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Geometry EOC Content Review Packet Name: Teacher: Period: Page 2 Review resources Review Resources Videos: Pearson Online www.pearsonsuccessnet.com (Covers Geometry topics in tutorials & videos. Aligns with the curriculum used in Geometry) Khan Academy https://www.khanacademy.com (Covers all these topics in 10-20 minute videos with some good visual representations.) Brightstorm Videos Online www.brightstorm.com (Search any topic you are interested in exploring further to see video clips) Page 31 FORMULA SHEET: Page 30 FORMULA SHEET Cont… Page 3 G.1.C Use deductive reasoning to prove that a valid geometric statement is true. Deductive Reasoning- Reasoning accepted as logical from agreed-upon assumptions and proven facts. Example Theorem: All right angles are congruent. If ∠1 and ∠2 are right angles, then ∠1≅∠2. Proof: ___________ are right angles because it is given. By the definition of _______ m∠1= 90 and ∠2 = 90. By the transitive property of equality, ______ =_______. Examples: 1. The measure of an angle is three more than twice its supplement. What is the measure of the angle? 2. Resources: 2.4 (Pearson Geometry Textbook) http://www.brightstorm.com/math/geometry/reasoning-diagonals-angles-andparallel-lines/deductive-reasoning/all Page 4 Page 29 G.1.D Write the converse, inverse, and contrapositive of a valid proposition and determine their validity. Original Statement: ________________________________________ Converse Statement: _______________________________________ Inverse Statement: _________________________________________ Contrapositive Statement: ___________________________________ Tips for doing well on the EOC Don’t skip any questions – answer them! Read the question carefully to get to what they are truly asking you. Re-read what you wrote and make sure it answers the prompt Valid Statement: ____________________________________ Invalid Statement: ___________________________________ Just think back to what you have learned, do your best, and you will be fine! Even if your answer feels a bit silly, write it down, it’s better than nothing! Example: Write the converse, inverse, and contrapositive of the following statement: If the weather is rainy, then the sidewalks will be wet. Do well! Take your time, be thorough and careful, and you will do well! You can do it! Calculators will be allowed on the Geometry EOC… …BRING YOUR OWN CALCULATOR! Examples: 1. If m and n are odd integers, then the sum of m and n is an even integer. State the converse and determine whether it is valid. 2. If a quadrilateral is a rectangle, the diagonals have the same length. State the contrapositive and determine whether it is valid. Resources: 2.2, 2.3,2.5 (Pearson Geometry Textbook) http://hotmath.com/hotmath_help/topics/converse-inversecontrapositive.html Page 28 Page 5 What will the Geometry EOC be like? Logical arguments & proofs: 6-8 points – 5-8 multiple choice – 0-1 short answer Proving & applying properties of 2D shapes: 24-26 points – 15-19 multiple choice – 2-4 completion – 1-3 short answer Coordinate planes & measurement: 7-9 points – 5-8 multiple choice – 1-3 completion problems – 0-1 short answer G.1.E Identify errors or gaps in a mathematical argument and develop counterexamples to refute invalid statements about geometric relationships. A Counterexample is: ______________________________________ _________________________________________________________ An Invalid statement is: _____________________________________ _________________________________________________________ Examples: 1. Give a counterexample that disproves each conjecture below. a. No triangles have two sides of the same length. b. No women have been elected U.S. senators. Details for scoring please!?!?!?! c. All basketball players are more than 6 feet tall. 2. Error Analysis: Identify errors in reasoning in the following proof: Given ∠ABC ≅ ∠PRQ, AB ≅ PQ, and BC ≅ QR, then ∆ABC ≅ ∆PQR by SAS. Resource: Look at Error Analysis problems in Pearson textbook Page 27 Page 6 G.1.F Explain the role of definitions, undefined terms, postulates (axioms), and theorems. Undefined terms: Can be described but cannot be given precise definitions. List the 3 undefined Geometric Terms: __________________, ________________, _________________ Postulates: Statements that we assume to be _____ without proof. Theorems: Statements that ______________________ within a deductive system. Examples: 1. 