Triangle LMN and triangle OPN are similar triangles. Find the angle measurements for x, y, and z.

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1 1 Use measurements of the two triangles below to find x and y. Are the triangles similar or congruent? Explain. 1a Triangle LMN and triangle OPN are similar triangles. Find the angle measurements for x, y, and z. E x B y M x y N 50 z O R L D U L 53 P The sum of the measures of angles in a triangle is 180. Therefore to find the missing angle measurements, add up the given measurements and subtract this number from b Select all of the following that describe the triangles shown below. x = 180 x = 180 x = x = y = 180 y = 180 y = y = 74 The triangles are similar because they have the same angle measurements. However, we cannot conclude that the triangles are congruent because we do not know their side lengths. 60 A) Both triangles are similar. B) Both triangles are congruent. C) The triangles are congruent but not similar. D) The triangles are similar but not congruent. E) The triangles are neither similar nor congruent. Page 1 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

2 2 Christopher wants to give his friend a baseball bat as a gift for his birthday. He finds a box that is 16 inches wide, 30 inches long, and 12 inches tall to wrap it in. If the bat is 33 inches long, will it fit in the box? If the bat is 36 inches long will fit in the box? Explain. 2a A 25-foot ladder is placed against a building. The bottom of the ladder is 7 feet from the building. If the top of the ladder slips down 4 feet, how many feet will the bottom slide out? 30 in. 12 in. in. 16 in. Since the bat is longer than the box, it will have to fit in the box at an angle. We can use the Pythagorean Theorem, a 2 + b 2 = c 2, to find the length of the hypotenuse and compare it with the length of the bat. 8.G.7 c 30 in. 16 in. 2b In this figure AB and CD are perpendicular. Select all of the following statements that are true. Justify your answers = c = c = c 2 34 = c Since the hypotenuse is 34 inches long, the bat will fit in the box. To find out if a 36-inch long bat will fit in the box, we have to compare it to the diagonal from the top of the box to the bottom of the box. A 12 D 14 C 15 B d 34 in. 12 in. in. A) The length of AD is 5 units. B) The perimeter of CDB is 37 units = d = d = d 2 The bat is 36 inches, and 36 2 = 1296, which is less than Therefore a 36-inch bat can fit in the box. 8.G.7 C) The length of AC is 11 units. D) The perimeter of ABC is 42 units. 8.G.7 Page 2 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

3 3 What is the length of line segment PQ shown below? 3a What is the length of line segment AB shown below? A (-6, 5) B (3, -7) We can use this segment and draw a right triangle. With the segment as the hypotenuse, we can use the Pythagorean Theorem a 2 + b 2 = c 2 to find its length. 3b 8.G.8 Given the points graphed below. Select all of the statements that are true. D (2, 9) c 6 units E (5, 5) 8 units F (2, 1) Counting the grid lines will give us the length of the legs of the right triangle. c 2 = c 2 = c 2 =100 c = ±10 PQ =10 units 8.G.8 A) The distance between D and E is 4 units. B) The distance between E and F is 5 units. C) The distance between D and F is 8 units. D) If the three points were connected in order, the resulting figure would be a right triangle. 8.G.8 Page 3 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

4 4 Determine whether the triangle shown below is a right triangle. 4a Determine whether the triangle with side lengths 9 cm., 40 cm., and 41 cm., is a right triangle If the lengths of the sides of a triangle satisfy the Pythagorean equation, a 2 + b 2 = c 2, then the triangle is a right triangle. Substitute the values into the equation and check if the equation is true. a 2 + b 2 = c = = = 576 False Since the equation is False, the triangle is not a right triangle. 4b 8.G.6 Select all of the statements that are true about the described triangles. A) A triangle with side lengths 8 cm., 15 cm, and 17 cm., is a right triangle. B) A triangle with side lengths 10 in., 20 in., and 24 in., is a right triangle. C) A triangle with side lengths 9 ft., 12 ft., and 18 ft., is a right triangle. D) A triangle with side lengths 12 m., 16 m., and 20 m., is a right triangle. 8.G.6 8.G.6 Page 4 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

