# CURRICULUM GUIDE. Honors Geometry

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1 CURRICULUM GUIDE Honors Geometry This level of Geometry is approached at an accelerated pace. Topics of postulates, theorems and proofs are discussed both traditionally and with a discovery approach. The usual topics and properties of triangles, polygons, congruency, and similarity are analyzed both geometrically and algebraically. Advanced topics include three-dimensional figures and trigonometry. Prerequisites: C+ or better in Honors Algebra 1 and approval by current instructor. Saint Patrick High School Author: Dan Kohl

2 HONORS GEOMETRY CURRICULUM TIMELINE First Quarter Chapter 1 Introduction To Geometry Chapter 2 Basic Concepts And Proofs Chapter 3 Congruent Triangles Second Quarter Chapter 4 Lines In The Plane Chapter 5 Parallel Lines And Related Figures Chapter 6 Lines And Planes In Space Third Quarter Chapter 7 Polygons Chapter 8 Similar Polygons Chapter 9 The Pythagorean Theorem Fourth Quarter Chapter 10 Circles Chapter 11 Area Chapter 12 Surface Area And Volume Chapter 13 Coordinate Geometry Extended (time permitting) Required Materials Course Textbook: Geometry for Enjoyment and Challenge (On loan from St. Rhoad, Milauskas, and Whipple Patrick) ( McDougal Littell, 1991) Scientific calculator, straight edge, pencil, colored pens or pencils, three 70 page notebooks or loose leaf paper for homework and notes, thirty sheets of loose leaf paper for quizzes

3 COMMON CORE STATE STANDARDS Addressed Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 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.

4 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 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. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

5 Chapter 1 Introduction to Geometry Recognize points, lines, segments, rays, angles, and triangles Measure segments and angles Classify angles and name the parts of a degree Recognize congruent angles and segments Recognize collinear and non collinear points Recognize when a point is between two other points Apply the triangle-inequality principle Correctly interpret geometric diagrams Write simple two-column proofs Identify bisectors and trisectors of lines segments and angles Write paragraph proofs Recognize that geometry is based on a deductive structure Identify undefined terms, postulates, and definitions Understand the characteristics and application of theorems Recognize conditional statements and the negation, the converse, the inverse, and the contrapositive of a statement Use the chain rule to draw conclusions Solve probability problems

6 Chapter 2 Basic Concepts and Proofs Recognize the need for clarity and concision in proofs Understand the concept of perpendicularity Recognize complementary and supplementary angles Follow a five-step procedure to draw logical conclusions Prove angles congruent by means of four new theorems Apply the addition properties of segments and angles Apply the subtraction properties of segments and angles Apply the multiplication and division properties of segments and angles Apply the transitive properties of angles and segments Apply the Substitution Property Recognize opposite rays Recognize vertical angles

7 Chapter 3 Congruent Triangles Understand the concept of congruent figures Accurately identify the corresponding parts of congruent figures Identify included angles and included sides Apply the SSS postulate Apply the SAS postulate Apply the ASA postulate Apply the principle of CPCTC Recognize some basic properties of circles Apply the formulas for the area and the circumference of a circle Identify medians of a triangle Identify altitudes of a triangle Understand why auxiliary lines are used in some proofs Write proofs involving steps beyond CPCTC Use overlapping triangles in proofs Name various types of triangles and their parts Apply theorems relating the angle measures and side lengths of triangles Use the HL postulate to prove right triangles congruent

8 Chapter 4 Lines in the Plane Use detours in proofs Apply the midpoint formula Organize the information in, and draw diagrams for, problems presented in words Apply one way of proving that two angles are right angles Recognize the relationship between equidistance and perpendicular bisection Recognize planes Recognize transversals Identify the pairs of angles formed by a transversal Recognize parallel lines Understand the concept of slope Relate the slope of a line to its orientation in the coordinate plane Recognize the relationships between slopes of parallel and perpendicular lines

