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Unit 5A.1 โ Ratios Student Learning Targets: ๏ท I can solve proportions. Notes: -1- Notes (Continued for 5A.1): -2- Assignment 5A.1 Solve each proportion. 1. 6 21 = ๐ฅ 31.5 2. 6 9 = 18.2 ๐ฆ 3. ๐ฅ 11 = 4 โ6 4. 11 55 = 20 20๐ฅ 5. 4๐ฅ 56 = 24 112 6. 6๐ฅ = 43 27 7. 4๐ฅ โ 5 โ26 = 3 6 2๐ฅ + 5 42 = 10 20 9. 3๐ฅ โ 1 2๐ฅ + 4 = 4 5 10. ๐+2 3 = ๐โ2 2 8. 11. 3๐ฅ โ 6 4๐ฅ โ 2 = 2 4 -3- 12. 7 9 = ๐งโ1 ๐ง+4 Unit 5A.2 โ Similar Triangles Student Learning Targets: ๏ท I can prove that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar (AA) using the properties of similarity transformations. ๏ท I can solve problems in context involving sides or angles of congruent or similar triangles. Angle-Angle (AA) Similarity โ If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Notes: -4- Notes (Continued for 5A.2): -5- Assignment 5A.2 Prove why the triangles shown are similar. Write a similarity statement. 1. 4. 2. 3. 5. 6. Find the indicated measure (show your work). Draw a picture if one is not provided for you. 7. Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1pm. The length of the shadow was 2 feet. Then he measured the length of the Sears Towerโs shadow and it was 242 feet at the same time. What is the height of the Sears Tower? 8. Hallie is estimating the height of the Superman roller coaster. She is 5 feet 3 inches tall and her shadow is 3 feet long. If the length of the shadow of the roller coaster is 40 feet, how tall is the roller coaster? -6- Assignment 5A.2 (Continued) 9. A local furniture store sells two versions of the same chair: one for adults and one for children. Find the value of x such that the chairs are similar. 10. The two sailboats shown are participating in a regatta. Find the value of x. 11. Adam is standing next to the Palmetto Building in Columbia, South Carolina. He is 6 feet tall and the length of his shadow is 9 feet. If the length of the shadow of the building is 322.5 feet, how tall is the building? 12. A cell phone tower casts a 100-foot shadow. At the same time, a 4-foot 6-inch post near the tower casts a shadow of 3 feet 4 inches. Find the height of the tower. 13. Angie is standing next to a statue in the park. If Angie is 5 feet tall, her shadow is 3 feet long and the statueโs shadow is 10 ½ feet long, how tall is the statue? 14. When Alonzo, who is 5โ11โ tall, stands next to a basketball goal, his shadow is 2โ long, and the basketball goalโs shadow is 4โ4โ long. About how tall is the basketball goal? -7- Unit 5A.3 โ Similar Polygons Student Learning Targets: ๏ท ๏ท ๏ท I can understand that in similar triangles corresponding sides are proportional and corresponding angles are congruent. I can find lengths of measures of sides and angles of congruent and similar triangles. I can decide whether two figures are similar using properties of transformations. Distance Formula: d = Notes: -8- Notes (Continued for 5A.3): -9- Assignment 5A.3 List all pairs of congruent angles and write a proportion that relates the corresponding sides. โ๐จ๐ฉ๐ช~โ๐๐๐ฟ โ๐ญ๐ฎ๐ฏ~โ๐ฑ๐ฒ๐ณ 1. 2. โ๐จ๐ฉ๐ช~โ๐น๐บ๐ป 3. Explain whether each pair of figures is similar. If so, write the similarity statement. 4. 5. 6. Each pair of polygons is similar. Find the missing values. 7. 8. โ๐ด๐ต๐ถ~โ๐ท๐ธ๐น, ๐๐๐๐ ๐ท๐น ๐๐๐ ๐ธ๐น -10- โ๐ฝ๐พ๐ฟ~โ๐๐๐, ๐๐๐๐ ๐ฅ ๐๐๐ ๐ฆ Assignment 5A.3 (Continued) 9. Find x, ED and FD. 10. Find x, ST and SU. Find each measure. 