5 5 On the coordinate grid below, perform the following series of transformations to the preimage, and then state whether the resulting images are congruent or similar. 5 On the coordinate grid below, perform the following series of transformations to the preimage, and then state whether the resulting images are congruent or similar. 1) Translate ΔABC 6 units right and 1 unit down and label the points A!, B!, and C!. 1) Reflect ΔDEF about the x-axis and label the points D!, E!, and F!. 2) Reflect Δ A " B " C " about the x-axis and label the points A!!, B!!, and C!!. 2) Rotate Δ D " E " F" 90 clockwise about the origin and label the points D!!, E!!, and F!!. A B C E D F A A! B B! C B!! C! C!! 8.G.2 A!! Preimages that are translated or reflected remain congruent to the resulting images; therefore the resulting images above are congruent. 8.G.2 Page 5 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

6 6 If the figure below were reflected about the y-axis, what would be the coordinates of the vertices of the resulting figure? 6a If the figure below were rotated 180 the to the right about the origin, what would be the coordinates of C!? (-7, -2) S (-1, -2) T R (-4, -6) The reflected figure with its vertices is shown below. (1, -2) T! R! (4, -6) (7, -2) S! 6b 8.G.3 Select all of the expressions that describe the translation from ( 3, 4) to (4, 1). A) 7 units left and 5 units up Notice that the x-coordinate of each vertex is the opposite, while the y-coordinate remained unchanged. B) 7 units right and 5 units up C) 7 units left and 5 units down D) 7 units right and 5 units down 8.G.3 8.G.3 Page 6 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

7 7 Parallel lines r and s are intersected by a transversal t. List each pair of corresponding, alternate interior, and alternate exterior angles. If m 2 is 37, what are the measures of angles 3 and 6? 7a Parallel lines m and n are intersected by a transversal l. Find the measures of angels a and b. m n t r 105 a b l s Corresponding angles: 1& 5, 2 & 6, 3& 7, 4& 8 Alternate Interior angles: 3& 6, 4& 5 Alternate Exterior angles: 1& 8, 2& 7 If m 2 is 37, then m 3 = 37 and m 6 = 37 because corresponding angles are congruent and alternate interior angles are congruent. 7b m Parallel lines m and n are intersected by a transversal t. Select all of the statements below that are true. n t A) 2 and 8 are alternate interior angles, therefore they are congruent. B) 1 and 6 are corresponding angles, therefore they are congruent. C) m 3, m 5, and m 7 are all equal to m 1. D) 4 and 7are alternate exterior angles, therefore they are congruent. Page 7 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

8 8 List each pair of supplementary and vertical angles. If m 8 = 50 and m 9 =115, find the measures of angles 2, 3, 6, and 10. 8a In the figure below, line r is parallel to line s. Find the measures of angles a, b, and c. s t r a 54 b s c r Supplementary angles: 1& 4, 1& 2, 2 & 3, 3 & 4, 5& 8, 5& 6, 6 & 7, 7 & 8, 9 & 10, 9 & 12, 10 & 11, 11 & 12 8b In the figure below, lines g and h are parallel. Select all of the statements that are true. g 132 a b c Vertical angles: 1& 3, 2 & 4, 5& 7, 6 & 8, 9 & 11, 10 & 12 If m 8 = 50 then m 6 = 50 because vertical angles are congruent. h A) m d =100. j d 100 e f k If m 9 =115 then m 10 = , or 65 because angles 9 and 10 are supplementary angles that add to 180. m = 180 because the sum of the angles in a triangle = 180. Therefore m 3= 65. B) Angles b and c are supplementary angles. C) Angles d and e are supplementary angles. D) Angles a and c are congruent because they are vertical angles. Since m 3= 65, then m 2 = , or 115 because angles 2 and 3 are supplementary angles that add to 180. E) m a = 48. Page 8 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

