# 1 - GEOMETRY: INTRODUCTION

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1 Geometry Unit Lesson Title Lesson Objectives 1 - GEOMETRY: INTRODUCTION Geometry and the World Recognize and describe connections between geometry, the world, and God Nature of Mathematics Restate important aspects of the nature of mathematics. Explore the relationship between the real world and the world of ideas The History of Geometric Mathematics Geometry's Effect on Me Mathematic System: Set Theory Review Mathematic System: Operations with Sets Geometry Undefined Terms: Points Geometry Undefined Terms: Lines Geometry Undefined Terms: Planes Defined Terms: Definitions Geometric Postulates Review of Algebraic Postulates Geometric Theorems Review of Properties of Algebra Recognize contributions of past mathematicians. Interpret the significance of major mathematical discoveries Develop an appreciation for the potential usefulness of geometry knowledge Review and practice the rules of set theory. Identify finite and infinite sets. Identify subsets of a given set Find the intersections and unions of sets (set operations). Solve word problems using set theory and set operations List properties and characteristics of the undefined term "point" List properties and characteristics of the undefined term "line" List properties and characteristics of the undefined term "plane" Define segment, ray, and collinear. Identify and name examples of segments, rays when prompted. Indicate whether two lines are collinear or not Apply postulates to solve word problems. Identify characteristics of postulates Review and practice the algebraic postulates Recall and relate geometric theorems on points, lines, and planes Review properties of algebra 2011 Glynlyon, Inc. 56

2 2 - GEOMETRY: LOGIC Logic and Reasoning Know the fundamental principles of logic and reasoning History of Logic and Reasoning Recall past discoveries and influential mathematicians Logic Define and identify types of logical statements. Recognize and use strategies of logic Conjunctions Classify a conjunction as true or false. Use a truth table to analyze conjunctions Disjunctions Classify a disjunction as true or false. Use a truth table to analyze disjunctions Negation Classify a negation as true or false Conditional or Implication Statements Use truth tables to judge conditional statements. Solve problems using conditional statements Converse, Inverse, Contrapositive Identify the converse, inverse, and contrapositive of conditional statements. Determine if a statement is true or false Inductive Reasoning Use inductive reasoning to draw reasonable conclusions. Identify statements as inductive or not inductive Deductive Reasoning Identify the major and minor premises of a syllogism. Draw conclusions from premises Using Deductive Reasoning Use deductive reasoning to prove basic theorems Proof Formats: Statement of the Theorem Identify the essential parts of a two-column proof. Rewrite statements in "if-then" form Proof Formats: The Figure Identify the appropriate figure for a proof Proof Formats: The Given Information Identify the "given" information in a two-column proof Proof Formats: To Prove Statement Proof Formats: The Plan of the Proof Indirect Proof Format: The Paragraph Proof Identify the statement to prove in a two-column proof Describe several strategies for planning a proof. Match statements with reasons Write the negation of a statement. Prove some simple statements using the indirect method, or contradiction 2011 Glynlyon, Inc. 57

3 3 - GEOMETRY: ANGLES AND PARALLELS Angle Definitions Identify and describe acute, right, and obtuse angles. Name an angle and its parts Angle Measurement Use a protractor to measure angles. Find the sum of angle measures Relationship Definitions Define and identify adjacent angles. Define and identify complementary angles. Define and identify supplementary angles. Define and identify vertical angles Angle Relationship Theorems (1) Use theorems about adjacent, complementary, supplementary and vertical angles to answer questions and complete proofs Angle Relationship Theorems (2) Use theorems about adjacent, complementary, supplementary and vertical angles to answer questions and complete proofs Construction: Copying Figures Copy a figure by using mathematical construction techniques Construction: Bisecting Figures Bisect figures by using mathematical construction techniques Basic Properties of Parallels Define and describe properties of parallelism of lines. Define and describe properties of parallelism of planes Transversals and Special Angles Calculate angle measures using transversals. Complete proofs by applying properties and theorems of tranversals More Proofs: Transversals and Special Angles Define and identify exterior and interior angles. Complete proofs using your knowledge of transversals Continued Proofs: Transversals and Special Angles Define and identify exterior and interior angles. Complete proofs using your knowledge of transversals More Proofs for Postulates 9 and 10 Practice proofs and questions that relate to parallels and transversals Construction: Perpendiculars Construct a line that is perpendicular to another line at a given point Construction: Tangents to Circle Construct a line that is tangent to a circle at a given point Construction: Parallels Construct a line that is parallel to a given line Classifying Triangles by Sides and Angles Identify triangles as scalene, isosceles, or equilateral. Identify triangles as acute, obtuse, or equiangular Exterior/Remote Interior Angles of Triangle Define exterior and remote interior angles of a triangle. Find the measures of exterior and remote interior angles Proofs Involving Triangles Define corollary. Define auxiliary line. Prove theorems and corollaries using auxiliary lines Other Polygons Categorize a shape as a polygon or non-polygon. Identify different kinds of polygons. Find the angle measures of polygons. Apply properties of polygons to solve problems 2011 Glynlyon, Inc. 58

