Construction of an Angle bisector

Midterm Multiple Choice Practice 1. Based on the construction below, which statement must be true? 1 1) m ABD m CBD 2 2) m ABD m CBD 3) m ABD m ABC 1 4) m CBD m ABD 2 2. Line segment AB is shown in the diagram below. Which two sets of construction marks, labeled I, II, III, and IV, are part of the construction of the perpendicular bisector of line segment AB? 1) I and II 2) I and III 3) II and III 4) II and IV 3. Given the similarity transformation shown below; identify the composition: 1) RC, 80 T D 1 ( ABCD) mn C"', 4 2) T D 1 ( ABCD) mn C"', 4 3) RC, 80 T DC "',4( ABCD) mn 4) RC, 80 D 1 ( ABCD) C"', 4 Construction of an Angle bisector

4. The line constructed that connects the vertex of a triangle that is perpendicular to the opposite side is called the: 1) Altitude 2) Median 3) Perpendicular bisector 4) Angle bisector 5. The midpoint of each side of a triangle connects to the opposite vertex to form a: 1) Altitude 2) Median 3) Perpendicular bisector 4) Angle bisector 6. Given the following diagram, which angles are NOT congruent? 1) 1 and 4 2) 4 and 5 3) 2 and 7 4) 3 and 5 7. The perpendicular bisector of a line segment is the endpoints of the line segment. 1) congruent to 2) perpendicular to 3) equidistant from 4) parallel to 8. Two lines perpendicular to the same line are to each other. 1) Perpendicular 2) Parallel 3) Neither 4) Both 9. Two lines parallel to the same line are to each other. 1) Perpendicular 2) Parallel 3) Neither 4) Both Same side supplementary angles

10. How many lines of symmetry are there for a regular hexagon? 1) 4 2) 6 3) 0 4) 2 11. In the diagram below, the vertices of are the midpoints of the sides of equilateral triangle ABC, and the perimeter of is 36 cm. What is the length, in centimeters, of? 1) 6 2) 12 3) 18 4) 4 12. In the diagram below of, medians,, and intersect at G. If, what is the length of? 1) 8 2) 10 3) 12 4) 16 13. In the diagram below of, medians,, and intersect at G. The length of is 12 cm. What is the length, in centimeters, of? 1) 24 2) 12 3) 6 4) 4 14. In the diagram below of and,, and. To prove that and are congruent by SAS, what other information is needed? 1) 2) 3) 4)

15. The diagonal is drawn in parallelogram ABCD. Which method cannot be used to prove that? 1) SSS 2) SAS 3) SSA 4) ASA 16. In the diagram below of, side is extended to point D,,, and. What is? 1) 5 2) 20 3) 25 4) 55 17. Transversal intersects and, as shown in the diagram below. Which statement could always be used to prove? 1) 2 4 2) 7 8 3) 3 and 6 are supplementary 4) 1 and 5 are supplementary 18. In the diagram below, is isosceles with. If and, what is? 1) 27 2) 28 3) 42 4) 70 19. Let P(2, 4) be a point on a figure, and let P be the corresponding point on the image. The figure is dilated by a scale factor of 4. What are the coordinates of P? 1) (-2, 0) 2) ( ½, 1) 3) (6, 8) 4) (8, 16) **SSA is NOT a triangle congruence *Vertical angles are not used to prove parallel lines

20. Which transformation best describes the image of an object viewed through a microscope? 1) dilation 2) reflection 3) rotation 4) translation 21. If the accompanying diagram, DE AC 1) 4.5 2) 3 3) 13.5 4) 12. Which of the following represents the length of BC? 22. AC is a diagonal of rectangle ABCD and EF joins the midpoints of AB and BC, respectively. If AC = 26, which of the following represents the length of EF? A E B 1) 52 2) 13 3) 10 4) 5 23. In the diagram below of right triangle ACB, altitude CD is drawn to hypotenuse AB. If AB = 36 and AC = 12, what is the length of AD? 1) 3 2) 4 3) 6 4) 32 24. Enlargement = dilation B F C

25. 26. AB C is a dilation of ABC from vertex A. CC = 4, AB = 8, AC = 10 and BC = 12. Which of the following represents the length of B C? 1) 11.2 2) 12 3) 16.8 4) 16 27. Which of the following transformations are not distance preserving? 1) Rotation 3) Translation 2) Reflection 4) Dilation 28. In the diagram of quadrilateral ABCD,,, and diagonal is drawn. Which method can be used to prove is congruent to? 1) AAS 2) SSA 3) SAS 4) SSS