2. 3. Resource: http://www.brightstorm.com/math/geometry/geometry-building-blocks/postulateaxiom-conjecture/all Additional Notes/Practice: Page 26 Additional Notes/Practice: Page 7 G.3.A Know and apply basic postulates and theorems about triangles and the special lines, line segments, and rays associated with a triangle. Examples: • Prove that the sum of the angles of a triangle is 180°. Triangle Sum Theorem Definition: Picture/Sketch: • Prove and explain theorems about the incenter, circumcenter, orthocenter, and centroid. Incenter Definition: Picture: Circumcenter Definition: Picture: Page 25 Page 8 Additional Notes/Practice: G.3.A Continued Orthocenter Definition: Picture: Centroid Definition: Picture: Example: 1. The rural towns of Atwood, Bridgeville, and Carnegie are building a communications tower to serve the needs of all three towns. They want to position the tower so that the distance from each town to the tower is equal. a. Where should they locate the tower? b. How far will it be from each town? 2. If the measures of three angles of a triangle are represented by (y + 30)°, (4y +30)°, and (10y – 30)°, then the triangle must be: A. obtuse B. isosceles C. scalene D. right Resources: 3.1, 3.2, 3.3, 3.4, 3.5 (Pearson Geometry Textbook) http://www.sparknotes.com/math/geometry2/theorems/section2.rhtml Page 24 Additional Notes/Practice: Page 9 G.3.B Determine and prove triangle congruence and other properties of triangles. Postulate: Side-Side-Side (SSS) Example: Postulate: Side-Angle-Side (SAS) Example: Postulate: Angle-Side-Angle (ASA) Example: Theorem: Angle-Angle-Side (AAS) Example: Using Corresponding Parts of Congruent Triangles (CPCTC) Example: Resources: 4.2, 4.3, 4.4, 4.5 (Pearson Geometry Textbook) http://www.mathopenref.com/congruenttriangles.html Page 10 G.3.D Know, prove, and apply the Pythagorean Theorem and its converse. Then: Page 23 G.6.F Solve problems involving measurement conversions within and between systems, including those involving derived units, and analyze solutions in terms of reasonableness of solutions and appropriate units. _____________________ Examples: 1. You want to make a triangular sail that has a base of 13 ft. 4 in. and a height of 12 ft. 2 in. How many square feet of material do you need? Examples: 1) 2. An airplane is flying at an altitude of 10,000 ft. The airport at which it is scheduled to land is 50 mi away. Find the average angle at which the airplane must descend for landing. Round your answer to the nearest degree. 2) 3) Classifying a Triangle Resources: 8.1 (Pearson Geometry Textbook) http://mathworld.wolfram.com/PythagoreanTheorem.html Resource: Throughout Pearson Geometry Textbook Page 22 G.6.E Use different degrees of precision in measurement, explain the reason for using a certain degree of precision, and apply estimation strategies to obtain reasonable measurements with appropriate precision for a given purpose. Page 11 G.3.E Solve problems involving the basic trigonometric ratios of sine, cosine, and tangent. Examples: 1. Find the value of each variable. Sin(A) = Cos(A) = Tan(A) = Examples: 1) Write each ratio for the following: 2. The International Space Station orbits 350 km above Earth’s surface. Earth’s radius is about 6370 km. Use the Pythagorean Theorem to find the distance from the space station to Earth’s horizon. Round your answer to the nearest 10 kilometers. (Diagram is not to scale.) Sin(A) = Cos(B) = 2) Find the value of x for the following: Resource: Throughout Pearson Geometry Textbook Tan(A) = Page 12 Page 21 G.3.F Know, prove, and apply basic theorems about parallelograms. G.3.C Continued Properties of Parallelograms: Examples: 3. What is the value of z? A. 2 2 B. 2 3 C. 4 2 D. 8 2 If…. Then…. Examples: 1) Find the value of the variables for each: 2) Find the value of the variables: 3) Find the measures of the numbered angles: Resource: 6.2, 6.3, 6.4, 6.5 (Pearson