## Learning Math: Geometry

# Parallel Lines and Circles

## Use dynamic geometry software to construct figures with given characteristics, such as segments that are perpendicular, parallel, or of equal length, and to examine the properties of parallel lines and circles. Look past formal definitions and discover the properties and relationships among geometric figures for yourself.

### In This Session

**Part A: **Introduction to Geometer’s Sketchpad

**Part B: **Parallel Lines

**Part C: **Circles

**Part D:**Thinking with Technology

**Homework**

For information on required and/or optional materials for this session, see the Notes tab.In this session, you will use dynamic geometry software to construct figures with given characteristics such as perpendicular segments, parallel segments, and segments that are equal in length. You will also use this software to discover properties of parallel lines and circles. You will look at some formal geometric definitions, but the emphasis will be on the discovery of properties and relationships among geometric figures.

For information on required and/or optional materials for this session, see **Note 1**.

**Learning Objectives**

In this session, you will do the following:

- Learn how to use dynamic geometry software to construct geometric figures and explore their properties
- Learn about the properties of angles formed when a pair of parallel lines is cut by a transversal
- Learn to use the properties of circles to make different constructions
- Consider the advantages and disadvantages of using dynamic software in studying geometry

### Key Terms

**Previously Introduced**

**Line: **A line has only one dimension: length. It continues forever in two directions (so it has infinite length), but it has no width at all. A line connects two points via the shortest path, and then continues on in both directions.

**Plane: **A plane is a flat, two-dimensional object. We often represent a plane by a piece of paper, a blackboard, or the top of a desk. In fact, none of these is actually a plane, because a plane must continue infinitely in all directions and have no thickness at all. A plane can be defined by two intersecting lines or by three non-collinear points.

**Point: **A point specifies only location; it has no length, width, or depth. We usually represent a point with a dot on paper, but the dot we make has some dimension, while a true point has dimension 0.

**Quadrilateral: **A quadrilateral is a polygon with exactly four sides.

**Ray: **A ray can be thought of as a half a line. It has a point on one end, and it extends infinitely in the other direction.

**Vertex: **A vertex is the point where two sides of a polygon meet.

**New in This Session**

**Central Angle: **A central angle is an angle with its vertex at the center of a circle.

**Circle: **A circle is the set of all points in a plane that are equidistant from a given point in the plane, which is the center of the circle.

**Diameter: **A circle’s diameter is a segment that passes through the center and has its endpoints on the circle.

**Inscribed Angle: **An inscribed angle is an angle whose vertex is on a circle and whose rays intersect the circle.

**Parallel Lines: **Parallel lines are two lines in the same plane that never intersect. Another way to think about parallel lines is that they are “everywhere equidistant.” No matter where you measure, the perpendicular distance between two parallel lines is constant.

**Radius: **The radius of a circle is the distance from the circle’s center to a point on the circle, and is constant for a given circle.

**Transversal: **A transversal is a line that passes through (transverses) two other lines. We often consider what happens when the two other lines are parallel to each other.

### Notes

**Note 1**

**Materials Needed: **

Many of the activities in this session are built around the use of a dynamic geometry construction software package called Geometer’s Sketchpad. Your school may already have a copy or own similar software, or you can download a free trial version from Key Curriculum Press. Go to http://www.keypress.com/sketchpad/ and download the Instructor’s Evaluation Edition.

**Optional**

- Two pieces of paper
- Overhead projector
- Two transparencies