Honors Geometry Application



Identification InformationCourse: Honors GeometryDuration: Honors Geometry is a year-long coursePrerequisite: Teacher recommendation, and a “B” or higher average in Algebra OneInstructors: Sandie Sullivan, Carrie Starnes, Ashley Ahlman Phone: (303) 982-7096E-mail Address: sasulliv@jeffco.k12.co.us cstarnes@jeffco.k12.co.us aahlman@jeffco.k12.co.usPart I: CURRICULUMSection A — Course DescriptionThe goal of Honors Geometry is to inspire students to enjoy mathematics, to prepare students to use mathematics in every day life, and to challenge students to create his or her own learning about mathematics and apply it to engineering, business, science, and beyond. Honors Geometry is an accelerated two-semester course designed to challenge the student with geometric concepts coupled with a strong algebraic emphasis. The topics that students will learn and be able to do during this study include, but are not limited to:Semester 1: Develop the structure and language of Euclidian geometry including proof; concepts of congruence, similarity, parallelism, perpendicularity, and proportion; rules of angle measurement in triangles and concepts and applications of coordinate geometry. Semester 2: Similarity of right triangles is extended to include the trigonometric ratios and then extended to any type of triangle by use of the Law of Sines and Law of Cosines; probability of geometric areas and its applications; surface area and volume of solid figures and applications; circle properties; and transformations of functions.The content included is both rigorous and enjoyable. Students are provided curricular content that has been taken directly from the CAP and the IB curriculum. Students are asked to read, write, and discuss their mathematical learning with their peers and with the teacher. This course has been designed to delicately balance district and state standards found in the Jefferson County CAP documents and the international and national curricula designed by the International Baccalaureate Organization. Ultimately, both curricular strands are designed to foster students’ curiosity, skill, and mastery in mathematics. Honors Geometry requires students to take their learning beyond the level of a standard geometry course. Key differences are highlighted below:Depth of Understanding: In Honors Geometry students dig into the “why” of geometric truths. It is not enough to simply know how to apply concepts, they must know and understand why they work, i.e. the relationship between the distance formula and the Pythagorean Theorem. Students derive the formula in the coordinate plane and in 3 dimensions using the Pythagorean Theorem. They also derive the equation relating the center of a circle and its radius with the center placed in a coordinate plane and applying the Pythagorean Theorem. Depth of Algebra Understanding with Applications: Algebra skills are strongly emphasized in Honors Geometry. Student skills in algebra are continuously challenged during the first semester with a steady diet of linear algebra review as well as increasingly complex demands in solving of both algebraic and geometric problems. By the end of second semester mastery of both linear and quadratic equation solving skills is expected, as well as the ability to solve both linear and quadratic proportions. Rather than just simple review, the focus is applying algebra to geometric problems. Rather than finding a missing angle in a triangle numerically, students must work with angles which are all algebraic expressions, and use their algebra skills to solve problems. Perimeter and area of polygons are found as algebraic expressions.Ability to Use Deductive Reasoning: Students must explain how they arrive at a conclusion. Beginning with properties of equality in equation solving, moving on to triangle proofs (paragraph, two column, and flowchart), quadrilateral and circle proofs, similarity and area and volume proofs students demonstrate their reasoning skills. Students explain to their peers the methods used to complete a solution. Students must justify clearly and concisely how they achieve their results.Use and Application of Technology: Students are given opportunities to experiment with technology and collect real-world data, including the regular use of TI-84 series calculators, GoMotion! Sensors, Geogebra and calculator applications. They are also encouraged to seek out geometric enlightenment by using websites such as , Kahn Academy, , etc.Upon successful completion of this course, students should have a firm grasp of the content studied and should be able to apply this content to real world projects. Throughout the course, students have an opportunity to work on various projects that illustrate their learning of particular mathematical content and department exams. All students will have a firm understanding of the TI-84 Family of graphing calculators and use this technology as a mathematical learning tool.Section B — Course Outline:The following is a brief synopsis of key points in the units of study in Honors Geometry. More complete descriptions are found in the Jefferson County CAP documents online in the Secondary Math PLC.Unit 1 ~ Structure and Language of Geometry – Student outcomes: students will know… Relationships between angles, lines and the postulates