Geometry Textbook) 4. Determine the length of the altitude of An equilateral triangle whose side lengths measure 8 units. Keep your answer written as an exact value. Page 13 Page 20 G.3.C Use the properties of special right triangles (30°–60°–90° and 45°–45°–90°) to solve problems. Rules: 45-45-90 Triangle: Examples: 1) What are the values of x and y? 2) What are the values of x? 30-60-90 Triangle: G.3.F Continued Page 14 G.4.B Determine the coordinates of a point that is described geometrically. Midpoint Formula: ________________________________ Distance Formula: _________________________________ Examples: 1. b. Determine the coordinates for the midpoint of a line segment whose endpoints are (6, 8) and (-4, -5). b. Determine the coordinates for the midpoint of a line segment whose endpoints are (-5, 8) and (-2, -9). 2. a. Points X, Y, Z are collinear. Y is the midpoint of XZ. The coordinates of point X are (-4, 5). The coordinates of point Y are (2, 4). Determine the coordinates of point Z. b. Points A, B, C are collinear. B is the midpoint of AC. The coordinates of point Z are (-3, 9). The coordinates of point B are (2, -2). Determine the coordinates of point C. 3. Three vertices of parallelogram ABCD are ( 4, 9 ), ( 7, 7 ), and ( 1, 1 ). Which of the following coordinates would be possible for the fourth vertex? A. ( 4, 0 ) B. ( 4, 1 ) C. ( -3, 2 ) D. ( -2, 3 ) Resources: 6.8 (Pearson Geometry Textbook) http://www.mathopenref.com/coordintro.html http://www.onlinemathlearning.com/coordinate-geometry.html Page 19 G.4.C Continued Example: 3. Classify the quadrilateral ABCD whose vertices lie on A (-11,12), B(4,15), C(7,7), and D(-3,5). Classify the quadrilateral and justify your answer. Page 15 Page 18 G.4.C Verify and apply properties of triangles and quadrilaterals in the coordinate plane. Triangle Properties: Obtuse: Acute: Scalene: Right: Isosceles: Equilateral: G.4.B Cont. Examples: 4. A kite is designed for an upcoming craft show. In order to get the design lined up properly, point S must be located. Point S is the midpoint of segment EI. What are the coordinates of point S? A. (2, 2) B. (5,10) C. (5.5, 9.5) D.(7.5, 7.5) 5. In parallelogram WXYZ, what are the coordinates of the point of intersection of WY and ZX? Examples: 1. The vertices of ΔABC are A( –1, –2), B( –1, 2), and C( 6, 0). Which conclusion can be made about the angles of ΔABC? mACB 90 A. mA mB B. C. mA mC D. mABC 60 2. Classify triangle FGH by its sides and angles. F(-12,3), G(0,0), H(-10,11) A. Scalene, Right B. Scalene, Obtuse C. Isosceles, Right D. Equilateral, Acute Resource: 6.8, 6.9 (Pearson Geometry Textbook) Page 16 G.3.G Know, prove, and apply theorems about properties of quadrilaterals and other polygons. Special Parallelograms: Page 17 G.3.G Cont. Examples: 1. LMNP is a rectangle. Find the value of x and the length of each diagonal: LN = 9x – 14 and MP = 7x + 4 1) A ___________ is a parallelogram with four congruent sides. Other Properties: 2. a. What is the length of the apothem of a regular hexagon with a side length 9in? 2) A _______________ is a parallelogram with four right angles. Other Properties: 3) A ______________ is a parallelogram with four congruent sides and four right angles. Other Properties: b. What is the length of the apothem of a regular hexagon with a side length 12in? 3. a. What is the sum of the interior angles of a regular nonagon? b. What is the sum of the interior angles of a regular pentagon? Geometric Properties of polygons: 1. If a figure is a trapezoid, then consecutive angles between a pair of parallel lines are ___________________________ 2. If a figure is a kite, then the diagonals are __________________ 3. The sum of the exterior angles of a polygon is ______________ 4. The sum of the interior angles of a polygon is _______________ when n is the _________________________________________ 5. The area of a regular polygon is found by: __________________ 6. The diagonals of a rectangle ______________ each other. 7. The diagonals of a rhombus are ______ and intersect at _______ 4. Solve for x, using the information from the trapezoid below. 5x + 5 3x + 7 Resources: 6.4, 6.5 (Pearson Geometry Textbook)