## Learning Math: Measurement

# Angle Measurement

## Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.

### In This Session

**Part A:** Angle Definition

**Part B:** Angles in Polygons

**Part C:** Geo-Logo

**Homework**

In this session, we will investigate angle measurement. We will review appropriate notation and describe angles in terms of the amount of turn. Looking at angles in this way highlights the fact that angle measure is dynamic rather than static — it changes as the amount of turn changes.

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

### Learning Objectives

In this session, you will do the following:

- Use the fact that there are 360 degrees in a complete turn to find the measures of angles in polygons
- Learn about the relationships between the angles within shapes
- Generalize a formula for finding the sum of the angles in any n-gon
- Use software to explore the differences between interior, exterior, and central angles
- Understand the relationships between turns, resulting angles, and the number of sides of a regular polygon

### Key Terms

**New in This Session:**

**Acute Angle: **An acute angle is an angle with a measurement greater than 0 degrees and less than 90 degrees.

**Adjacent Angles: **Adjacent angles are angles that share a common vertex and a common side between them.

**Central Angle: **A central angle, for regular polygons, has its vertex at the center of the polygon, and its rays go through any two adjacent vertices.

**Complementary Angles: **Complementary angles are such that the sum of their angle measures equals 90 degrees.

**Congruent Angles: **An exterior angle is an angle outside a polygon that lies between one side and an adjacent extended side.

**Exterior Angle:**An interior angle, or vertex angle, is an angle that lies between two sides inside a polygon.

**Interior (Vertex) Angle:** An interior angle, or vertex angle, is an angle that lies between two sides inside a polygon.

**Irregular Polygon:**An irregular polygon is any polygon that is not regular. (see regular polygon)

**Obtuse Angle: **An obtuse angle is an angle with a measurement greater than 90 but less than 180 degrees.

**Polygon:**A polygon is a two-dimensional geometric figure with these characteristics: It is made of straight line segments; each segment touches exactly two other segments, one at each of its endpoints; and it is closed — in other words it divides the plane into two distinct regions, one inside and the other outside the polygon.

**Regular Polygon: **A regular polygon has sides that are all the same length and angles that are all the same size.

**Right Angle: **A right angle is an angle that measures 90 degrees.

**Supplementary Angles: **Supplementary angles are such that the sum of their angle measures equals 180 degrees.

### Notes

**Note 1**

**Materials Needed:**

- Straightedge
- Protractor
- Angle ruler
- Bendable straws
- Geo-Logo software (optional)
- Power Polygons (a set of geometric shapes that are often used in upper elementary and middle school classrooms; the set consists of 15 different plastic polygons, labeled with letters from A to O) (optional)

Power Polygons can be obtained from:

- Power Polygons (a set of geometric shapes that are often used in upper elementary and middle school classrooms; the set consists of 15 different plastic polygons, labeled with letters from A to O) (optional)

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