and theorems that support those relationships and how to prove them using deductive reasoning. Methods and tools for constructions Students will know and understand…Geometry concepts can be proven using logical reasoning Constructions help us visualize geometric problems and determine effective solutions.Proof requires constructing viable arguments and critiquing the reasoning of others.Students will be able to…Make formal geometric constructions with a variety of tools and methods Prove theorems regarding lines and angles.Create algebraic equations based on geometric postulates for lines and angles and solve. Make formal arguments to justify reasoning.Unit 2 ~ Congruence – Student outcomes: students will know… Corresponding parts of congruent triangles are congruent SAS, ASA, AAS, SSS, and HL postulates to show congruence. Right triangle congruence (hypotenuse-leg, hypotenuse-angle, leg-angle, leg-leg). Methods to create and prove triangle congruence using constructions. Methods for using coordinates and transformations to create and verify triangle congruence. Methods for proof (paragraph, flowchart and two column)Students will know and understand…Constructions help us visualize geometric problems and determine effective solutions.Triangles can be proven congruent by comparing corresponding parts.The coordinate plane is used to identify and verify properties of Geometric figures.Algebraic properties and modeling are connected to Geometric properties. Construct viable arguments and critique the reasoning of others. Students will be able to…Use the definition of congruence in terms of rigid motions (through technology, graph paper, or tracing paper) to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Prove theorems about triangles Use congruence criteria for triangles to solve real-world problems. Use congruence criteria for triangles (SAS, AAS, ASA, SSS, HL) to prove relationships in geometric figures (i.e. angle congruence and segment congruence based on triangle congruence postulates). Create and solve algebraic equations in one variable based on congruence. Use coordinates to prove simple geometric theorems algebraically (e.g. triangle mid-segment theorem or SSS). Use multiple methods (coordinates, constructions, transformations) to prove two figures are congruent. Unit 3 ~ Quadrilaterals & Coordinate Geometry – Student outcomes: students will know… The properties of specific quadrilaterals (parallelogram, rectangle, square, rhombus, kite, trapezoid).Polygon sum theorem. Methods for using coordinates to verify congruence. Distance formula to find side lengths. Connection between coordinates and transformations and congruence.Students will know and understand…Congruence is proven using properties of shapes.Angles and side measures are used to distinctly classify shapes.The coordinate plane is used to identify and verify properties of Geometric figures. Students will be able to…Use coordinates to prove simple geometric theorems algebraically (for example, prove that a figure in a plane is a square, rectangle, or other quadrilateral). Use coordinates and the distance formula to compute perimeters of polygons and areas of triangles and rectanglesCreate equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales. Unit 4A ~ Similarity – Student outcomes: students will know… AA, SSS, and SAS triangle similarity conjectures.Properties of dilations: 1. Dilation for line segments longer or shorter in the ratio given by the scale factor and 2. Line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged). The distance formula is based on the Pythagorean Theorem.Problem solving strategies for proportions. Students will know and understand…If similarity exists, then conjectures can be made about angles measures and side lengths of geometric figures.Similar figures can be used to model real life situations and to find measures indirectly. Students will be able to…Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. Prove theorems involving similarity of triangles (Includes a line parallel to one side of a triangle divides the other two proportionally, and the converse) Create proportional equations, based on similarity, and use them to solve problems. Solve design problems by designing an object or structure that satisfies certain constraints, such as working with a grid system based on ratios (i.e., the enlargement of a picture using a grid and ratios and proportions). Unit 4B ~ Right Triangle Similarity – Student outcomes: students will know… The Pythagorean Theorem and its converse.Ways to simplify with radical numbers and expressions.30-60-90 and 45-45-90 right triangle conjectures.Geometric mean.Sine, Cosine, and Tangent ratios.Angles of depression and elevation.Law of Sines and law of Cosines.Students will know and understand…Right triangles are used in multiple settings to find measures indirectly.Students will be able to…Prove the Pythagorean Theorem using triangle similarity.Define trigonometric ratios and solve problems involving right triangles. Explain 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. Explain and use the relationship between the sine and cosine of complementary angles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Create proportional equations, based on similarity, and use them to solve problems. Unit 5~ Probability – Student outcomes: students will know… Venn diagrams and set notation for intersections, unions, and complements. Independent events – the occurrence of one event has no effect on the occurrence of the other event; P(A and B) = P(A) ? P(B) . Dependent events – the occurrence of one event affects the occurrence of the other event; P(A and B) = P(A) P(B given A).Conditional probability- P(B given A)= P(A and B)/P(A)Experimental and theoretical probability. Procedures and reasoning to create scatter plots. Types of scatter plots correlations – positive, negative, no relatively. Residuals – the difference between an observed y-value and its predicted y-value(found on the line of best fit).Insurance and riskStudents will know and understand…Probability helps us to make inferences and predict the outcome of an event in order to make informed decisions.The relationship among events affects probability.Functions may be used to describe data; if the data suggests a linear relationship, the relationship can be modeled with a regression line, and its strength and direction can be expressed through a correlation coefficient. Students will be able to…Describe events as subsets of a sample space using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events. Find the probability of two independent events. Find the probability of two dependent events. Analyze* the cost of insurance as a method to offset the risk of a situation (PFL).Unit 6 ~ Measurement, Area & Volume – Student outcomes: students will know… Area formulas developed in earlier grades for triangles, parallelograms, rectangles, squares, trapezoid, and circle. Area formulas for kites, sectors, and annulus (washer)Formulas and methods for area of regular polygons.Methods for finding area of shaded regions and compound shapes.Volume formulas for prisms and cylinders (Bh) and pyramids and cones (1/3 Bh).Strategies for finding surface area of 3-D figures.Formula for finding the volume and surface area of spheres.Methods for finding the volume of compound 3-D figures (to include figures with missing portions such as a tire).Methods for measuring density in multiple contexts.Applications of area and volume such as density of a material, packaging, GIS, and creating a floor plan.Methods for simplifying polynomial expressions.Methods for solving literal equations.The relationship between linear dimensions, two dimensions, and three dimensions is x to x^2 to x^3.Students will know and understand…Area is the measure of two dimensional spaces and can be found using indirect measurement and formulas.Area of irregular figures can be found using area of common figures.There is often more than one correct model for any given situation.Algebraic properties and modeling are connected to Geometric properties and formulasStudents will be able to…Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone (informal can be dissection arguments, Cavalieri’s Principle, or informal limit arguments). Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply concepts of density based on area and volume in modeling situations (e.g. persons per square mile, BTUs per cubic foot). Write polynomial expressions to represent area and volume in equivalent forms to solve problems. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Create and solve quadratic equations that represent area and volume problem solving situations through factoring or other methods. Find and explain the change in the volume or area when a linear measurement is changed. Unit 7 ~ Circles – Student outcomes: students will know… Precise definition of a circle based on the undefined notions of point and distance around a circular arc. Theorems about chords, tangents, and secants. The definition and how to label the following: central angle, minor arc, major arc, semicircle, intercepted arc, and congruent arcs. Standard equation of a circle. Methods to complete the square. Methods for graphing circles. Real-world examples of circles, arcs, and sectors.Students will know and understand…Relationships exist among angles, segments, lengths, circumference, and area of circles.A circle can be modeled with a graph, equation, table, or description. Students will be able to…Identify and describe relationships among inscribed angles, radii, and chords Create algebraic equations based on properties and theorems for circles and use them to solve problems. Construct the inscribed and circumscribed circles of a triangle. Prove properties of angles for a quadrilateral inscribed in a circle. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Derive the formula for the area of a sector. Graph circles using the equation. Fluently move from the equation of a circle to the graph.Unit 8 ~ Transformations – Student outcomes: students will know… Function families (linear, exponential, quadratic)Key features of linear, exponential and quadratic functions (i.e. slope, intercepts, roots/zeros, maxima, and minima, symmetries. Methods for finding equations of line. Symbolic representations for transformations of linear (y=x) and quadratics (y=x2). Vertical stretch and compression.Students will know and understand…Functions can be represented with graphs, tables, words, and algebraically. Transformations of functions are represented using graphs, tables, and equations.Properties and equivalence govern the methods for solving systems. Students will be able to…Compare linear, quadratic, and exponential models and solve problems. Distinguish between situations that can be modeled with linear functions and with exponential functions. Graph a quadratic function and determine the zeros through graphing and factoring. Construct and rewrite linear, quadratic, and exponential expressions in equivalent forms to solve problems. Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales. Solve systems of equations graphically and algebraically. Part II: ASSESSMENT and EVALUATION of STUDENT PERFORMANCEFormative Assessment — Students use think-pair-share and other reflective techniques to have an opportunity to learn from their peers. Reading, writing, and discussion are critical in allowing students to demonstrate learning outside of formative assessment. Students are regularly asked to write reflections of his or her progress and learning in journals, which teachers read periodically. Students also work in teams on challenging problems where students create solutions to present to the class. This collaboration is crucial to preparing students for the world outside of CAP, IB, and AP as most professionals work in teams to solve problems. In addition, brief quizzes with no grade attached are administered periodically for teachers to gauge student understanding. Teachers meet in a formal Professional Learning Community to discuss the data collected and the journal comments. The data and journals help the PLC participants reflect and create action plans for curricular improvement. Teachers create higher order questions that help facilitate discussion and drive student learning. These questions focus on the higher order thinking skills addressed in the Instructional Strategies section. Summative Assessment — Unit exams, and formal graded quizzes and quests, are used as a cumulative evaluation tool for each unit, the semester, and the entire year. Exams have questions taken from IB exams pertaining to the unit. All exams require both short and long constructed responses, requiring students to justify his or her solutions. Each summative exam has an opportunity for students to write about his or her learning using formal writing skills. Students are also asked to reflect on his or her successes and struggles during the unit. The use of summative assessment is crucial to this course because both AP and IB have summative assessments as the culminating activity at the end of the curriculum. Student test information is crucial in driving changes and improvements in curriculum and alignment. Grading Scale:A: 90% and aboveB: 80 to 89%C: 70 to 79%D: 60 to 69%F: below 60%Students unable to maintain at least a “C” average at the end of first semester are moved from Honors Geometry and into a regular Geometry class.Grades are composed of the following categories:Grade CompositionExams – 55%Daily Work– 15%Quizzes – 15%Final Exam – 15%Daily Work - This category comprises both in-class work and homework. In-class assignments usually are worth 10-15 points and embody investigations to discover relationships, practice of concepts or challenges to move understanding to a higher level. Homework assignments may be as few as 2-3 rich questions that require students to extend their thinking to 15 problems that build in intensity of understanding.Quizzes - During an average semester, approximately 8-12 quizzes are given, each ranging in value from 10 to 60 points. The types of questions range from basic problems showing understanding of concepts, to analysis level questions that require a high level of mathematical thinking.Quests - During a semester, four to five 100-point quests are administered. To get a clearer understanding of a “typical” unit quest, please note the attached unit quest from the congruence unit. Final Exam - The final exam is a comprehensive exam of all topics discussed during the semester and contains both a calculator and non-calculator component.Part III: INSTRUCTIONAL STRATEGIESThe focus of Honors Geometry is to have students engage in their learning. On a daily basis, students are given opportunities to discuss rich mathematics with small groups or with partners to deepen his or her understanding of the complexities of the subject. Students are given opportunities to experiment with technology and collect real-world data, including the regular use of TI-84 series calculators, Geogebra, and other calculator applications. These tools are used throughout the course to allow students to discover and construct his or her own understanding of geometric concepts. Students create his or her own calculator programs and will use applications including Inequalz, Cabri Jr, and GeoMaster on the TI-84 to deepen understanding of the curriculum. Students will complete projects in groups of three or more that are created to encourage students to work collaboratively. On a daily basis, students present their discoveries to their peers whether in large or small groups and argue the merits of their calculations in think-pair-share and other small group discussion venues. The learning activities used in Honors Geometry are designed to provide students with the opportunity to: analyze data regularly and make conclusions provided the data, apply their learning in class to real world situations, and