11. BE and AD 12. QP and MP 13. WR and RT 14. RQ and QT -11- Assignment 5A.3 (Continued) Identify the similar triangles. Then find each measure. 15. JK 16. ST 17. WZ and UZ 18. HJ and HK Verify that the dilation is a similarity transformation. 19. Original: A(-6, 3), B(3, 3), C(3, -3) 20. Original: (-6, -3), (6, -3), (-6, 6) Image: X(-4, -2), Y(2, 2), Z(2, -2) 2) Image: (-2, -1), (2, -1), (-2, -12- Unit 5A.4 โ Dilations Student Learning Targets: ๏ท Given a line segment, a point not on the line segment, and a dilation factor, I can construct a dilation of the original segment. (10-8) ๏ท I can recognize that the length of the resulting image is the length of the original segment multiplied by the scale factor and that the original and dilated image are parallel to each other (10-8) ๏ท I can use coordinate geometry to divide a segment into a given ratio. Distance Formula: d = Notes: -13- Notes (Continued for 5A.4): -14- Assignment 5A.4 Use line AB and a ruler to construct a line with the given dilation factor x with endpoint C. 1. 2. ๐=3 ๐ = 0.5 โ๐ถ โ๐ถ 3. ๐ = 1.5 4. ๐ = 2.25 โ๐ถ โ๐ถ Determine whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation and x. 5. from Q to Qโ Bโ 6. from B to 7. from W to Wโ Wโ 8. from W to 9. from W to Wโ 10. from W to Wโ -15- ๐จ ๐ฉ Assignment 5A.4 (Continued) 11. from K to Kโ 12. from K to Kโ DISCOVERY ACTIVITY Graph segment AB and then find point C on the line such that AC and CB form the indicated ratio. 13. Graph endpoint A and the point B that is on the line. Find endpoint C such that AB and BC form the indicated ratio. A(3, 2), B(6, 8); 2:1 ratio 14. Endpoint:A(2, -1), point on line:B(4, 2); 1:3 ratio -16- Unit 5B.1- Parallel Lines Student Learning Targets: I can prove and use theorems about lines and angles, including but not limited to: ๏ท ๏ท ๏ท ๏ท Vertical angles are congruent. When parallel lines are cut by a transversal congruent angle pairs are created. When parallel lines are cut by a transversal supplementary angle pairs are created. Points on the perpendicular bisector of a line segment are equidistant from the segmentโs endpoints. Notes: -17- Notes (Continued for 5B.1): -18- Assignment 5B.1 Find the measure of each numbered angle, and name the theorems that justify your work. 1. ๐โ 2 = 26 2. ๐โ 4 = 3(๐ฅ โ 1), ๐โ 5 = ๐ฅ + 7 In the figure, ๐โ ๐ = ๐๐๐. Find the measure of each angle. Tell which postulate(s) or theorem(s) you used. 3. โ 6 4. โ 7 5. โ 5 Find the value of the variable(s) in each figure. Explain your reasoning. 6. 7. Find each measure. 8. XW 9. LP 10. AC -19- Assignment 5B.1 (Continued) Copy and complete the proof using the list of properties to the right. Answer Bank 11. ๐โ 3 = ๐โ 3 Given Substitution Definition of complementary angles Reflexive property Definition of congruent angles Transitive Symmetric 12. GivenAnswer Bank Definition of linear pair Corresponding Angles Vertical Angles Alternate interior angle theorem Substitution -20- Assignment 5B.1 (Continued) 13. Prove โ 1 โ โ 8 1 2 3 4 5 6 7 8 -21- Unit 5B.2- Triangles Student Learning Targets: I can prove and use theorems about triangles including, but not limited to: ๏ท Prove that the sum of the interior angles of a triangles = 180. ๏ท Prove that the base angles of an isosceles triangle are congruent. Prove that if two angles of a triangle are congruent, the triangle is isosceles. ๏ท Prove the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. ๏ท Prove the medians of a triangle meet at a point. Notes: -22- Notes (Continued for 5B.2): -23- Assignment 5B.2 Find the measures of each numbered angle. 1. 2. Find each measure. 3. ๐โ ๐ต๐ด๐ถ 4. ๐โ ๐๐ ๐ 5. ๐๐ 6. ๐ถ๐ต 7. Write a paragraph explaining why the segment joining midpoints of two sides of a triangle is parallel to the third side. -24- Assignment 5B.2 (Continued) 8. Write a two-column proof for the following. 9. Write a two column proof. ฬ ฬ ฬ โ ๐๐ ฬ ฬ ฬ ฬ ฬ , ๐ฝ๐ฟ ฬ โ ๐๐ฟ ฬ ฬ ฬ ฬ , ๐๐๐ ๐ฟ ๐๐๐ ๐๐๐ก๐ ๐พ๐ ฬ ฬ ฬ ฬ ฬ Given: โ ๐ฝ โ โ ๐, ๐ฝ๐พ Prove: โ๐ฝ๐ฟ๐พ โ โ๐๐ฟ๐ 10. Prove that the base angles of an isosceles triangle are congruent. -25- Unit 5B.3- Prove Theorems Involving Similarity Student Learning Targets: ๏ท I can prove that a line constructed parallel to one side of a triangle intersecting the other two sides of the triangle divides the intersected sides proportionally. ๏ท I can prove that a line that divides two sides of a triangle proportionally is parallel to the third side. I can prove that if three sides of one triangle are proportional to the corresponding sides of another triangle, the triangles are similar. I can prove the Pythagorean Theorem using similarity. ๏ท ๏ท Notes: -26- Notes (Continued for 5B.3): -27- Assignment 5B.3 1. A triangle is intersected by a segment parallel to one side. Prove that the result is a proportional division of the sides. 2. Prove the Pythagorean Theorem using similarity. -28- Assignment 5B.3 (Continued) 3. Prove that if three sides of one triangle are proportional to the corresponding sides of another triangle, the triangles are similar. 4. Prove that a line that divides two sides of a triangle proportionally is parallel to the third side. -29- Unit 5B.4 โSolving Problems Using Congruency and Similarity Student Learning Targets: ๏ท ๏ท ๏ท I can find lengths of measures of sides and angles of congruent and similar triangles. I can solve problems in context involving sides or angles of congruent or similar triangles. I can prove conjectures about congruence or similarity in geometric figures using congruence and similarity criteria. Notes: -30- Notes (Continued for 5B.4): -31- Assignment 5B.4 Find the value of the variable that yields congruent triangles. Explain. 1. 2. 3. 4. 5. A high school wants to hold a 1500-meter regatta on Lake Powell but is unsure if the lake is long enough. To measure the distance across the lake, the crew members locate the vertices of the triangles below and find the measures of the lengths of triangle HJK as shown below. a. Explain how the crew team can use the triangles formed to estimate the distance FG across the lake. -32- Assignment 5B.4 (Continued) b. Using the measures given, is the lake long enough for the team to use as the location for their regatta? Explain your reasoning. 6. The length of George Washingtonโs face at Mt. Rushmore is 60 feet. Describe a method for determining the length of his nose using similar triangles. Justify your reasoning. Write a two-column proof. 7. 8. -33- Assignment 5B.4 (Continued) 9. The pattern shown is created using regular polygons. a. What two polygons are used to create the pattern? b. Name a pair of congruent triangles. c. Name a pair of corresponding angles. d. If CB=2 inches, what is AE? Explain. e. What is the measure of angle D? Explain. 10. Write a two-column proof. -34- Unit 5B.5- Parallelograms Student Learning Targets: I can prove and use theorems about parallelograms including, but not limited to: ๏ท Opposite sides of a parallelogram are congruent. ๏ท Opposite angles of a parallelogram are congruent. ๏ท The diagonals of a parallelogram bisect each other ๏ท Rectangles are parallelograms with congruent diagonals. Notes: -35- Notes (Continued for 5B.5): -36- Assignment 5B.5 Determine whether each quadrilateral is a parallelogram. Justify your answer. 1. 2. 3. 4. Find the value of each variable in each parallelogram. 5. 6. 7. 8. -37- Assignment 5B.5 (Continued) The quadrilateral WXYZ is a rectangle. Use it to answer questions 9-14. 9. If ๐๐ = 2๐ฅ + 3 and ๐๐ = ๐ฅ + 4, find ๐๐. 10. If ๐๐ = 3๐ฅ โ 5 and ๐๐ = 2๐ฅ + 11, find ๐๐. 11. If ๐โ ๐๐๐ = 2๐ฅ โ 7 and ๐โ ๐๐๐ = 2๐ฅ + 5., find ๐โ ๐๐๐. 12. If ๐๐ = 4๐ฅ โ 9 and ๐๐ = 2๐ฅ + 5, find ๐โ ๐๐๐. 13. If ๐โ ๐๐๐ = 3๐ฅ + 6 and ๐โ ๐๐๐ = 5๐ฅ โ 12, find ๐โ ๐๐๐. 14. If ๐โ ๐๐๐ = ๐ฅ โ 11 and ๐โ ๐๐๐ = ๐ฅ โ 9, find ๐โ ๐๐๐. 15. Four jets are flying in formation. Three of the jets are shown in the graph. If the four jets are located at the vertices of a parallelogram, what are the three possible locations of the missing jet? 16. Write a two-column proof. -38- Assignment 5B.5 (Continued) 17. Write a paragraph proof showing that a rectangle is a parallelogram with congruent diagonals. -39- Unit 5C.1 โ Basic Trig Functions Student Learning Targets: ๏ท I can understand that the ratio of two sides in one triangle is equal to the ratio of the corresponding two sides of all other similar triangles. ๏ท I can define sine, cosine, and tangent as the ratio of sides in a right triangle. ๏ท I can demonstrate the relationship between sine and cosine in the acute angles of a right triangle. ๏ท I can explain the relationship between the sine and cosine in complementary angles. Pythagorean Theorem: ๐๐ + ๐๐ = ๐๐ Notes: -40- Notes (Continued for 5C.1): -41- Assignment 5C.1 Find the reduced ratio of the following trig functions. 1. SIn J 3. 2. Sin J Cos J Cos J Tan J Tan J Sin L Sin L Cos L Cos L Tan L Tan L SIn J 4. Sin J Cos J Cos J Tan J Tan J Sin L Sin L Cos L Cos L Tan L Tan L Fill in the sides of the special right triangles and express each trigonometric ratio as a reduced fraction. 5. 6. 30° 60° -42- Assignment 5C.1 (Continued) 7. 8. 5 12 45° 9. 4 10. 10 11. Find the second acute angle of a right triangle given that the first acute angle has measure of 39°. Complete the following statement: 1 1 If sin 30°=2 , then the cos_____ = 2 12. Explain why the sine of x is the same regardless of which triangle is used to find it in the figure. x -43- Unit 5C.2 โ Pythagorean Theorem and Trig Ratios Student Learning Targets: ๏ท I can use the Pythagorean Theorem and trigonometric ratios to find missing measures in triangles in contextual situations. Pythagorean Theorem: ๐๐ + ๐๐ = ๐๐ Notes: -44- Notes (Continued for 5C.2): -45- Assignment 5C.2 1. Leah wants to see a castle in an amusement park. She sights the top of the castle at an angle of elevation of 38°. She knows that the castle is 190 feet tall. If Leah is 5.5 feet tall, how far is she from the castle to the nearest foot? 2. A hockey player takes a shot 20 feet away from a 5-foot goal. If the puck travels at a 15° angle of elevation toward the center of the goal, will the player score? 