29. In the diagram of below,. Which reasons can be used to prove? 1) reflexive property and addition postulate 2) reflexive property and subtraction postulate 3) transitive property and addition postulate 4) transitive property and subtraction postulate 30. A father who is 6 ft. tall is standing next to his son. The father casts a 9 ft shadow. If the son casts a shadow that is 6 ft long then how tall is he? 1) 3 ft 3) 5 ft 2) 4 ft 4) 6 ft 31. A statue that is 12 ft tall casts a shadow that is 15 ft long. Determine the length of the shadow that a 8 ft statue casts. 1) 11.5 ft 3) 7 ft 2) 4.5 ft 4) 10 ft 32. Quadrilateral MNOP is a trapezoid with. If is the image of MNOP after a reflection over the x-axis, which two sides of quadrilateral are parallel? 1) and 2) and 3) and 4) and 33. Given that ABCD is a parallelogram, a student wrote the proof below to show that a pair of its opposite angles are congruent. What is the reason justifying that? 1) Opposite angles in a quadrilateral are congruent. 2) Parallel lines have congruent corresponding angles. 3) Corresponding parts of congruent triangles are congruent. 4) Alternate interior angles in congruent triangles are congruent.

34. Which of the following must be true about the diagonals of a rectangle? A. The diagonals are perpendicular B. The diagonals are congruent C. The diagonals bisect each other (1) A, only (3) B and C, only (2) A and C, only (4) A, B, and C 35. In the diagram, m A = x + 20, m B = 3x, and BCD is an exterior angle formed by extending AC to point D, and m BCD = 120. Find the value of x. (1) 10 (3) 35 (2) 25 (4) 40 36. In the accompanying figure, ABCD is a parallelogram, m A = 4x + 15, and m C = 9x 40. Find the measure of angle D. D C 9x - 40 (1) 11 (3) 59 (2) 16 (4) 121 37. Which of the following is not a degree of rotational symmetry for a regular pentagon? (a) 36 o (b) 72 o (c) 144 o (d) 288 o 38. In rectangle ABCD, ADB BCD. E is the midpoint of BD. Which of the following rigid motion(s) maps AD onto CB? (a) r (c) R o BD 90 (b) T (d) R E,180 DB -->Only in a rhombus or a square A 4x+15 B

39. Which is an angle that is complementary to BOC? (a) BOE (b) COD (c) DOE (d) BOA 40. If the letter P is rotated 90 degrees, which is the resulting figure? (a) Df (b) P P (c) (d) 41. In the diagram below, PQ RS and transversal TU intersects PQ and RS at V and W, respectively. If m TVQ 5x 22 and m VWS 3x 10, find m WVQ. (a) 16 o (b) 58 o P Counter clockwise! (c) 24 o (d) 122 o 42. Two triangles are similar, and the ratio of each pair of corresponding sides is 3:1. Which statement regarding the two triangles is not true? (1) Their areas have a ratio of 9:1 (2) Their altitudes have a ratio of 3:1 (3) Their perimeters have a ratio of 3:1 (4) Their corresponding angles have a ratio of 3:1. **In similar triangles angles are congruent 2 43. The ratio of the sides of two similar triangles is 4 :1. If the area of the larger triangle is 144cm, find the area of the smaller triangle. (1) 36 (2) 18 (3) 12 (4) 9

44. Given the diagram below, DE BC. AE x, CE 3, AD 6, and DB 2. What is the value of x? (1) 7 (2) 9 (3) 4.5 (4) 1 45. In the diagram below,. Which statement is not true? (1) (2) (3) (4) Bottom = Right Right Bottom 46. If a quadrilateral ABCD is dilated by a scale factor of 2 and angle A = 72 o then angle A equals A) 72 o C) 144 o B) 36 o D) Can t be determined **Angle measures do not change in dilations

Short Answer Practice 47. Construct an equilateral triangle given the following side length 48. Using a compass and straightedge, determine if ABC is an equilateral triangle. Explain your reasoning. 49. Using a compass and straightedge, construct the bisector of the angle shown below. [Leave all construction marks.] 50. Using a compass and straightedge, construct an equilateral triangle with as a side. Using this triangle, construct a 30 angle with its vertex at A. [Leave all construction marks.] A C B

51. A) Using a compass and straightedge, construct the perpendicular bisector of. B) Construct a 45 angle with its vertex at the midpoint of AB. [Leave all construction marks.] 52. Given the diagram, find the center point of dilation for triangle ABC and triangle A B C 53. Determine the scale factor of the figure below given center O.