evaluate not only their own mathematics but that of other students as well. In addition, students are asked to create mathematical models to use for real world data and synthesize new ways to use their knowledge beyond the course through assignments that require students to research questions like, “How is similarity used to design structures? Describe your opinion of the usefulness or uselessness of similarity.” Part IV: RESOURCES —Textbook: Discovering Geometry, First Edition. Key Curriculum Press, 2007. Additional Resources:Discovering Geometry: More Projects & Explorations.International Baccalaureate/Advanced Placement questionsRocky Mountain Middle School Math and Science Partnership Algebra Problems (McMillan, Bartuska, & Hicks)Texas Instruments Activities (education.)Projects designed by Honors Geometry team College Preparatory Mathematics, Geometry. CPM, 2000.American Mathematics Competition Exam Honors GeometrySample Projects and Unit Assessments Unit 1 Project Assessment Honors GeometryName ________________________As a review of our first unit in Honors Geometry, you will prove your knowledge (deductively!) Please choose the project below that most appeals to you and create one of the following:Create a complex puzzle using parallel lines and transversals. This puzzle should be created in such a way that someone could solve for many different angle measurements using the angle relationships we have discussed in class (i.e. vertical angles, alternate interior angles, etc.) Some of the angles should be solved using algebra (5x – 22 = 3x + 78). You must use at least 8 lines or line segments and are encouraged to use more! Produce an accurate answer key for your puzzle.On graph paper, create any picture using your knowledge of linear equations. Your picture must use at least 8 lines. Use horizontal and vertical lines as well as lines with a slope and y-intercept. Write all the equations of lines used in the drawing and label them on your picture. Color your picture. You may also add shapes to your picture that are not lines. You must produce a separate sheet that contains all of the equations of the lines to turn in with the picture.Using transformations, create a complex pattern created by rotations, reflections and translations. On a separate sheet of paper you need to accurately describe the transformations of each shape (i.e. this shape was rotated 90? counterclockwise then shifted two units left.)This project is worth 40 points: 20 for the mathematics and 20 for creativity. You are encouraged to make a product that I would consider using for a geometry class of the future or for your class when we begin working on our semester review. 666750100965Example 2 - Rotation of 180o3620892147955 Escher drawing of handsUnit 2 Project Assessment Unit 7 Project Assessment The Great GoodyThe Problem: Given any three non-collinear points, find the equation of the circle that contains them.Given any three non-collinear points, find the equation of the circle that contains them.Geometry Solution:The perpendicular bisectors of two chords of a circle intersect at the center of the circle.I. Problem Set-Up:You need to select three non-collinear points to use in your solution. Your first point will be in Quadrant I, so choose a point with x and y coordinates between 5 and 25; for example (6, 21).First Point: ( _______, _______ )Let your second point be in Quadrant II, again with values between 5 and 25, or -5 and -25, but select different values than you did for the first point.Second Point: ( _______, _______ )Your third point can be in Quadrant III or Quadrant IV. Use the same range of values as above, but again make sure the values are different from your first two points.Third Point: ( _______, _______ )Create a scatterplot of the three points you have selected on your graphing calculator. Set up your window so all three points can be seen.II. Algebraic Solution:Show all of your work neatly on another sheet of paper, but write the indicated steps of your solution below.Equations of the two perpendicular bisectors.y = _______________________y = _______________________Center of the circle (rounded to the nearest thousandth.)( _______, _______ )Radius of the circle (rounded to the nearest thousandth.)r = __________________Equation of the circle.___________________________________________________________III. Checking your solution:Solve the equation of your circle for y and enter the two resulting equations into Y? and Y? of your graphing calculator.Graph the equations and, if you have solved the Great Goody correctly, you should see a circle going through the three points of your scatterplot! (Again, you may need to adjust your window to see the entire circle. Also, if the figure does not look “circular”, try a ZSquare from the Zoom menu of your calculator.)If the circle does not go through your three points, find your algebra mistake and try it again!If you have the capabilities to take a screen shot of your calculator, use this as an opportunity to do so. I would love to see what your final product looks like! If you have questions about taking a screen shot from your calculator let me know.Unit 3 Team Challenge:Quadrilaterals & Writing Equations of LinesPART A: Writing Equations of LinesFind the x and y-intercepts of the following equation: . Then rewrite