## Dynamic Explorations

### Chapter 0

• Daisy Designs
Learn how to create the daisy designs shown on page 11 of Discovering Geometry. These JavaSketches will help you better understand how to use a compass or geometry software to make your own daisy designs.

### Chapter 1

• Spiral Designs
Learn how to create a spiral design. This exploration will help you better understand how to use compass and straightedge or geometry software to complete the Spiral Designs Project on page 35 of Discovering Geometry.
• Protractor
Learn how to use a protractor using this interactive electronic version. This will help you understand how to use your own protractor to do Example A on page 39 and the exercises at the end of Lesson 1.2 of Discovering Geometry.
• Three Types of Angles
Explore the differences between right, acute, and obtuse angles. This dynamic sketch gives infinitely many examples, helping you write definitions for the Investigation on page 49 of Discovering Geometry.
• Treasure Hunt
Use this exploration to solve Example B on page 82 of Discovering Geometry. Then extend the scenario and explore different types of solutions that could come up for similar locus problems.

### Chapter 2

• Special Angles on Parallel Lines
Investigate the properties of the angles that are formed when a transversal cuts two parallel lines. Then formulate the Parallel Lines Conjecture (C-3) on page 129 of Discovering Geometry.
• The Sierpinski Triangle
Investigate the Sierpinski triangle and its properties. This exploration is similar to the Exploration Patterns in Fractals on pages 137-139 of Discovering Geometry.

### Chapter 3

• Triangle Centers
Investigate the behavior of the incenter, circumcenter, and orthocenter of a triangle as the triangle changes from acute to obtuse. Use this sketch to extend the investigations on pages 178-180 in Lesson 3.7 of Discovering Geometry or to help you solve Exercises 10 and 11 on page 183.
• The Centroid
Use the sketches on this page to investigate the medians of a triangle, state the Median Concurrency Conjecture (C-14), and state the Centroid Conjecture (C-15). This activity can be used as a replacement or extension of Investigation 1 on pages 185???186 in Lesson 3.8 of Discovering Geometry.
• The Euler Line
Discover how the points of concurrency of a triangle relate to a special line, the Euler line. This exploration can be used to replace the Exploration The Euler Line on pages 191-192 of Discovering Geometry.

### Chapter 4

• Triangles in a Circle
Investigate the properties of a triangle with one vertex at the center of a circle and the other two vertices on the circumference of the circle. This will help you to solve Exercise 12 on page 210 of Discovering Geometry.
• The Triangle Inequality
The shortest path between two points is a segment. See how this property is related to the lengths of the three sides of a triangle. Then formulate the Triangle Inequality Conjecture (C-20) on page 216 of Discovering Geometry.
• Random Triangles
When a segment is randomly cut into three segments, sometimes a triangle can be formed from the segments, and sometimes it can't. If you have Fathom Dynamic Data??? software, you can use the Random Triangles demonstration in the sample activities at the Fathom Resources Center to help you complete the project on page 220 of Discovering Geometry.
• Changing Triangle
Investigate the properties of a triangle with a median as one vertex moves parallel to the base. This will help you solve Exercise 22 on page 235 of Discovering Geometry.

### Chapter 5

• The Exterior Angle Sum of a Polygon
Investigate the sum of the exterior angles of a polygon. Then formulate the Exterior Angle Sum Conjecture (C-32) and the Equiangular Polygon Conjecture (C-33) on page 263 of Discovering Geometry.
• Properties of Kites
Investigate the properties of kites. Then formulate the Kite Angles Conjecture (C-34), the Kite Diagonals Conjecture (C-35), the Kite Diagonal Bisector Conjecture (C-36), and the Kite Angle Bisector Conjecture (C-37) on page 269 of Discovering Geometry.
• Properties of Parallelograms
Use this dynamic exploration to discover some special properties of parallelograms. Then formulate the Parallelogram Opposite Angles Conjecture (C-45), the Parallelogram Consecutive Angles Conjecture (C-45), the Parallelogram Opposite Sides Conjecture (C-46), and the Parallelogram Diagonals Conjecture (C-47) on pages 281???282 of Discovering Geometry.
• Resultant Vector
See how an airplane's engine velocity combines with the wind's velocity to determine the resulting speed and direction of the plane. This sketch is a dynamic version of the illustration on page 283 of Discovering Geometry.
• Properties of Special Parallelograms
Discover some properties of rhombuses, rectangles, and squares, and formulate five conjectures (C-48 to C-52). This exploration can be used to replace or extend the investigations on pages 291-294 of Discovering Geometry.

### Chapter 6

• Chord Properties
Discover several properties of chords and formulate five conjectures (C-55 to C-59). This exploration can be used to replace or extend the investigations on pages 317-320 of Discovering Geometry.
• Intersecting Chords
Use this exploration to discover a relationship between the measure of an angle formed by two intersecting chords and the intercepted arcs. This will help you complete the mini-investigation (Exercise 16) on page 339 of Discovering Geometry.
• Intersecting Secants
Discover a relationship between the measure of an angle formed by two intersecting chords and the intercepted arcs. This will help you complete the mini-investigation (Exercise 9) on page 343 of Discovering Geometry.

### Chapter 8

• Area Formula for Parallelograms
Discover how the area formula for a parallelogram is based upon the area formula for rectangles. This exploration can be used to replace or extend the investigation on pages 424-425 of Discovering Geometry.
• Areas of Triangles, Trapezoids, and Kites
Use the area formula for rectangles and parallelograms to discover or demonstrate the formulas for the areas of triangles, trapezoids, and kites. This exploration gives you an alternative way to complete the investigations on pages 429???430 of Discovering Geometry.

### Chapter 9

• The Theorem of Pythagoras
Learn more about the Pythagorean Theorem and see whether or not it works for triangles that are not right triangles. This exploration will help you understand Lesson 9.1 on pages 478-480 of Discovering Geometry.

### Chapter 11

• Similar Triangles
Investigate shortcuts for triangle similarity, including AA, AAA, SS, SSS, SAS, SSA, ASA, and SAA. These dynamic sketches will help you understand which shortcuts work under all circumstances. This exploration can be used to replace or extend the investigations starting on pages 589-591 in Lesson 11.2 of Discovering Geometry.
• Similar Polygons
Investigate the proportional relationships between the sides of similar polygons, including the Proportional Parts Conjecture (C-94) for similar triangles. This exploration can be used to replace or extend Investigation 1 on page 603 in Lesson 11.4 of Discovering Geometry.
• Similar Solids
Investigate proportional relationships between similar polygons, including the Proportional Areas Conjecture (C-96) and Proportional Volumes Conjecture (C-97). This exploration can be used to replace or extend the investigations on pages 608-609 and pages 615-616 in Lessons 11.5 and 11.6 of Discovering Geometry.

### Chapter 12

• The Law of Cosines
Validate for yourself the Law of Cosines, given on page 661 of Discovering Geometry. This law answers the question, "What happens to the Pythagorean Theorem for acute triangles or obtuse triangles?"