot be proven congruent, write not possible. a) _ ASA or AAS b) SSS c) not possible d) ASA or AAS _ e) not possible f) HL g) HL h)sas or HL or SSS i) SAS j) ASA or AAS k) AAS or ASA i) not possible 68) Name a pair of overlapping congruent triangles. State whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL. ΔWZX ΔXYW by AAS 69) Which two sides must be congruent to use ASA to prove DMX JCX? DX JX 70) Which two angles must be congruent to use SAS to prove ADX CBX? D B
71) Are the triangles congruent? If so, write a congruence statement and justify your reasoning. If not, write not congruent. yes! ΔABC ΔEDC by HL 72) Given: AB DC AD BC Prove: ADC CBA Statements Reasons 1) AB DC 1) Given 2) AD BC 2) Given 3) BAC DCA 3) If lines parallel, then alternate interior angles congruent. 4) BCA DAC 4) If lines parallel, then alternate interior angles congruent. 5) AC AC 5) Reflexive Property of Congruence 6) ADC CBA 6) ASA 73) Given: EI and HF bisect each other at G. Prove: E I Statements Reasons 1) EI and HF bisect each other at G. 1) Given 2) EG GI 2) Definition of segment bisector 3) HG GF 3) Definition of segment bisector 4) EGF HGI 4) Vertical angles are congruent. 5) EGF IGH 5) SAS 6) E I 6) CPCTC
74) Given: MJ JK Prove: Statements KL ML JK ML JM KL Reasons 1) MJ JK 1) Given 2) KL ML 2) Given 3) JK ML 3) Given 4) J and L are right angles 4) Definfition of perpendicular 5) MJK and KLM are right triangles 5) Definition of right triangles 6) MK MK 6) Reflexive Propety of Congruence 7) MJK KLM 7) HL 8) JM KL 8) CPCTC 75) Given: AED is isosceles AB CD Prove: BEC is isosceles Statements 1) AED is isosceles with vertex E Reasons 1) Given 2) AB CD 2) Given 3) AE ED 3) Definition of isosceles triangle 4) A D 4) Base Angles of Isosceles Triangle Congruent 5) EAB EDC 5) SAS 6) BE CE 6) CPCTC 7) BEC is isosceles 7) Definition of isosceles triangle 76) Find the area and perimeter of ΔABC. Area = [ 1 (4)(4)] + 2 [1 (4 3)(4)] = 8 + 8 3 units 2 Perimeter = 8 + 4 2 + 4 + 4 3 units = 12 + 4 2 + 4 3 units