Acknowledgments 

ii  
The University of Chicago School mathematics Project 

iv  


vi  
To the Student 

1  (3) 


4  (60) 


6  (5) 


11  (6) 


17  (5) 


22  (8) 

InClass Activity: Drawing a Box in Perspective 


29  (1) 


30  (5) 

The Need for Undefined Terms 


35  (6) 

Postulates for Euclidean Geometry 


41  (5) 


46  (18) 


52  (2) 


54  (2) 


56  (2) 


58  (6) 

The Language and Logic of Geometry 


64  (58) 


64  (5) 


69  (7) 


76  (5) 


81  (6) 

Unions and Intersections of Figures 


87  (8) 

Inclass Activity: Defining Polygons 


94  (1) 


95  (6) 

Using an Automatic Drawer: The Triangle Inequality 


101  (6) 


107  (15) 


113  (2) 


115  (1) 


116  (2) 


118  (3) 


121  (1) 


122  (60) 

Angles and Their Measures 


124  (8) 

Inclass Activity: Rotating a Triangle 


131  (1) 


132  (5) 


137  (7) 

Algebra Properties Used in Geometry 


144  (6) 


150  (5) 


155  (6) 


161  (6) 

Drawing parallel and Perpendicular Lines 


167  (15) 


172  (2) 


174  (2) 


176  (2) 


178  (4) 

From Reflections To Congruence 


182  (60) 


184  (7) 

Inclass Activity: Reflecting Polygons 


190  (1) 


191  (6) 

Miniature Golf and Billiards 


197  (6) 

Composing Reflections over Parallel Lines 


203  (7) 

Composing Reflections over Intersecting Lines 


210  (6) 


216  (6) 


222  (6) 

When are Figures Congruent? 


228  (14) 


233  (2) 


235  (1) 


236  (2) 


238  (4) 


242  (56) 

Corresponding Parts of Congruent Figures 


244  (6) 


250  (6) 

OneStep Congruence Proofs 


256  (7) 

Inclass Activity: Circle Constructions 


262  (1) 

Proofs Using Transitivity 


263  (7) 

Inclass Activity: Investigating Perpendicular Bisectors 


269  (1) 


270  (6) 

Auxiliary Figures and Uniqueness 


276  (5) 

Sums of Angle Measures in Polygons 


281  (17) 


288  (2) 


290  (1) 


291  (1) 


292  (6) 


298  (64) 

Reflectionsymmetric Figures 


300  (9) 

Inclass Activity: Angle Bisectors, Medians, ⊥Bisectors of a Triangle 


308  (1) 


309  (7) 


316  (7) 

Inclass Activity: Conjectures about Quadrilaterals 


322  (1) 


323  (6) 


329  (6) 


335  (6) 

Inclass Activity: Symmetry Lines in Polygons 


340  (1) 


341  (6) 

Regular Polygons and Schedules 


347  (15) 


352  (2) 


354  (2) 


356  (2) 


358  (4) 


362  (72) 


364  (6) 

Triangle Congruence Theorems 


370  (8) 

Proofs Using Triangle Congruence Theorems 


378  (6) 


384  (5) 

The SSA Condition and HL Congruence 


389  (8) 


397  (8) 

Inclass Activity: Congruent Parts of Parallelograms 


404  (1) 

Properties of Parallelograms 


405  (6) 

Sufficient conditions for Parallelograms 


411  (5) 


416  (18) 


423  (2) 


425  (1) 


426  (2) 


428  (5) 


433  (1) 


434  (60) 


436  (5) 

Fundamental Properties of Area 


441  (6) 

Areas of irregular Regions 


447  (6) 


453  (6) 


459  (7) 

Inclass Activity: Creating Squares from Right Triangles 


465  (1) 


466  (8) 

Inclass Activity: Estimating the Circumference of a Circle 


473  (1) 

Arc Length and circumference 


474  (6) 