the equation in slope-intercept form. Write an equation of the line that passes through the points: (-1,2) and (3,-4).Write an equation of the line that is parallel to the line: , and passes through the point (7,1).Write an equation of the line that is perpendicular to the line: and passes through the point (3,-1).3797300148590PART B: ReviewIf a polygon has 42 sides, how many diagonals will it have from one vertex? First, complete the table below, find the rule for the nth term, and then answer the question.3456…nQuadrilateral DRCK is a rhombus. What is the slope of ?PART C: PolygonsAnswer true or false to the following questions._______ The diagonals of a parallelogram are congruent._______ The diagonals of a kite bisect each other._______ The diagonals of a rhombus bisect each other. _______ The consecutive interior angles of a trapezoid are supplementary._______ The vertex angles of a kite are congruent.Fill in the blank.Each interior angle of a regular undecagon measures ______________.The sum of the measures of the interior angle of a dodecagon is ______________.The measure of one exterior angle of a regular octagon is ______________.A regular polygon has ______________ sides if it has one interior angle measure of .A regular polygon has ______________ sides if it has one exterior angle measure of .300355029845Find each lettered angle measure:m = ______s = ______n = ______t = ______p = ______r = ______A regular hexagonal mirror frame is to be built from strips of 2-inch wide pine lattice. At what angles a and b should the lattice be cut?4540250375920194945The figure below is a kite. Find the value of x.464820226060Solve for the values of x, y, z in the parallelogram below.331470114300Trapezoid ABCD contains midsegment If inches and inches, find the length of Find the perimeter of given QRST is a rectangle.41529090805294640398145Solve for x if and . B is the midpoint of and D is the midpoint of .One side of a kite is 1 centimeter less than 5 times the length of another. If the perimeter is 94 centimeters, find the length of each side of the kite.904240179705Quadrilateral ABCD is a parallelogram. Use the diagram below to answer the following questions:(Not drawn to scale)In parallelogram ABCD, Set up an equation and solve for x.In parallelogram ABCD, Set up an equation and solve for s. Solve for x. (HINT: This is a quadratic equation - it needs to be set equal to zero and factored – there may be two possible answers)32829500PART D: Flowchart ProofsGiven: ABCD is a parallelogram and Show: bisect each other. -19431068580Complete the following flowchart proof to prove the following conjecture: The diagonal of a parallelogram divides the parallelogram into two congruent triangles. Given: Parallelogram SOAK with diagonal 2929890-635 Show: 41910130810Honors GEOMETRYNAME _________________________________BLOCK ________ DATE________________UNIT 3 Quest – QUADRILATERALS & WRITING EQUATIONS OF LINES PART A – Review2564130313055The dots at the vertices in the diagram below represent the first four hexagonal numbers. Complete the table, and then find a rule that will help you find any hexagonal number. Then find the 15th hexagonal number.1234…15nHexagonal Numbers1615?…??332613026670Find the slopegiven square.PART B – WRITING EQUATIONS OF LINESWrite an equation of the line through the points (1, 2) and (3, -4).Write an equation of the line that is perpendicular to the line and that passes through the point (0, -3).Write an equation of the line that is parallel to the line and passes through the point (9,-1).3025140163195Write the following equation in slope-intercept form and then graph: Ace Rent a Car charges a flat fee of $15 and $0.24 per mile for their cars. Acme Rent a Car charges a flat fee of $24 and $0.12 per mile for their cars. Write a system of equations and solve it to find out where the rental companies charge the same total amount. PART C – POLYGONSFind each lettered measure3848101543059. a = _____, b = ______, x = ______, y = ______3093085-190547625013970.3413760157480MJKL is a rhombus. If , what is the measure of ABCD is a square. ABE is an equilateral triangle.1855470142240x = __________53721083820Find the sum of the interior measures of a polygon with 15 sides.How many sides does a regular polygon have if each exterior angle measures One interior angle of a regular polygon measures How many sides does the polygon have?4152902165353051810115570Find the lettered angle measures if 186690107950Find the values of x and y given parallelogram DEGF. Find the measure of ∠EDF.3821431251460In the figure below, is the midsegment of trapezoid ABCD. Find the length of segments BC , EF and AD.3257550438785are base angles of isosceles trapezoid JKLP. If and , find the value of x, then find the measure of each plete the following flowchart proof to prove the rhombus angles conjecture: 403479040640Given: Rhombus DENI with diagonal -415290229870Prove: Diagonal bisects Use the figure below to give the most specific name for the quadrilateral. Then use coordinate geometry to prove that your figure matches that description.28536901054102929890403860For A(1, –1), B(–1, 3), and C(4, –1), find a location of a fourth point, D, so that a parallelogram is formed using A, B, C, D in any order as vertices. ................
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