9 9 A farm building is shaped like the figure below. Find the volume of the figure. Use 3.14 for π. Round to the nearest tenth. 9a The top of a stone post at the entry to an estate is in the shape of a cone as shown below, with a height of 2.6 ft. and radius of 1.4 ft. Find the volume of stone needed to make the top of the post. Use 3.14 for π. Round to the nearest tenth. The figure is made up of a cone and a cylinder. The total volume is therefore the volume of the cone + the volume of the cylinder. Volume of a cone = πr 2 h 3 Volume of a cylinder = πr 2 h r is the radius and h is the height of the cone and the cylinder. Substitute these values into the equation and solve. We ll use 3.14 for π. Since the diameter of both figures is 50 ft., their radii are 25 ft. each. Vcone + Vcylinder = V total 9b 8.G.9 The figure below shows a large cylinder with a height of 40 mm. and a diameter of 36 mm. It contains a smaller cylinder that has a diameter of 10 mm. πr 2 h + πr 2 h = V total 3! 12 $ (3.14)(25) 2 # & " 3 % + (3.14)(25) 2 (8) = V total = V total = V total The volume of the total figure is 23,550 ft 3. Select all of the following statements that are true. Use 3.14 for π. Values are rounded to the nearest tenth. A) The volume of the large cylinder is mm 3. B) The volume of the small cylinder is 3140 mm 3. 8.G.9 C) If the small cylinder is removed, the remaining volume would be mm 3. D) If the small cylinder is removed, the remaining volume would be 37,554.4 mm 3. 8.G.9 Page 9 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

10 Setting (gas mark) Temperature ( C) The table below shows the thermostat setting on a British gas oven and the corresponding temperatures in degrees Celsius.

10 10 Setting (gas mark) Temperature ( C) The table below shows the thermostat setting on a British gas oven and the corresponding temperatures in degrees Celsius Vince buys gasoline once per week. He keeps track of the price of a gallon of gasoline each time he fills up. The table shows the first five prices in Vince s records. Week Price Per Gallon $1.87 $1.90 $1.95 $1.99 $2.02 A) Make a scatterplot of the data where x represents the thermostat setting (in gas marks) and y represents the temperature (in degrees Celsius). line of best fit A) Make a scatterplot of the data where x represents the weeks and y represents the price per gallon (in dollars). Price Per Gallon (in dollars) Week B) Describe the correlation of the data. The data has a positive correlation. B) Describe the correlation of the data. C) An oven set to gas mark 10 heats to a temperature of 260 C. Does this fit the trend shown by your scatterplot? Explain. This does fit the trend because the coordinate (10, 260) will lie close to the line of best fit. C) If the price of gasoline continues to rise at the same rate, predict the price of a gallon of gasoline for Week 7. Explain how you made your prediction. 8.SP.1 8.SP.1 Page 10 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

11 11 The two-way table below shows the results when 80 students were asked if they have a television in their bedroom and if they have internet access in their bedroom. 11a The two-way table below shows the results when 25 adults were asked whether they had at least one child under the age of 18 and whether they had at least one pet. Internet No Internet Total Television No Television Total At Least One Pet At Least One Child Under 18 No Child Under 18 Total No Pets Total A) What is the relative frequency that a student has internet access in their bedroom, given that they have a television in their room? A) What is the relative frequency that an adult has at least one pet, given that they have at least one child under the age of 18? Of the 80 students asked, 30 of them have television. 10 of this 30 have internet. The relative frequency is therefore 10 or B) What did more than half of all students surveyed have, or not have in their bedroom. More than half of all students surveyed have internet access in their bedroom. Also, more than half of all students do not have a television in their bedroom. 11b B) What did more than half of all adults surveyed have in terms of children under 18 or pets? 8.SP.4 The two-way table shows the results when 120 eighth graders were asked if they receive an allowance and if they have chores at home. No Allowance Total Allowance Chores No chores Total SP.4 Select all of the following statements that are true. A) Less than half of the students who receive an allowance also have chores. B) Less than half of the students surveyed don t have chores. C) The relative frequency of students who have chores given they receive an allowance is SP.4 Page 11 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