Inclass Activity: Estimating the Area of a circle 


479  (1) 


480  (14) 


485  (2) 


487  (1) 


488  (2) 


490  (4) 

ThreeDimensional Figures 


494  (68) 

Points, Lines, and Planes in Space 


496  (4) 

Parallel and Perpendicular Lines 


500  (5) 


505  (6) 


511  (6) 


517  (8) 

Reflection Symmetry in Space 


525  (6) 

Viewing Solids and surfaces 


531  (6) 

Inclass Activity: Building Cubes 


536  (1) 


537  (7) 

Inclass Activity: Coloring Map 


543  (1) 

Maps and the FourColor Theorem 


544  (18) 


552  (2) 


554  (2) 


556  (2) 


558  (4) 

Surface Areas And Volumes 


562  (58) 

Surface Areas of Prisms and Cylinders 


564  (6) 

Inclass Activity: Lateral Area of Regular Pyramids 


569  (1) 

Surface Area of Pyramids and Cones 


570  (6) 

Fundamental Properties of Volume 


576  (6) 

Multiplication, Area, and Volume 


582  (5) 

Volumes of Prisms and Cylinders 


587  (6) 


593  (6) 

Inclass Activity: Building a Prism from Pyramids 


598  (1) 

Volumes of Pyramids and Cones 


599  (5) 


604  (5) 

The Surface Area of a Sphere 


609  (11) 


613  (2) 


615  (1) 


616  (1) 


617  (3) 

Indirect and Coordinate Proofs 


620  (64) 

The Logic of Making Conclusions 


622  (6) 


628  (6) 


634  (5) 


639  (6) 


645  (6) 


651  (5) 


656  (6) 

Inclass Activity: Connecting Midpoints in a Triangle 


661  (1) 


662  (6) 

ThreeDimensional Coordinates 


668  (16) 


675  (2) 


677  (2) 


679  (1) 


680  (4) 


684  (52) 


686  (6) 

Inclass Activity: Drawing SizeChange Images 


691  (1) 


692  (6) 

Properties of Size Changes 


698  (7) 


705  (6) 


711  (6) 

The Fundamental Theorem of Similarity 


717  (7) 


724  (12) 


729  (2) 


731  (1) 


732  (1) 


733  (3) 

Similar Triangles and Trigonometry 


736  (62) 

The SSA Similarity Theorem 


738  (6) 

The AA and SAS Similarity Theorem 


744  (7) 

The SideSplitting Theorem 


751  (8) 

Inclass Activity: Similarity in Right Triangles 


758  (1) 

Geometric Means in Right Triangles 


759  (6) 


765  (7) 

Inclass Activity: Ratios of Legs in Right Triangles 


771  (1) 


772  (5) 

The Sine and Cosine Ratios 


777  (7) 

More Work with Vectors and Area 


784  (14) 


789  (2) 


791  (1) 


792  (2) 


794  (4) 

Further Work With Circles 


798  (64) 

Chord Length and Arc Measure 


800  (6) 

The Inscribed Angle Theorem 


806  (7) 

Locating the Center of a Circle 


813  (5) 

Angles Formed by Chords or Secants 


818  (5) 

Tangents to Circles and Spheres 


823  (6) 

Angles Formed by Tangents 


829  (7) 

Inclass Activity: Investigating Lengths of Chords. Secants, and Tangents 


835  (1) 

Lengths of Chords, Secants, and Tangents 


836  (6) 

The Isoperimetric Inequality 


842  (6) 

The Isoperimetric Inequality in three Dimensions 


848  (14) 


853  (2) 


855  (1) 


856  (2) 


858  (4) 
Appendix A: Theorems 

862  (6) 
Appendix B: Postulates 

868  (2) 
Appendix C: Formulas 

870  (2) 
Selected Answers 

872  (35) 
Glossary 

907  (9) 
Index 

916  (11) 
List of Symbols 

927  (1) 
Photo Acknowledgments 

928  