Geometry

Scope & Sequence

Semester 1 

Semester 2 

Topic 1: Foundations of Geometry 

  • 1.4: Inductive Reasoning 

  • 1.5 Conditional Statements (converse only) 

  • 1.6 Deductive Reasoning (concept only) 

  • 1.7 Writing Proofs (two-column only; emphasize algebra proofs) 

  • Standards: HSG.CO.C.9, 10, 11. 

Topic 2: Parallel and Perpendicular Lines 

  • 1.1 Measuring Segments and Angles 

  • 2.1 Properties of Parallel Lines 

  • 2.2 Proving Lines are Parallel 

  • 2.3 Parallel Lines and Triangles 

  • 2.4 Slopes of Parallel and Perpendicular Lines 

  • Standards: HSG.CO.A.1, 3; HSG.CO.C.9, 10; HSG.MG.A.1, 3; HSG.GPE.B.5. 

Topic 3: Transformations (transform y = x^2 and y = |x| in tandem) 

  • 3.1 Reflections 

  • 3.2 Translations 

  • 3.3 Rotations (omit Example 4) 

  • 3.4 Classification of Rigid Motions 

  • 3.5 Symmetry 

  • Standards: HSG.CO.A.2, 3, 4, 5; HSG.CO.B.6 

Topic 4: Triangle Congruence 

  • 4.1 Congruence 

  • 4.2 Isosceles and Equilateral Triangles 

  • 4.3 Proving and Applying the SAS and SSS Congruence Criteria 

  • 4.4 Proving and Applying the ASA and AAS Congruence Criteria 

  • 4.5 Congruence in Right Triangles 

  • 4.6 Congruence in Overlapping Triangles 

  • Standards: HSG.CO.A.5; HSG.CO.B.6, 7, 8; HSG.SRT.B.5 

Topic 6A: Quadrilaterals and Other Polygons 

  • 1.3 Midpoint & Distance 

  • 9.1 Polygons in the Coodinate Plane 

  • 6.1 Polygon Angle-Sum Theorems 

  • 6.2 Kites and Trapezoids 

  • Standards: HSG.GPE.B.4, 6, 7; HSG.SRT.B.5 

Topic 6B: Quadrilaterals and Other Polygons 

  • 6.3 Properties of Parallelograms 

  • 6.4 Proving a Quadrilateral is a Parallelogram 

  • 6.5 Properties of Special Parallelograms 

  • 6.6 Identifying Special Parallelograms 

  • Standards: HSG.CO.C11HSG.SRT.B.5 

Topic 7: Similarity 

  • 7.1 Dilations 

  • 7.2 Similarity Transformations 

  • 7.3 Proving Triangles are Similar 

  • 7.4 Similarity in Right Triangles 

  • 7.5 Proportions in Triangles 

  • Standards: HSG.CO.A.2, 5; HSG.CO.C.10; HSG.SRT.A.1, 2, 3; HSG.SRT.B.4, 5 

Topic 8: Right Triangles and Trigonometry 

  • 8.1 Right Triangles and the Pythagorean Theorem 

  • 8.2 Trigonometric Ratios 

  • 8.5 Problem Solving with Trigonometry (Example 3 optional) 

Topic 10: Circles 

  • 10.1 Arcs and Sectors 

  • 10.2 Tangent Lines to a Circle (Example 5 optional) 

  • 10.3 Chords 

  • 10.4 Inscribed Angles 

  • Standards: HSG.C.A.2; HSG.C.B.5; HSG.CO.A.1 

Topic 11: Two and Three-Dimensional Models 

  • 11.2 Volumes of Prisms and Cylinders (also cross sectionsconcept only) 

  • 11.3 Pyramids and Cones (omit Example 1) 

  • 11.4 Spheres 

  • Standards: HSG.GMD.A.1, 2(+), 3; HSG.MG.A.1, 2 

Topic 5: Relationships in Triangles (as time allows) 

  • 1.2 Basic Constructions 

  • 5.1 Perpendicular and Angle Bisectors 

  • 5.2 Bisectors in Triangles 

  • 5.3 Medians and Altitudes 

  • 5.4/5 Inequalities in Triangles 

  • Standards: HSG.C.A.3; HSG.CO.C.9, 10; HSG.CO.D.12; HSG.SRT.B.5 

 

Essential Learning

Semester 1 

 

Topic 1:  Foundations of Geometry 

               

1.4        Inductive Reasoning 

  • Use inductive reasoning to identify patterns and make predictions based on data. 
  •  Use inductive reasoning to provide evidence that conjectures are true or provide counterexamples to disprove them. 

1.5        Conditional Statements (converse only) 

  • Write conditional and biconditional statements. 

1.6        Deductive Reasoning (concept only) 

  • Use deductive reasoning to draw a valid conclusion based on a set of given facts. 

1.7        Writing Proofs 

  • Use deductive reasoning to prove geometric theorems about lines and angles. 

 

Topic 2: Parallel and Perpendicular Lines

1.1        Measuring Segments and Angles 

  • Use the ruler and segment addition postulates. 
  • Use the protractor and the angle addition postulates. 
  • Identify congruent segments and congruent angles. 