12 You Try Solutions: 1b Select all of the following that describe the triangles shown below. 1a Triangle LMN and triangle OPN are similar triangles. Find the angle measurements for x, y, and z M x y N 50 z O L L is similar to O therefore z = 53. P A) Both triangles are similar. B) Both triangles are congruent. C) The triangles are congruent but not similar. y = 50 because similar angles are congruent. The sum of the measures of angles in a triangle is 180. Therefore to find the missing angle measurements, add up the given measurements and subtract this number from 180. D) The triangles are similar but not congruent. E) The triangles are neither similar nor congruent. x = 180 x = 180 x = x = 77 Page 12 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

13 2a A 25-foot ladder is placed against a building. The bottom of the ladder is 7 feet from the building. If the top of the ladder slips down 4 feet, how many feet will the bottom slide out? 2b In this figure AB and CD are perpendicular. Select all of the following statements that are true. Justify your answers. A D Use the Pythagorean Theorem to find the height against the building before the ladder slipped down ft. h C 15 A) The length of AD is 5 units. B 7 ft = h = h = h 2 24 = h B) The perimeter of CDB is 37 units. C) The length of AC is 11 units. D) The perimeter of ABC is 42 units. The ladder starts out at 24 ft. After it slips, its height is now 24 ft. 4 ft., or 20 ft. high on the building. Use this new height with the Pythagorean Theorem to find out how far the ladder is from the building. 3a What is the length of line segment AB shown below? 8.G.7 A (-6, 5) 25 ft. 20 ft. b = b = b = b 2 15 = b 12 units c 9 units B (3, -7) The ladder is now 15 ft. from the building, so it slid out 15ft. 7 ft., or 8 ft. 8.G.7 Use the Pythagorean Theorem to solve for c. c 2 = c 2 = c 2 = 225 c = 15 units 8.G.8 Page 13 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

14 3b Given the points graphed below. Select all of the statements that are true. 4b Select all of the statements that are true about the described triangles. D (2, 9) E (5, 5) A) A triangle with side lengths 8 cm., 15 cm, and 17 cm., is a right triangle. F (2, 1) B) A triangle with side lengths 10 in., 20 in., and 24 in., is a right triangle. C) A triangle with side lengths 9 ft., 12 ft., and 18 ft., is a right triangle. D) A triangle with side lengths 12 m., 16 m., and 20 m., is a right triangle. A) The distance between D and E is 4 units. 8.G.6 B) The distance between E and F is 5 units. C) The distance between D and F is 8 units. D) If the three points were connected in order, the resulting figure would be a right triangle. 8.G.8 4a Determine whether the triangle with side lengths 9 cm., 40 cm., and 41 cm., is a right triangle. Substitute the values into the Pythagorean equation and check if the equation is true. The longest side of a right triangle is the hypotenuse, so c = 41 cm. a 2 + b 2 = c = = = 1681 True Since the equation is true, the triangle is a right triangle. 8.G.6 Page 14 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

15 5 On the coordinate grid below, perform the following series of transformations to the preimage, and then state whether the resulting images are congruent or similar. 6a If the figure below were rotated 180 the to the right about the origin, what would be the coordinates of C!? 1) Reflect ΔDEF about the x-axis and label the points D!, E!, and F!. 2) Rotate Δ D " E " F" 90 clockwise about the origin and label the points D!!, E!!, and F!!. C! D!!! D! F E! E E!! F!! The coordinates of C! are ( 1, 1). D F Preimages that are reflected or rotated remain congruent to the resulting images; therefore, the resulting images above are similar and congruent. 6b 8.G.3 Select all of the expressions that describe the translation from ( 3, 4) to (4, 1). 8.G.2 A) 7 units left and 5 units up B) 7 units right and 5 units up C) 7 units left and 5 units down D) 7 units right and 5 units down 8.G.3 Page 15 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