4 4 - GEOMETRY: CONGRUENT TRIANGLES AND QUADRILATERALS Defining Congruent Triangles Define congruent triangles. Identify corresoponding parts of congruent triangles. Judge whether two triangles are congruent or not Proving Triangles Congruent (1) Prove that triangles are congruent using side & angle postulates Proving Triangles Congruent (2) Prove that triangles are congruent using side & angle postulates Proving Right Triangles Congruent Prove that right triangles are congruent using the Hypotenuse-Leg Theorem Independent Triangles (1) theorems on non-overlapping triangles. Prove that line segments are congruent using triangle congruence theorems on nonoverlapping triangles Independent Triangles (2) theorems on non-overlapping triangles. Prove that line segments are congruent using triangle congruence theorems on nonoverlapping triangles Overlapping Triangles (1) theorems on overlapping triangles. Prove that line segments are congruent using triangle congruence theorems on overlapping triangles Overlapping Triangles (2) theorems and properties of isosceles triangles. Prove that line segments are congruent using triangle congruence theorems and properties of isosceles triangles Isosceles Triangles (1) theorems. Prove that line segments are congruent using triangle congruence theorems. Prove that angles are congruent using properties of isosceles triangles. Prove that line segments are congruent using properties of isosceles triangles Isosceles Triangles (2) Prove that line segments are congruent using triangle congruence theorems. Prove that line segments are congruent using isosceles triangles. Prove that angles are congruent using triangle congruence theorems. Prove that angles are congruent using isosceles triangles Construction of Triangles Construct right triangles Construction of Triangles Construct right triangles Inequality Theorem in One Triangle Part1 Use angle measures to prove when one side of a triangle is longer than another side Inequality Theorem in One Triangle Use angle measures to prove when one side of a triangle is longer Part2 Inequalities in Two Triangles Quadrilateral Parallelograms Theorems Part1 Quadrilateral Parallelograms Theorems Part2 Triangles that Use Parallelograms in Proofs Parallelograms: Rectangles Parallelograms: Rhombus Trapezoids-Definitions and Proofs than another side Determine when sides of two different triangles are equal. Determine when one side of a triangle is greater than or less than another side. Use properties of parallelograms to prove statements Use properties of parallelograms to prove statements Use parallelograms to prove statements about triangles Prove statements involving the rectangle Prove statements involving the rhombus Prove statements involving trapezoids 2011 Glynlyon, Inc. 59

5 5 - GEOMETRY: SIMILAR POLYGONS Algebra and Ratios Express ratios in their simplest forms. Use geometric figures to find a ratio Algebra Properties and Proportions Know the definition of a proportion. Identify the means and extremes of a proportion. Solve proportions in one variable, including in the context of word problems Properties of Proportions Solve proportions in two variables. Relate proportions to geometric figures Meaning of Similarity Define similarity. Identify similar triangles. State key properties of similarity Meaning of Similarity-Theorems Prove when triangles are similar. Know important facts about similar triangles Meaning of Similarity-Proofs Prove when triangles are similar. Know important facts about similar triangles Theorems about Similar Polygons Know facts about similar polygons. Use facts about similarity to calculate side measures of similar polygons Theorems about Special Segments in Find segment measure in triangles using special relationships and Triangles proportions Theorems about Similar Right Triangles Find the geometric mean of two numbers. Use the altitude of a right triangle to create proportions. Solve for unknown segment measures Pythagorean Theorem Solve for missing sides of a right triangle. Determine whether 3 segments form a right triangle or not Theorem about Right Triangles Find the side measures of right triangles by applying special properties of right triangles Theorem about Right Triangles Find the side measures of right triangles by applying special properties of right triangles Using Triangles in Rectangular Solids Apply the Pythagorean theorem when solving for parts of rectangular solids Using Triangles in a Regular Square Pyramid Identify the parts of a regular square pyramid. Apply the Pythagorean theorem to solve for side lengths and other measures of a regular square pyramid Trigonometry-Sine Ratio State the sine ratio of a given angle. Use a table of sine values to solve for a missing value Trigonometry-Cosine Ratio State the cosine ratio of a given angle. Use a table of cosine values to solve for a missing value Trigonometry-Tangent Ratio State the tangent ratio of a given angle. Use a table of tangent values to solve for a missing value Using Similar Triangles in Indirect Use properties of similar triangles to measure lengths indirectly Measure Using Trigonometry in Indirect Use properties of trigonometric ratios to measure lengths indirectly Measure 2011 Glynlyon, Inc. 60