2.1        Properties of Parallel Lines 

  • Define parallel lines using the undefined terms point and line. 
  • Prove theorems about lines and angles. 
  • Use theorems to find the measures of angles formed by parallel lines and a transversal. 

2.2        Proving Lines Parallel  

  • Prove that two lines cut by a transversal are parallel using the converses of parallel line angle relationship theorems. 
  • Use properties of parallel lines and transversals to solve real-world and mathematical problems. 
  • Write and use flow proofs. 

2.3        Parallel Lines and Triangles 

  • Use line constructed parallel to another line to solve problems and prove theorems. 
  • Use the sum of the angle measures in a triangle to solve problems. 

2.4        Slopes of Parallel and Perpendicular Lines 

  • Show that two lines in the coordinate plane are parallel by comparing their slopes, and solve problems.  
  •  Show that two line in the coordinate plane are perpendicular by comparing their slopes, and use that information to solve problems. 

 

Topic 3: Transformations

3.1        Reflections 

  • Find a reflected image and write a rule for a reflection. 
  • Define reflection as a transformation across a line or reflection with given properties and perform reflections on and off a coordinate grid. 

3.2        Translations  

  • Translate a figure and write a rule for a translation. 

3.3        Rotations 

  • Rotate a figure and write a rule for a rotation. 
  • Find the image of a figure after a composition of rigid motions. 

3.4        Classification of Rigid Motions 

  • Specify a sequence of transformations that will carry a given figure onto another. 
  • Use geometric descriptions of rigid motions to transform figures. 

3.5        Symmetry 

  • Describe the rotation and/or reflections that carry a polygon onto itself.  
  • Predict the effect of a given rigid motion on a figure. 
  • Identify types of symmetry in a figure. 

 

Topic 4: Triangle Congruence

 

4.1        Congruence 

  • Relate congruence to rigid motions. 
  • Demonstrate that two figure are congruent by using one or more rigid motions to map one onto the other. 

4.2        Isosceles and Equilateral Triangles 

  • Use properties of and theorems about isosceles and equilateral triangles to solve problems. 
  • Identify congruent triangles using properties of isosceles and equilateral triangles.   

4.3        Proving and Applying the SAS and SSS Congruence Criteria 

  • Prove triangle congruence by SAS and SSS criteria and use triangle congruence to solve problems. 
  • Understand that corresponding parts of congruent triangles are congruent and use CPCTC to prove theorems and solve problems. 

4.4        Proving and Applying the ASA and AAS Congruence Criteria 

  • Prove that two triangles are congruent using ASA and AAS criteria and apply ASA to solve problems. 
  • Prove that when all corresponding sides and angles of two polygons are congruent, the polygons are congruent.   

4.5        Congruence in Right Triangles 

  • Prove and use the HL Theorem.   
  • Use congruence criteria for triangles to solve problems and to prove relationships in geometric figures. 

4.6        Congruence in Overlapping Triangles 

  • Apply congruence criteria to increasingly difficult problems involving overlapping triangles and multiple triangles. 
  • Prove triangles are congruent by identifying corresponding parts and using theorems. 

 

 

Topic 6A:  Quadrilaterals and Other Polygons, plus 9.1 Polygons in the Coordinate Plan

 

1.3        Midpoint & Distance 

  • Use the midpoint formula to find the midpoint of a segment drawn on a coordinate plane. 
  •  Use the distance formula to find the length of a segment drawn on the coordinate plane. 

9.1        Polygons in the Coordinate Plane 

  • Use coordinate geometry to classify triangles and quadrilaterals on the coordinate plane.  
  • Solve problems involving triangles and polygons on the coordinate plane  

6.1        Polygon Angle-Sum Theorems 

  • Show that the sum of the exterior angles of a polygon is 360 degrees and use that to solve problems. 
  • Show that the um of the interior angles of a polygon is the product of 180 degrees and two less than the number of sides, and use that to solve problems. 

6.2        Kites and Trapezoids 

  • Use properties of the diagonals of a kite to solve problems. 
  • Use properties of isosceles trapezoids to solve problems. 
  • Use the relationship between the lengths of the bases and the midsegment of a trapezoid to solve problems.

 

Semester 2 

  

Topic 6B:  Quadrilaterals and Other Polygons, plus 9.1 Coordinates in the Coordinate Plane

 

6.3        Properties of Parallelograms 

  • Show that the consecutive angles of a parallelogram are supplementary and opposite angle are congruent.  
  • Show that opposite sides of a parallelogram are congruent. 
  • Show that diagonals of a parallelogram bisect each other. 

6.4        Proving a Quadrilateral is a Parallelogram 

  • Demonstrate that a quadrilateral is a parallelogram based on its sides and diagonals. 
  • Demonstrate that a quadrilateral is a parallelogram based on its angles.  