54. Given the diagram below, Drawing 2 is the image of Drawing 1 with a scale factor of r = 2 1 centered at O1, and Drawing 3 is the image of Drawing 2 with a scale factor of r = 2 3 centered at O2, find the following: a) Find the center of dilation going from drawing 1 to drawing 3 and label O 3. b) Find the dilation factor of drawing 1 to drawing 3 55. Find the values of x and y: 56. A) Using a compass and a straight edge, find the midpoints of AB, BC, and AC and label them as D, E, and F respectively. B) Find the centroid of the triangle. B Orthocenter Incenter Centroid Circumcenter altitudes angle bisectors medians perpendicular bisectors *the median is a line from the vertex to the midpoint of the opposite side A C

57. Construct a square inscribed in the given circle: 58. Explain how you would know if a triangle was isosceles. 59. Answer the following questions using the figures below: a) What transformation was performed? b) Find the center of rotation. 60. Given triangle XYZ, translate it using vector KL, and label its image X Y Z. O

61. Given the diagram below, answer the following questions: a) Find the scale factor of the dilation already drawn. b) Using the triangle A B C create a third triangle dilated by 1.5cm centered at point O. c) Find the scale factor, in centimeters, from triangle ABC to triangle A B C. 62. Answer the following questions using the figures below: a) What transformation was performed? b) Find the line of reflection.

63. Given: <1 is supplementary to <2 Prove: l 1 l 2 Statement <1 is supplementary to <2 Given <1 is supplementary to <3 Linear pairs form supplementary angles <1 + <2 = <1+<3 Substitution <1=<1 Reflexive property <2 =<3. Subtraction Line 1 is parallel to line 2. Reason If 2 lines are cut by a transversal and there are a pair of corresponding angles then there are parallel lines 64. The triangle on the right has been mapped to the triangle on the left by a 120 o rotation about point P. a) Identify all six pairs of corresponding parts (vertices and sides). b) Write the transformation in function notation. 65. Rotate the triangle ABC 60 around point F using a compass and straightedge.

66. Perform the following construction using construction tools. r m ( ABC) 67. A student has performed the rotation R O,50 o(bc) = B C. What have they done wrong? 68. In the diagram of below, A is the midpoint of, B is the midpoint of, C is the midpoint of, and and are drawn. If and, what is the length of? m C A B 50 C' B' B C They rotated clockwise instead of counterclockwise

69. Given: ABCD is a rectangle, M is the midpoint of AB Prove: DMC is isosceles Statement Reason ABCD is a rectangle, M is the Midpoint of AB AD = CB Given In a rectangle, opposite sides are congruent D A M B C AM = MB. A midpoint divides a segment into two congruent parts <A and <B are right angles. In a rectangle, all angles are right angles <A = <B. All right angles are congruent AMD = BMC. SAS = SAS DM = CM. DMC is isosceles. corresponding parts of congruent triangles are congruent In an Isosceles triangle opposite sides are congruent 70. Given: Parallelogram FLSH, diagonal,, Prove: Statement Reason Parallelogram FLSH, diagonal FGAS LG l FS, HA l FS. Given <LGS and <HAF are right angles. Perpendicular lines form right angles <LGS = <HAF. All right angles congruent LS = FH. In a parallelogram, opposite sides congruent LS is parallel to FH in a parallelogram opposite sides parallel <LSG = < HFA if parallel lines are cut by a transversal alternate interior angles are equal LGS = HAF AAS = AAS