6 6 - GEOMETRY: CIRCLES Characteristics of Circles Identify and define the parts of a circle. Calculate measures of parts of a circle Characteristics of Spheres Identify and define the parts of a sphere. Calculate measures and relate other basic shapes, such as circle and triangle, to solve problems involving spheres Tangents Know and identify tangent lines. Apply properties of tangent lines to answer questions involving circles and polygons Arcs Define and identify major and minor arcs. Use the definitions of major and minor arcs to find angle and arc measures Chords Prove theorems that relate to tangents, arcs, and chords of a circle. Practice finding the measures of major and minor arcs Theorems (1) Prove theorems that relate to tangents, arcs, and chords of a circle. Practice finding the measures of major and minor arcs Theorems (2) Prove theorems that relate to tangents, arcs, and chords of a circle. Practice finding the measures of major and minor arcs Special Angles Type 1 Identify and define inscribed and intercepted arcs. Use properties of inscribed angles and intercepted arcs to solve problems and complete proofs Special Angles Type 2 Identify angles formed by intersecting secants. Solve for angle and arc measures when secant lines intersect inside a circle Special Angles Type 3 Solve for angle and arc measures when secant lines intersect outside a circle Special Segments Find the lengths of chords, secants, and tangents Construction: Circles Construct a circle circumscribed by a triangle. Construct a circle circumscribing a triangle 2011 Glynlyon, Inc. 61

7 7 - GEOMETRY: AREA AND VOLUME Area Concepts of Polygons Recognize that polygons can be broken into non-overlapping triangles. Find the area of a polygon by breaking it into triangles Area of Rectangles Find the area of a rectangle. Solve problems involving areas of rectangles Area of Parallelograms Find the area of a parallelogram. Solve problems involving areas of paralelograms Area of Triangles and Rhombuses Find the area of a triangle. Find the area of a rhombus Area of Trapezoids Find the area of a trapezoid Area of Regular Polygons Define and identify regular polygons. Find the area and other measures of regular polygons Area Comparison of Polygons Find area and linear measures such as side length of regular polygons that are similar Construction: Polygons Construct a rectangle, parallelogram, hexagon, and octagon Circles: Circumference and PI Find the circumference of a circle when given the radius. Find the radius of a circle when given the circumference Circles: Area of Circles Find the area of a circle. Find the area of a circle that is similar to another circle Circles: Area of Sectors Find the area of a sector, or "slice" of a circle. Find the arc length of a sector Circles: Area of Segments Find the area of a segment of a circle. Find the area of unusual shapes using the areas of sectors and segments Solids: Prisms Find the surface area and volume of a prism Solids: Pyramids Find the surface area and volume of a pyramid Solids: Cylinders Find the surface area and volume of a cylinder Solids: Cones Find the surface area and volume of a cone Solids: Spheres Find the surface area and volume of a sphere Construction: Dividing a Segment Divide a segment into a given number of equal segments Construction: 4th Proportion Construct a line segment that is in proportion to the other three Construction: ti Geometric Mean Construct t a line segment that t is the geometric mean of two given line segments 2011 Glynlyon, Inc. 62

8 8 - COORDINATE GEOMETRY Symmetry Find points of symmetry. Find lines of symmetry. Find planes of symmetry Ordered Pairs: Points in a Plane Plot points on a coordinate plane Graphs of Algebraic Sentences Review and practice graphing linear equations. Review and practice graphing linear inequalities Distance Formula Review and practice using the distance formula to find the distance betwwen two points. Find the lengths and perimeters of geometric shapes by using the distance formula Equation of a Circle Find equation for a circle in the coordinate plane Midpoint Formula Find the midpoint of line segments. Solve problems by using the midpoint formula Slope Calculate slope of a line. Test points to determine whether they are collinear (on the same line) Parallel and Perpendicular Lines Determine if lines are parallel, perpendicular, or neither (skew). Use properties of lines to prove theorems Equations of Lines Find the equation of a line given two points. Find the equation of a line given a point and a slope Figures in the Coordinate Plane Find properties and measures of shapes using the coordinate plane. Use coordinate techniques to prove geometric statements Proofs with Coordinate Geometry (1) Proofs with Coordinate Geometry (2) Prove theorems about plane figures using coordinate geometry Prove theorems about plane figures using coordinate geometry 9 - GEOMETRY: TRANSFORMATIONS Introduction: Rigid Motion, or Isometry Define isometry and the three types of rigid motion. Find the image points of a shape after a rigid motion Isometry: Reflection Find the image of a shape after a reflection Isometry: Translation Find the image of a shape after a translation Isometry: Rotation Find the image of a shape after a rotation Dilation: Congruence and Similarity il it Tell the difference between a contraction ti and an expansion. Find the image of points after a dilation Product Transformation Find the result of combining multiple transformations Inverse and Identity Transformation Symmetry Review Identify the inverse of a transformation Find points of symmetry. Find lines of symmetry. Find planes of symmetry 10 - GEOMETRY REVIEW History of Geometry Geometry as a System Geometry Proofs Angle Relationships and Parallels Congruent Triangles and Quadrilaterals Similar Polygons Circles Area and Volume Coordinate Geometry Review Unit 1 (Geometry: Introduction) Review Unit 2 (Geometry: Logic) Review Unit 2 (Geometry: Logic) Review Unit 3 (Geometry: Angles and Parallels) Review Unit 4 (Geometry: Congruent Triangles and Quadrilaterals) Review Unit 5 (Geometry: Similar Polygons) Review Unit 6 (Geometry: Circles) Review Unit 7 (Geometry: Area and Volume) Review Unit 8 (Geometry: Coordinate Geometry) 2011 Glynlyon, Inc. 63

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