6.5        Properties of Special Parallelograms 

  • Prove that the diagonals of rhombuses are perpendicular bisectors of each other and angle bisectors of the angles of the rhombus.  
  • Prove that the diagonals of a rectangle are congruent. 
  • Use properties of rhombuses, rectangles, and squares to solve problems. 

6.6        Conditions of Special Parallelograms 

  • Identify rhombuses, rectangles, and squares by the characteristics of diagonals of parallelograms. 

 

 

Topic 7: Transformations

7.1        Dilations 

  • Dilate figures on and off the coordinate plane. 
  • Understand how distances and links in a dilation are related to the scale factor and center of dilation  

7.2        Similarity Transformations 

  • Understand that two figures are similar if there is a similarity transformation that maps one figure to the other. 
  • Identify a combination of rigid motions and dilation that maps one figure to a similar figure.  
  • Identify the coordinates of an image under a similarity transformation. 

7.3        Proving Triangles Similar 

  • Use dilations and rigid motions to prove triangles are similar. 
  • Prove and do use the AA~,SSS ~, and SAS ~ theorems to prove triangles are similar.         

7.4        Similarity in Right Triangles 

  • Use similarity of right triangles to solve problems. 
  • Use length relationships of the sides of right triangles and an altitude drawn to the hypotenuse to solve problems. 

7.5        Proportions in Triangles 

  • Use the side splitter theorem in the triangle midsegment theorem to find lengths of sides in segments of triangles. 
  • Use the triangle angle bisector theorem to find length of side in segments of triangles  

 

Topic 8:  Right Triangles and Trigonometry

 

8.1        Right Triangles and Pythagorean Theorem  

  • Prove the Pythagorean theorem using similar right triangles. 
  • Understand and apply the relationship between side links in a 45-45-90 and a 30-60-90 triangle.   

8.2        Trigonometric Ratios 

  • Define and calculate sine, cosine, and tangent ratios. 
  • Use trigonometric ratios to solve problems.           

8.5        Problem Solving with Trigonometry 

  • Distinguish between angles of elevation and depression. 

 

Topic 10: Circles

 

10.1      Arcs and Sectors 

  • Calculate the length of an arc when the central angle is given in degrees calculate the length of an arc when the central angle is given in degrees or radians.  
  • Calculate the area of sectors and segments of circles. 

10.2      Tangent Lines to a Circle

  • Identify lines that are tangent to a circle using angle measures and segment lengths. 
  • Solve problems involving tangent lines. 

10.3      Chords 

  • Prove and apply relationships between chords, arcs, and central angles. 
  • Find links of chords given the distance from the center of the circle and use this information to solve problems 

10.4      Inscribed Angles 

  • Identify and apply relationships between the measures of inscribed angles, arcs, and central angles.  
  • Identify and apply the relationships between an angle formed by a chord and a tangent to its intercepted arc. 

 

Topic 11: Two- and Three-Dimensional Models

 

11.2      Volumes of Prisms & Cylinders 

  • Do use Euler’s formula to calculate the number of vertices, faces, and edges in polyhedrons. 
  • Describe cross sections of polyhedrons. 
  • Describe rotations of polygons about an axis. 

11.3      Pyramids & Cones 

  • Understand how the volume formulas for pyramids and cones apply to oblique pyramids and cones. 
  • Model three dimensional figures as pyramids and cones to solve problems. 

 

11.4      Spheres 

  • Use Cavalieri’s Principle to show how the volume of a hemisphere is related to the volume of a cone and a cylinder. 
  • Calculate volumes and surface areas of spheres and composite figures. 

 

 

Topic 5:  Relationships in Triangles

 

1.2        Basic Constructions 

  • Construct copies of segments and angles, perpendicular bisectors of segments, and bisectors of angles. 
  • Apply construction to solve problems. 

5.1        Perpendicular and Angle Bisectors 

  • Prove the Perpendicular Bisector Theorem, the Angle Bisector Theorem, and their converses.  
  • Use the Perpendicular Bisector Theorem to solve problems. 
  • Use the Angle Bisector Theorem to solve problems. 

5.2        Bisectors in Triangles 

  • Prove that the point of concurrency of the perpendicular bisectors of a triangle, called the circumcenter, is equidistant from the vertices. 
  • Prove that the point of concurrency of the angle bisectors of a triangle, called the incenter is equidistant form the sides. 

5.3        Medians and Altitudes 

  • Identify special segments in triangles and understand theorems about them.  
  • Find and use the point of concurrency of the medians of a triangle to solve problems and prove relationships in triangles.  
  • Find the point of concurrency of the altitudes of a triangle. 

5.4        Inequalities in One Triangle 

  • Prove that the side lengths of a triangle are related to the angle measures of the triangle.   
  • Use the angle measures of a triangle to compare the side lengths of the triangle. 
  • Use the Triangle Inequality Theorem to determine if three given side lengths will form a triangle and to find a range of possible side lengths for a third side given two side lengths. 

5.5        Inequalities in Two Triangles 

  • Prove the Hinge Theorem and use the Hinge Theorem to compare side lengths.   
  • Prove the Converse of the Hinge Theorem and use the Converse of the Hinge Theorem to compare angle measures.