71. Given: Quadrilateral ABCD, diagonal,,,, Prove: ABCD is a parallelogram. Statement reason AFEC, AE=FC, BF l AC, DE l AC, <1=<2 Given <BFA and <DEC are right angles perpendicular lines form right angles <BFA=<DEC all right angles are congruent FE=FE Reflexive property AF=CE subtraction property BFA = DEC AAS=AAS <BAF=<DCE, BA=CD corresponding parts of congruent triangles are congruent BA ll CD if two lines are cut by a transversal and a pair of alternate interior angles are congruent, then there are parallel lines ABCD is a parallelogram a quadrilateral with one set of sides both congruent and parallel is a parallelogram 72. The diagram below shows rectangle ABCD with points E and F on side. Segments and intersect at G, and. Prove: Statement reason Rectangle ABCD, <ADG=<BCG given AD=BC in a rectangle, opposite sides are congruent <A and <B are right angles in a rectangle, all angles are right angles <A= <B all right angles congruent ADF = BCE ASA=ASA AF=BE corresponding parts of congruent triangles are congruent EF =EF reflexive property AE=BF subtraction property

73. Given:,,,, Prove: Statement Reason AFCD, AB l BC, DE l EF, BC ll FE, AB =DE given <B and <E are right angles perpendicular lines form right angles <B = <E All right angles are congruent <BCA=<EFD if parallel lines are cut by a transversal, alternate interior angles are congruent ABC = DEF AAS=AAS AC=FD corresponding parts of congruent triangles are congruent ABCD is a rectangle 74. Given: BE CF Prove: DE AF D A C F E B Statement Reason ABCD is a rectangle, BE=CF given DC=AB in a rectangle, opposite sides are congruent FE=FE reflexive property CE=BF addition postulate <C and <B are right angles in a rectangle, all angles are right angles <C=<B all right angles congruent DCE = ABF SAS=SAS DE=AF corresponding parts of congruent triangles are congruent

75. In the diagram, and intersect at E. If m AED = 9x + 10 and m BEC = 2x + 52, find the value of x. 76. In the diagram, transversal intersects parallel lines and PQ m RAN = 3x + 24 and m RBQ = 7x 16, find the value of x. at A and B, respectively. If 77. In the diagram, parallel lines and are intersected by transversal at G and H, respectively. If m AGH = 4x + 30 and m GHD = 7x 9, what is the value of x? 78. In the accompanying figure, AB intersects CD. Solve for x and y.

79. In the diagram of shown below, is drawn from vertex B to point A on, such that. In,,, and. In, and. a) Find. b) Find. c) Find the length of. 80. Construct the image of QRS after a dilation with a scale factor of 2 with Q as the center of dilation. 81. Construct the image of ABC after a dilation with center O and a scale factor of 3. 82. Construct the image of ABC after a dilation with center O and a scale factor of ½.

83. A B C is the image of ABC under a dilation. Use a straightedge to determine the center of dilation. 84. A student who is 72 inches tall wants to find the height of a flagpole. He measures the length of the flagpole s shadow and the length of his own shadow at the same time of day, as shown in his sketch below. Explain the error in the student s work. The proportion is incorrect because 48 and 128 should be flipped so it is big over small for both sides of the proportion 85. A girl 160 cm tall, stands 360 cm from a lamp post at night. Her shadow from the light is 90 cm long. How high is the lamp post? 160 cm 90 cm 360 cm 86. Given: NPVR is a parallelogram Prove: NOW ~ SWT Statement Reason NPVR is a parallelogram Given NP ll RV in a parallelogram, opposite sides are parallel <NOT=<OTS when parallel lines are cut by a transversal <ONS=<TSN alternate interior angles are congruent NOW SWT AA=AA

87. XY Z is a dilation of XYZ from vertex X. YY = 2.4, XY = 6, YZ = 10 and XZ = 8. What is the length of Y Z? 88. Given BAT with coordinates B(-3,1), A(1,4) and T(5,-3). Perform a dilation of BAT from center O(0,0) and a scale factor of 2. Graph and determine the coordinates of the images of points B, A and T. 89. Given the diagrams below, state whether or not a dilation that maps A to A and B to B exists. Explain. a) Yes, AB ll A'B' b) No, AB ll A'B'

90. Given: D is the midpoint of AC AC BD Prove: ABC is isosceles Statement Reason D is he midpoint of AC, AC l BD given AD = CD a midpoint divides a segment into 2 congruent parts <ADB and <CDB are right <'s perpendicular lines form right angles <ADB=<CDB all right angles are congruent BD = BD reflexive property ADB. = CDB SAS= SAS <A = <B corresponding parts of congruent triangles are congruent ABC is isosceles isosceles triangles have 2 congruent sides 91. Write the following composition of transformations in function notation: 92. Complete the chart to the right. Sequence of rigid motions Composition in function notation Triangle congruence statement