16 7a Parallel lines m and n are intersected by a transversal l. Find the measures of angels a and b. 8a In the figure below, line r is parallel to line s. Find the measures of angles a, b, and c. m n r a 54 b a b l s c m b =105 because alternate exterior angles are congruent. m a =105 because a corresponds with b and corresponding angles are congruent. 7b m Parallel lines m and n are intersected by a transversal t. Select all of the statements below that are true. n t m a = = 126, because supplementary angles add to 180. m b = 66, because vertical angles are congruent. m c = 60. It is congruent (vertical) to the angle in the triangle that when added to 54 and 66 has the sum of 180, since the angles in a triangle add to b In the figure below, lines g and h are parallel. Select all of the statements that are true. g h 132 a b c d 100 e f j k A) 2 and 8 are alternate interior angles, therefore they are congruent. B) 1 and 6 are corresponding angles, therefore they are congruent. C) m 3, m 5, and m 7 are all equal to m 1. D) 4 and 7are alternate exterior angles, therefore they are congruent. A) m d =100. B) Angles b and c are supplementary angles. C) Angles d and e are supplementary angles. D) Angles a and c are congruent because they are vertical angles. E) m a = 48. Page 16 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

17 9a The top of a stone post at the entry to an estate is in the shape of a cone as shown below, with a height of 2.6 ft. and a radius of 1.4 ft. Find the volume of stone needed to make the top of the post. Use 3.14 for π. Round to the nearest tenth. 10 Vince buys gasoline once per week. He keeps track of the price of a gallon of gasoline each time he fills up. The table shows the first five prices in Vince s records. Week Price Per Gallon $1.87 $1.90 $1.95 $1.99 $2.02 A) Make a scatterplot of the data where x represents the weeks and y represents the price per gallon (in dollars). Substitute the values for the height h and radius r of the cone into the equation for the volume of the cone. Volume of a cone = πr 2 h 3 = (3.14)(1.4) 2! 2.6 $ # & " 3 % = 5.3 ft 3 8.G.9 Price Per Gallon (in dollars) line of best fit 9b The figure below shows a large cylinder with a height of 40 mm. and a diameter of 36 mm. It contains a smaller cylinder that has a diameter of 10 mm Week B) Describe the correlation of the data. The data has a positive correlation. C) If the price of gasoline continues to rise at the same rate, predict the price of a gallon of gasoline for Week 7. Explain how you made your prediction. Select all of the following statements that are true. Use 3.14 for π. Values are rounded to the nearest tenth. A) The volume of the large cylinder is mm 3. By Week 7 the price of a gallon of gasoline would be $2.08. If a line of best fit is drawn for the graph, the point (7, 2.08) would lie near the line. B) The volume of the small cylinder is 3140 mm 3. C) If the small cylinder is removed, the remaining volume would be mm 3. 8.SP.1 D) If the small cylinder is removed, the remaining volume would be 37,554.4 mm 3. 8.G.9 Page 17 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

18 11a The two-way table below shows the results when 25 adults were asked whether they had at least one child under the age of 18 and whether they had at least one pet. 11b At Least One Pet At Least One Child Under 18 No Child Under 18 Total No Pets Total A) What is the relative frequency that an adult has at least one pet, given that they have at least one child under the age of 18? The relative frequency is B) What did more than half of all adults surveyed have in terms of children under 18 or pets? They had at least 1 pet. Also, more than half of all adults surveyed did not have a child under SP.4 The two-way table shows the results when 120 eighth graders were asked if they receive an allowance and if they have chores at home. No Allowance Total Allowance Chores No chores Total Select all of the following statements that are true. A) Less than half of the students who receive an allowance also have chores. B) Less than half of the students surveyed don t have chores. C) The relative frequency of students who have chores given they receive an allowance is SP.4 Page 18 of 18 MCC@WCCUSD (WCCUSD) 09/29/15

Triangle LMN and triangle OPN are similar triangles. Find the angle measurements for x, y, and z.

1 Use measurements of the two triangles elow to find x and y. Are the triangles similar or ongruent? Explain. 1a Triangle LMN and triangle OPN are similar triangles. Find the angle measurements for x,