9 Chapter 5 Parallel Lines and Related Figures Write indirect proofs Apply the Exterior Angle Inequality Theorem Use various methods to prove lines parallel Apply the Parallel Postulate Identify the pairs of angles formed by a transversal cutting parallel lines Apply six theorems about parallel lines Solve crook problems Recognize polygons Understand how polygons are named Recognize convex polygons Recognize diagonals of polygons Identify special types of quadrilaterals Identify some properties of parallelograms, rectangles, kites, rhombuses, squares, and isosceles trapezoids Prove that a quadrilateral is a parallelogram Prove that a quadrilateral is a rectangle Prove that a quadrilateral is a kite Prove that a quadrilateral is a rhombus Prove that a quadrilateral is a square Prove that a quadrilateral is an isosceles trapezoid

10 Chapter 6 Lines and Planes in Space Understand the basic concept relating to planes Identify four methods of determining a plane Apply two postulates concerning lines and planes Recognize when a line is perpendicular to a plane Apply the basic theorem concerning the perpendicularity of a line and a plane Recognize line parallel to planes, parallel planes, and skew lines Use properties relating parallel lines and planes

11 Chapter 7 Polygons Apply theorems about the interior angles, the exterior angles, and midlines of a triangle Apply the No-Choice Theorem and the AAS theorem Use some important formulas that apply to polygons Recognize regular polygons Use a formula to find the measure of an exterior angle of an equiangular triangle

12 Chapter 8 Similar Polygons Recognize and work with ratios Recognize and work with proportions Apply the product and ratio theorems Calculate geometric means Identify the characteristics of similar figures Use several methods to prove that triangles are similar Use the concept of similarity to establish the congruence of angles and the proportionality of segments Solve shadow problems Apply three theorems frequently used to establish proportionality

13 Chapter 9 The Pythagorean Theorem Simplify radical expressions and solve quadratic equations Begin problems involving circles Identify the relationships between the parts of a right triangle when an altitude is drawn to the hypotenuse Use the Pythagorean Theorem and its converse Use the distance formula to compute lengths of segments in the coordinate plane Recognize groups of whole numbers known as Pythagorean triples Apply the principle of Reduced Triangles Identify the ratio of the side lengths in a triangle Identify the ratio of the side lengths in a triangle Apply the Pythagorean Theorem to solid figures Understand three basic trigonometric relationships Use trigonometric ratios to solve right triangles

14 Chapter 10 Circles Identify the characteristics of circles, chords, and diameters Recognize special relationships between radii and chords Apply the relationship between congruent chords of a circle Identify different types of arcs, determine the measure of an arc, and recognize congruent arcs Relate congruent arcs, chords, and central angles Identify secant and tangent lines and segments Distinguish between two types of tangent circles Recognize common internal and common external tangents Determine the measures of central, inscribed, tangent-chord, chord-chord, secant-secant, secant-tangent, and tangent-tangent angles Recognize congruent inscribed and tangent chord angles Determine the measure of an angle inscribed in a semi-circle Apply the relationship between the measures of a tangent-tangent angle and its minor arc Recognize inscribed and circumscribed polygons Apply the relationship between opposite angles of an inscribed quadrilateral Identify the characteristics of an inscribed parallelogram Apply the three power theorems Determine circle circumference and arc length

15 Chapter 11 Area Understand the concept of area Find the areas of rectangles and squares Use the basic properties of area Find the areas of parallelograms Find the areas of triangles Find the areas of trapezoids Use the measure of a trapezoid s median to find its area Find the areas of kites Find the areas of equilateral triangles Find the areas of other regular polygons Find the areas of circles Find the areas of sectors Find the areas of segments Find the ratios of areas by calculating and comparing areas Find the ratios of areas by applying properties of similar figures Find the areas of figures using Hero s formula and Brahmagupta s formula

16 Chapter 12 Find the surface areas of prisms Find the surface areas of pyramids Find the surface areas of circular solids Surface Area and Volume Find the volumes of right triangular prisms Find the volumes of other prisms Find the volumes of cylinders Use the area of a prism s or cylinder s cross section to find the solid s volume Find the volumes of pyramids Find the volumes of cones Solve problems involving cross sections of pyramids and cones Find the volumes of spheres

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