93. A wall casts that is 4 feet tall casts a shadow that is 6 feet long. How tall is a tree that casts a 24 ft shadow? 94. At noon, Jimmy is standing 22 feet from a flagpole. The sun shines down on Jimmy and casts a shadow 5 feet long. How tall is the flagpole if it casts a 27 foot shadow? 95. Solve for x. 96. Given: j ll k ll m Solve for each of the following if BC = x + 2; BA = 9; EF = x + 3; ED = 12. a) x = b) BC = c) EF =

97. Given: Isosceles triangle ACE with AC AE BDE FDC Statements Reasons Prove: CBD ~ EFD Isosceles triangle ACE with Given AC = AE, <BDE = <<FDC <C = <E in a triangle, angles opposite congruent sides are congruent <CDB +<BDE = 180 Linear pairs form supplementary angles <EDF +<FDC = 180 <CDB+<BDE=<EDF+<FDC substitution property <CDB=<EDF subtraction CBC EFD AA=AA 98. Given: AC and BD intersect at E AB CD Statements Prove: A B AEB ~ CED D E C AC and BD intersect ate, AB ll CD <A= <C <B=<D Reasons given if parallel lines are cut by a transversal, alternate interior angles are congruent AEB CED 99. Solve for x and y y = x = 5 2 3 5y 2 10 3x + 2 x

100. Perform the following similarity transformation: D 1 O, (T AN ( URB)) 2 O 101. A similarity transformation for triangle ABC is described by (D O,2 (r DG ( ABC))). Locate and label the image of triangle ABC under the similarity.

102. Dilate circle A by a scale factor of 2 with O being the center of dilation. 103. Dilate circle C with radius CA from center O with a scale factor r = 1 2. 104. Given: AB and FD bisect each other. Statements Prove: ADE EFB AB and FD bisect each other Reasons Given AE = EB, DE=EF A bisect or divides a segment into 2 congruent parts <AED=<BEF vertical angles are congruent ADE = EFB SAS=SAS

105. For each given pair of triangle, determine if the triangles are similar and provide your reasoning. If the triangles are similar, write a similarity statement relating the triangles. a. b. c. 106. The coordinates of triangle ABC are A(5,4), B(3,4) and C(1,1). State the coordinates of triangle A B C after R 0,90. 107. State the coordinates of the image of trapezoid ABCD after a transformation T 3,-4.

108. State the coordinate of the image of triangle XYZ after a transformation r x-axis. 109. Name and draw in a translation vector that takes ABC to A B C. 110. Translate triangle ABC across vector DE.

111. Translation, rotation, reflection What are the rigid motions? 112. What transformation is NOT a rigid motion? Dilation 113. State 4 degrees of rotational symmetry for each of the following regular polygons. a) Octagon b) Hexagon 114. True or False? ABC is dilated with a scale factor of r =4. The image is A' B' C'. a) ABC A' B' C' b) ABC A' B' C' c) AB 4( A' B' ) d) 4( AB) A' B' e) 1 ( A' B') AB 4 f) BC AB B' C' A' B' g) BA B' C' BC B' A' h) 4( B) B' 115. Alexandra is placing a pond in the backyard of their house. Alexandra would like the pond, the bench, and the swing set, to be equidistant from each other. A sketch of the yard appears below, with the centers of the bench and swing set listed as B and S. Using a compass and a straightedge show all possible location(s) for the pond and mark with an X. Front Yard

Do Now Day 1 1. Dilate circle A by a scale factor of 1/2 if A is the center of dilation. 2. Dilate circle A by a scale factor of 2 if O is the center of dilation.

Do Now Day 2 1. Determine the number of degrees of rotational symmetry. 2. Solve for x in the following if AB CD. 3. Solve for x in the following if AB CD.

1. Given: FE BC AB ED AF CD Prove: AF DC Do Now Day 3 Statements Reasons 2. Given: <ADC and <BEA are right angles Prove: ADC~ AEB Statements Reasons

Do Now Day 4 For each given pair of triangles, determine if the triangles are similar and provide your reasoning. If the triangles are similar, write a similarity statement relating the triangles. a. b. c.

Name: Teacher: Geometry Per: