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Geometry Curriculum Paterson Charter School for Science and Technology 2014-2015 Unit 2 Summary: Similarity, Right Triangles and Trigonometry Essential Questions Big Ideas What are the criteria for triangles to be similar? How does this relate to transformations? How can a line drawn parallel to one of the sides of a triangle be used to solve problems involving triangles? How can the median be used? How can the side of a triangle be partitioned into segments of a given ratio? How can this information be used to solve problems involving similar triangles? How does the perpendicular bisector of a segment relate to isosceles triangles? How can constructions be used to verify this? What are the properties of isosceles triangles and how can they be used to solve problems? What are the criteria for triangles to be congruent? How does this relate to transformations? How does congruency relate to similarity? How can similarity and congruence be used to solve problems and/or prove statements about or properties of triangles? Geometry Curriculum Unit 2: Similarity, Right Triangles and Trigonometry Unit 2 Summary: Similarity, Right Triangles and Trigonometry Learning Outcomes (CCSS) G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: G.SRT.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G.SRT.1bThe dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Define trigonometric ratios and solve problems involving right triangles G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* Apply trigonometry to general triangles G.SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G.SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Geometry Curriculum Unit 2 Similarity, Right Triangles and Trigonometry Unit 2 Summary: Similarity, Right Triangles and Trigonometry 25. 26. 27. 28. 29. Local Standards (LS) #24 are taught in this unit. Verify experimentally the properties of dilations given by a center and a scale factor. Know that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Know the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Use the idea of dilation transformations to develop the definition of similarity. Given two figures determine whether they are similar and explain their similarity based on the equality of corresponding angles and the proportionality of a l l corresponding pairs of sides. 30. 31. 32 33 34 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. Identify and use properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems about triangles: a line parallel to one side of a triangle divides the other two proportionally and its converse. Prove the Pythagorean Theorem using triangles similarity Recognize the characteristics of congruent figures recognize corresponding figures and their corresponding parts. Prove two triangles congruent using the SSS, SAS, ASA Postulate and AAS Theorem. Use triangle congruence and corresponding parts of congruent triangles to prove that parts of two triangles are congruent. Prove geometric figures, other than triangles, are similar and/or congruent. Use properties of isosceles triangles to show that right triangles are congruent and prove right triangles are congruent using the Hypotenuse-Leg Theorem Identify congruent overlapping triangles and prove two triangles congruent using other congruent triangles. Identify characteristics of midsegments and use properties of midsegments in order to solve problems. Identify and use properties of perpendicular bisectors, medians and altitudes to solve problems. Find the sum of the measures of the interior and exterior angles of a polygon Use relationships among sides and angles and diagonals of parallelograms in order to solve problems Determine properties of similar polygons in order to identify and apply them in problem solving. Apply use of AA Similarity Postulate, SSS Similarity Theorem and SAS Similarity Theorem to prove triangles similar. Use triangle similarity to find indirect measurements. Find and use relationships in similar right triangles in order to solve problems. Use geometric software to explore the trigonometric ratios of sine, cosine, and tangent. Use the sine, Cosine and Tangent ratio to find missing side lengths and angle measures in a right triangle. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems . Geometry Curriculum Unit 2: Similarity, Right Triangles and Trigonometry Unit 2 Summary: Similarity, Right Triangles and Trigonometry Prior Knowledge Expected Draw, construct, and describe geometrical figures and describe the relationships between them. Understand congruence and similarity using physical models, transparencies, or geometry software. Assessment Summary 1. (Scope and Sequence on the next page) Geometry Curriculum Unit 2 Similarity, Right Triangles and Trigonometry Unit 1 Scope and Sequence: Number System Fluency Periods* CCSS LS Student Friendly Language Resources Textbook Links 1 G.SRT. 1 S25 Verify experimentally the properties of dilations given by a center and a scale factor. Other Pearson Concept Byte 9-6 Illustrative Mathematics: Dilating a Line Engage NY Module 2: Lessons 1-11 Assessmen Skill Check 25 t 1 G.SRT. 1a S26 Know that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Textbook Pearson Concept Byte 9-6 Links Illustrative Mathematics: Dilating a Line Other Engage NY Module 2: Lessons 1-11 Assessmen Skill Check 26 t 1 G.SRT. 1b Geometry Curriculum S27 Know the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Textbook Pearson Concept Byte 9-6 Links Illustrative Mathematics: Dilating a Line Other Engage NY Module 2: Lessons 1-11 Unit 2: Similarity, Right Triangles and Trigonometry Assessmen Skill Check 27 t Textbook Links 1 G.SRT.2 S28 Use the idea of dilation transformations to develop the definition of similarity. Pearson 9-7 Inside Mathematics Performance Task: Hopewell Geometry Rhombuses Illustrative Mathematics: Are they Similar? Other Engage NY Module 2: Lessons 12-24 Assessmen Skill Check 28 t Textbook 1 G.SRT.2 S29 Given two figures determine whether they are similar and explain their similarity based on the equality of corresponding angles and the proportionality of corresponding sides. Links Pearson 9-7 Inside Mathematics Performance Task: Hopewell Geometry Rhombuses Illustrative Mathematics: Are they Similar? Geometry Curriculum Unit 2 Similarity, Right Triangles and Trigonometry 1 1 G.SRT.3 G.SRT.4 Geometry Curriculum S30 S31 Other Engage NY Module 2: Lessons 12-24 Assessment Skill Check 29 Textbook Pearson 9-7 Links IIustrative Mathematics Similar triangles Identify and use properties of similarity transformations to establish the AA criterion for two triangles to be similar. Other Engage NY Module 2: Lesson 12-24 Assessment Skill Check 30 Textbook Pearson 7-5 Links Illustrative Mathematics: Joining two midpoints of sides of a triangle Pythagorean Theorem Tangent Line to Two Circles Bank Shot Extensions, Bisections and Dissections in a Rectangle Folding a square into thirds Other Engage NY Module 2: Lesson 1-11 Assessment Skill Check 31 Prove theorems about triangles: a line parallel to one side of a triangle divides the other two proportionally and its converse. Unit 2: Similarity, Right Triangles and Trigonometry 1 G.SRT.4 Textbook Pearson 8-1 Links Illustrative Mathematics: Joining two midpoints of sides of a triangle Pythagorean Theorem Tangent Line to Two Circles Bank Shot Extensions, Bisections and Dissections in a Rectangle Folding a square into thirds Other Engage NY Module 2: Lesson 1-11 Assessment Skill Check 32 S32 Prove the Pythagorean theorem using triangle similarity. Textbook Pearson Links 1 1 G.SRT.5 G.SRT.5 Geometry Curriculum S33 S34 Recognize the characteristics of congruent figures recognize corresponding figures and their corresponding parts Prove two triangles congruent using the SSS, SAS, ASA Postulate and AAS Theorem. Other Engage NY Module 2: Lessons 12-24 Assessment Skill Check 33 Textbook Pearson 4-2, 4-3 Links Unit 2 Similarity, Right Triangles and Trigonometry 1 1 G.SRT.5 G.SRT.5 35 36 Use triangle congruence and corresponding parts of congruent triangles to prove that parts of two triangles are congruent. Use properties of isosceles triangles to show that right triangles are congruent and prove right triangles are congruent using the Hypotenuse-Leg Theorem Other Engage NY Module 2: Lessons 12-24 Assessment Skill Check 34 Textbook Pearson 4-4 Links Other Engage NY Module 2: Lessons 12-24 Assessment Skill Check 35 Textbook Pearson 4-5, 4-6 Links Other Engage NY Module 2: Lessons 12-24 Assessment Skill Check 36 Textbook Links 1 1 G.SRT.5 G.SRT.5 Geometry Curriculum 37 38 Prove geometric figures, other than triangles, are similar and/or congruent. Identify congruent overlapping triangles and prove two triangles congruent using other congruent triangles. Other Engage NY Module 2: Lessons 12-24 Assessment Skill Check 37 Textbook Pearson 4-7 Links Unit 2: Similarity, Right Triangles and Trigonometry Other Engage NY Module 2: Lessons 12-24 Assessment Skill Check 38 Textbook Links 1 G.SRT.5 39 Other Engage NY Module 2: Lessons 12-24 Assessment Skill Check 39 Textbook Links 1 G.SRT.5 40 Identify characteristics of midsegments and use properties of midsegments in order to solve problems. Other Engage NY Module 2: Lessons 12-24 Assessment Skill Check 40 Textbook 1 G.SRT.5 41 Links Identify and use properties of perpendicular bisectors, medians and altitudes to solve problems. Other Assessment 1 G.SRT.5 Geometry Curriculum 42 Engage NY Module 2: Lessons 12-24 Skill Check 41 Find the sum of the measures of the interior and exterior angles of a Textbook polygon Links Unit 2 Similarity, Right Triangles and Trigonometry Other Engage NY Module 2: Lessons 12-24 Assessment Skill Check 42 Textbook Links 1 G.SRT.5 43 Use relationships among sides and angles and diagonals of parallelograms in order to solve problems Other Engage NY Module 2: Lessons 12-24 Assessment Textbook 1 G.SRT.5 44 Determine properties of similar polygons in order to identify and apply them in problem solving. Links Other Engage NY Module 2: Lessons 12-24 Assessment Textbook 1 G.SRT.5 45 Links Apply use of AA Similarity Postulate, SSS Similarity Theorem and SAS Similarity Theorem to prove triangles similar. Other Assessment 1 Engage NY Module 2: Lessons 12-24 Skill Check 45 Textbook G.SRT.5 46 Use triangle similarity to find indirect measurements. Links Geometry Curriculum Unit 2: Similarity, Right Triangles and Trigonometry Other Engage NY Module 2: Lessons 12-24 Assessment Skill Check 46 Textbook 1 G.SRT.6 47 Links Find and use relationships in similar right triangles in order to solve problems. Other Assessment Engage NY Module 2: Lessons 12-24 Skill Check 47 Textbook Links 1 2 2 G.SRT.6 G.SRT.7 G.SRT.8 48 49 50 Use geometric software to explore the trigonometric ratios of sine, cosine, and tangent Other Engage NY Module 2: Lessons 12-24 Assessment Skill Check 48 Textbook Pearson 8-3 Links Use the Sine, Cosine and Tangent ratio to find missing side lengths and angle measures in a right triangle. Other Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Assessment Skill Check 49 Textbook Pearson 8-1, 8-2, 8-3, 8-4 CB 8-4 Links Other Geometry Curriculum Engage NY Module 2: Lessons 12-24 Skill Check 50 Unit 2 Similarity, Right Triangles and Trigonometry Assessment Geometry Curriculum Engage NY Module 2: Lessons 12-24 Unit 2: Similarity, Right Triangles and Trigonometry