Lesson a: They are congruent by ASA or AAS. b: AC 9.4 units and DF = 20 units

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1 Lesson a: They are congruent by ASA or AAS. b: AC 9.4 units and DF = 20 units 7-7. Relationships used will vary, but may include alternate interior angles, Triangle Angle Sum Theorem, etc.; a = 26, b = 65, c = 26, d = width = 60 mm, area = 660 mm A quadrilateral a: x = 18, y = 9 3 b: x = 24 2, y = 24 c: x = 8 = , y = 16 = a: (6, 13) b: Not possible, these curves do not intersect. Selected Answers 2014 CPM Educational Program. All rights reserved. 1

2 Lesson Using the Pythagorean Theorem, AB = 8 and JH = 5. Then, since 3 6 = 4 8 = 5 10, ΔABC ~ ΔHGJ because of SSS ~ cm Line L: y = 1 6 x + 6 ; line M: y = 2 3 x +1; point of intersection: (6, 5) a: 3m = 5m 28, m = 14 b: 3x x 8 = 180, x = 15 c: 2(n + 4) = 3n 1, n = 9 units d: 2(3x +12) = 11x 1, x = 5 units Rotating about the midpoint of a base forms a hexagon (one convex and one nonconvex). Rotating the trapezoid about the midpoint of either of the non-base sides forms a parallelogram a: 10 units b: ( 1, 4) c: 5 units, it must be half of AB because C is the midpoint of AB. Lesson (Day 1) a: The 90 angle is reflected, so m XYZ = 90º. Then m YZY = 180º M(0, 7) b: They must be congruent because rigid transformations (such as reflection) do not alter shape or size of an object. c: XY X Y, XZ XZ, YZ Y Z, Y Y, YXZ Y XZ, and YZX Y ZX c 2 and a 2 + b a: The triangles are similar by SSS ~. b: The triangles are similar by AA ~. c: Not enough information is provided. d: The triangles are congruent by AAS or ASA a: It is a parallelogram; opposite sides are parallel. b: 63.4 ; They are equal. c: AC : y = 1 2 x + 1 2, BD : y = x + 5 ; No d: (3, 2) a: No solution, lines are parallel. b: (0, 3) and (4, 11) Selected Answers 2014 CPM Educational Program. All rights reserved. 2

3 Lesson (Day 2) Side length = 50 units; diagonal is 50 2 = 100 = 10 units a: It is a rhombus. It has four sides of length 5 units. b: HJ : y = 2x + 8 and GI : y = 1 2 x + 3 c: They are perpendicular. d: (6, 1) e: 20 square units a: P(scalene) = 1 4 b: P(isosceles) = 2 4 = 1 2 c: P(side of the triangle is 6 cm) = 2 4 = a: 6n 3 = n +17, n = 4 b: 7x x +14 = 180, so x = Then 5y 2 = 7(18.5) 19, so y = 22.5 c: 5w w = 180, w = 18 d: k 2 = (15)(25) cos120, k = The graph is a parabola with roots ( 3, 0) and (1, 0), and y-intercept at (0, 3) m Selected Answers 2014 CPM Educational Program. All rights reserved. 3

4 Lesson a: = 10 sides b: Regular decagon If the diagonals intersect at E, then BE = 12 mm, since the diagonals are perpendicular bisectors. Then ΔABE is a right triangle and AE = = 9 mm. Thus, AC = 18 mm Yes, she is correct. Show that the lengths on both sides of the midpoint are equal and that (2, 4) lies on the line that connects ( 3, 5) and (7, 3) a: See flowchart at right. b: Not similar because corresponding sides do not have the same ratio. c: See below. AA ~ ΔFED ~ ΔBUG SSS ~ (a) and (c) are correct because if the triangles are congruent, then corresponding parts are congruent. Since alternate interior angles are congruent, then AB // DE AB = , BC = , therefore C is closer to B. Selected Answers 2014 CPM Educational Program. All rights reserved. 4

5 Lesson a: x = 8.5 b: x = 11 c: x = a: = 5 sides b: = feet from the point on the street closest to the art museum a: a n = n = (n 1) b: a n = 6( 1 2 )n = 3( 1 2 )n a: (0.7)(0.7) = 0.49 = 49% b: (0.3)(0.7) = 0.21 = 21% a: Similar (SSS ~) b: Congruent (ASA or AAS ) c: Similar, because if the Pythagorean Theorem is used to solve for each unknown side, then 3 pairs of corresponding sides have a common ratio; thus, the triangles are similar (SSS ~). d: Similar (AA ~) but not congruent since the two sides of length 12 are not corresponding Possible response: Rotate the second triangle 180 and then translate it to match the sides with the first triangle. Selected Answers 2014 CPM Educational Program. All rights reserved. 5

6 Lesson x 1 = x + 8, x = 3 ; 5y + 2 = 22, y = a: 0.8 b: 1200(0.8) 3 = $ c: 1200(0.8) 2 = $ a: a = 36, r = 54, m = 54, y = 72, z = 108 b: Possible response: y and z are supplementary interior same side angles a: It is a parallelogram, because MN // PQ and NP // MQ. b: (1, 5) E is a midpoint Given Vertical angles are congruent. AEB CED Definition of midpoint. Definition of midpoint. ΔAEB ΔCED SAS a: 50%; The sum must be 100% b: central angle for red = 0.4(360 ) = 144, white = 0.1(360 ) = 36, blue = 0.5(360 ) = 180 c: Yes; there could be more than three sections to the spinner, but the ratio of the areas for each color must match the ratios for the spinner in part (b). Selected Answers 2014 CPM Educational Program. All rights reserved. 6

7 Lesson a: Congruent (SSS ) b: Not enough information c: Congruent (ASA ) d: Congruent (HL ) See answers in problem a: 83 b: a: Yes; HL b: 18, 4 c: tan18 = AD 4, AD 12.3 units d: 49.2 square units a: Parallelogram because the opposite sides are parallel. b: AC : y = 4 3 x ; BD : y = 2 3 x a: , since 64 = 8, then 68 must be a little greater. b: (1) 2.2, (2) 9.2, (3) 7.1, (4) a: 2x + 52 = 180, 64 b: 4x 3 + 3x +1 = 180, 26 c: sin 77 x = sin 72 8, x 8.2 d: 5x + 6 = 2x + 21, x = 5 Lesson square units No. Using the Pythagorean Theorem and the Law of Cosines, the perimeter of the triangle is 26.3 feet a: x cm, tangent b: x 7.86 mi, Law of Sines c: x 15.3', Law of Cosines a: Congruent (SAS ) and x = 2 b: Congruent (HL ) and x = A = 24 square units a: 20 4 = 1 5 b: 4 5 ; Since the sum of the probabilities of finding the ring and not finding the ring is 1, you can subtract = 4 5. c: No, his probability is still 20 4 = 1 5 because the ratio of the shaded region to the whole sandbox is unchanged a 50 = 130 Selected Answers 2014 CPM Educational Program. All rights reserved. 7

8 Lesson a: See diagram at right. b: Since corresponding parts of congruent triangles are congruent, 2y + 7 = 21 and y = y-intercept: (0, 6), x-intercept: (4, 0) A T 21 P m a = 132º, m b = 108º, and m c = 120º, m a + m b + m c = 360º % AB CD and AB // CD (given), so BAC DCA (alt. int. angles). AC CA (Reflexive Property) so ΔABC ΔCDA (SAS ). BCA DAC ( s parts ). Thus, BC // AD (if alt. int. angles are congruent, then the lines cut by the transversal are congruent) Because alternate interior angles are congruent, the angle of depression equals the angle formed by the line of sight and the ground. Then tan θ = 52 38, θ a: ΔADC; AAS or ASA b: ΔSQR; HL c: No solution, only angles are congruent. d: ΔTZY; SAS and vertical angles e: ΔGFE; alternate interior angles equal and ASA f: ΔDEF; SSS Selected Answers 2014 CPM Educational Program. All rights reserved. 8

9 Lesson ZY WX, YZX WXZ, ZYW YWX (alt. int. s), ΔXWM ΔZYM, ASA, YM WM and XM ZM, Δ parts Typical responses: right angles, congruent diagonals, 2 pairs of opposite sides that are congruent, all sides congruent, congruent adjacent sides, diagonals that bisect each other, congruent angles, etc a: The triangles should be by SSS but 80º 50º. b: The triangles should be by SAS but 80º 90º and 40º 50º. c: The triangles should be by SAS but d: Triangle is isosceles but the base angles are not equal. e: The large triangle is isosceles but base angles are not equal. f: The triangles should be by SAS but sides The triangles described in (a), (b), and (d) are isosceles a: 12 b: 15 c: See reasons in bold below. Statements 1. AD // EH and BF // CG Given Reasons 2. a = b If lines are parallel, alternate interior angle measures are equal. 3. b = c If lines are parallel, corresponding angle measures are equal. 4. a = c Substitution 5. c = d Vertical angle measures are equal. 6. a = d Substitution This problem is similar to the Interior Design problem (7-21). Her sink should be located feet from the right front edge of the counter. This will make the perimeter 25.6 feet, which will meet industry standards. Selected Answers 2014 CPM Educational Program. All rights reserved. 9

10 Lesson a: (4.5, 3) b: ( 3, 1.5) c: (1.5, 2) a: ΔSHR ~ ΔSAK because ΔSHR can be dilated by a factor of 2. b: 2HR = AK, 2SH = SA, SH = HA c: 6 units a: 4 b: 1 c: a: ΔCED; vertical angles are equal, ASA b: ΔEFG; SAS c: ΔHJK; HI + IJ = LK + KJ, J J, SAS. d: Not, all corresponding pairs of angles equal is not sufficient See answers in problem No. Her conclusion in Statement #3 depends on Statement #4, and thus must follow it a: Must be a quadrilateral with all four sides of equal length. b: Must be a quadrilateral with one pair of opposite sides that are parallel. Selected Answers 2014 CPM Educational Program. All rights reserved. 10

11 Lesson a: Yes, each side has the same length ( 29 units). See graph at right. b: BD is y = x ; AC is y = 5 x c: The slopes are 1 and 1. Therefore the diagonals are perpendicular Multiple answers are possible. Any order is valid as long as Statement #1 is first, Statement #6 is last, and Statement #4 follows both Statements #2 and #3. Statements #2, #3, and #5 are independent of each other and can be in any order as long as #2 and #3 follow Statement # a: 6 b: 3 c: a: Yes, by SAS ~. b: FGH FIJ, FHG FJI c: Yes, because corresponding angles are congruent. (Triangle Midsegment Theorem) d: 2(4x 3) = 3x +14, so x = 4 and GH = 4(4) 3 = 13 units a: It is a right triangle because the slopes of AB and AC ( 4 3 and 4 3, respectively) are opposite reciprocals. b: B is at ( 2, 0). It is an isosceles triangle because B AB must be a straight angle (because it is composed of two adjacent right angles) and because BC B C (because reflections preserve length) a: feet b: x 7 yds c: x d: feet a: Must be: trapezoid. Could be: isosceles trapezoid, parallelogram, rhombus, rectangle, and square. b: Must be: parallelogram. Could be: rhombus, rectangle, and square. Selected Answers 2014 CPM Educational Program. All rights reserved. 11

12 Lesson a: (8, 8) b: (6.5, 6) a: X and Y b: Y and Z a: Must be: none. Could be: right trapezoid, rectangle, square. b: Must be: none. Could be: kite, rhombus, square a: If a polygon is a parallelogram, then its area equals its base times its height. b: If the area of a polygon is the base times its height, then the polygon is a parallelogram. This is always true. c: If a polygon is a triangle, then its area equals one half its base times its height. Arrow diagram: Polygon is a triangle area of the polygon equals one-half base times height. d: If a polygon s area is one-half its base times its height, then the polygon is a triangle. This is always true a: = 20 sides b: It can measure 90 (which forms a square). It cannot be 180 (because this polygon would only have 2 sides) or 13 (because 13 does not divide evenly into 360 ) The expected value is 1 4 ($3) ($1) = $1.50 per spin, so each player should pay $1.50 so that there is no net gain or loss over many games a: They are perpendicular because their slopes are opposite reciprocals. b: It could be a kite, rhombus, or square because the diagonals are perpendicular and at least one diagonal is bisected. Selected Answers 2014 CPM Educational Program. All rights reserved. 12