3. A search and rescue team is airlifting people from the scene of a boating accident when they observe another person in need of help. If the angle of depression to this other person in need of help. If the angle of depression to this other person is 42° and the helicopter is 18 feet above the water, what is the horizontal distance from the rescuers to this person to the nearest foot? 4. A lifeguard is watching a beach from a line of sight 6 feet above the ground. She sees a swimmer at an angle of depression of 8°. How far away from the tower is the swimmer? -46- Assignment 5C.2 (Continued) 5. The cross bar of a goalpost is 10 feet high. If a field goal attempt is made 25 yards from the base of the goalpost that clears the goal by 1 foot, what is the smallest angle of elevation at which the ball could have been kicked to the nearest degree? 6. A digital camera with a panoramic lens is described as having a view with an angle of elevation of 38°. If the camera is on a 3-foot tripod aimed directly at a 124-foot-tall monument, how far from the monument should you place the tripod to see the entire monument in your photograph? 7. A teenager whose eyes are 5โ above ground level is looking into a mirror on the ground and can see the top of a building that is 30โ away from the teenager. The angle of elevation from the center of the mirror to the top of the building is 75°. How far away from the teenagerโs feet is the mirror? How tall is the building? 8. To estimate the height of a tree she wants removed, Mrs. Long sights the treeโs top at a 70° angle of elevation. She then steps back 10 meters and sights the top at a 26° angle. If Mrs. Longโs line of sight is 1.7 meters above the ground, how all is the tree to the nearest meter? -47- Assignment 5C.2 (Continued) 9. Austin is standing on the high dive at the local pool. Two of his friends are in the water as shown. If the angle of depression to one of his friends is 40°, and 30° to his friend who is 5 feet beyond the first, how tall is the platform? 10. While traveling across flat land, you see a mountain directly in from of you. The angle of elevation to the peak is 3.5°. After driving 14 miles closer to the mountain, the angle of elevation is 9° 24โฒ 36". Explain how you would set up the problem, and find the approximate height of the mountain. -48- Unit 5C.3 โ Trig Functions and the Pythagorean Identity Student Learning Targets: ๏ท ๏ท Given sin (ฮธ), cos (ฮธ), or tan (ฮธ) for 0< ฮธ <90, I can find sin (ฮธ), cos (ฮธ), or tan (ฮธ) I can prove ๐ ๐๐2 ๐ + ๐๐๐ 2 ๐ = 1 for right triangles in the first quadrant. Notes: -49- Notes (Continued for 5C.3): -50- Assignment 5C.3 Let ๐ฝ be an acute angle of a right triangle. Find the values of the other two trig functions, (๐๐๐๐ฝ, ๐๐๐ ๐ฝ, ๐๐๐๐ฝ ). 5 1. ๐ ๐๐๐ = 6 7 2. ๐ก๐๐๐ = 3 3. ๐๐๐ ๐ = 5 12 ๐๐๐ ๐ = ๐ก๐๐๐ = ๐ ๐๐๐ = ๐๐๐ ๐ = ๐ ๐๐๐ = ๐ก๐๐๐ = Work through the following problems to discover a Pythagorean Trig Identity Use Angle L 4. 4 3 5 ๐ ๐๐๐ = ___________ ๐๐๐ ๐ = __________ ๐ ๐๐2 ๐ =___________ ๐๐๐ 2 ๐ = __________ ๐ ๐๐2 ๐ + ๐๐๐ 2 ๐ = _________ Use Angle J 5. 6 13 ๐ ๐๐๐ = ___________ ๐๐๐ ๐ = __________ ๐ ๐๐2 ๐ =___________ ๐๐๐ 2 ๐ = __________ ๐ ๐๐2 ๐ + ๐๐๐ 2 ๐ = _________ What does ๐ ๐๐2 ๐ + ๐๐๐ 2 ๐ ALWAYS equal? _________________ -51-