### Unit 2: Geometry and Measurement

In this unit, you will explore the properties of basic geometric two-dimensional and three-dimensional shapes, as well as the units of measurements of geometric and physical properties of objects.

**Completing this unit should take you approximately 32 hours.**

Upon successful completion of this unit, you will be able to:

- identify the mathematical properties of two- and three-dimensional figures;
- calculate ratio and proportions;
- solve problems involving length, area, volume, or mass using scientific notation; and
- solve problems involving unit conversion (length, mass, time, miles per hour, dollars per gallon, etc.).

### 2.1: Lines and Angles

This section introduces basic geometry definitions and conventions of naming lines, rays, and angles.This lecture series defines the simplest geometric shapes: lines and rays. Watch the videos and complete the interactive exercises.

Watch this lecture series and complete the exercises.

- Often, the angles are described based on their degree measure: they can be acute, right, or obtuse. Watch this lecture series and complete the exercises to practice identifying different types of angles.
We can also classify angles based on their relationship to another angle. Vertical angles are congruent, supplementary angles add up to 180 degrees, and complementary angles add up to 90 degrees. Watch this lecture series to see the examples of different angle pairs. Complete the interactive exercises.

Complete these exercises and check your answers.

### 2.2: Triangles

This section focuses on a common geometric shape: the triangle. You will learn about different kinds of triangles and their properties.

Watch this lecture series, which discusses how to classify triangles by lengths of their sides and measures of their angles.

Right triangles have a unique relationship between the lengths of their sides, known as the Pythagorean theorem. Watch these videos and complete the interactive exercises.

Proving if some shapes, especially triangles, are congruent is an important part of the study of geometry. This lecture series discusses the three theorems that establish the congruence of triangles.

Complete these exercises and check your answers.

Complete these exercises and check your answers.

### 2.3: Quadrilaterals and Circles

This section discusses other common two-dimensional shapes: quadrilaterals and circles.

A quadrilateral is a two-dimensional shape with four sides and four angles. There are a lot of different kinds of quadrilaterals: familiar ones like squares and rectangles and less familiar ones like parallelograms and trapezoids. Quadrilaterals are classified based on the number of parallel lines, equal sides, and angles. Watch this lecture series and complete the interactive exercises.

Read this chapter, which summarizes all properties of various quadrilaterals, including the properties of their diagonals.

This video defines the familiar shape of a circle and related concepts such as radius and diameter.

Complete these exercises and check your answers.

### 2.4: Three-Dimensional Shapes

In this section, you will explore common three-dimensional shapes and related concepts.

Watch these videos and complete the interactive exercises to review the basics of solids (three-dimensional shapes).

### 2.5: Symmetry and Geometric Transformations

This section discusses the concepts of symmetry and geometric transformations.

Symmetry is an intuitive concept, but in geometry, it has a formal definition. Watch this lecture series and complete the interactive exercises.

These resources assume a basic familiarity with the coordinate plane. To review the coordinate plane and related terminology, watch the videos in section 3.4 of Unit 3. Transformation is another term commonly used, but it has a specific meaning in geometry. This lecture series will help you identify different kinds of transformations.

Translation is a type of rigid transformation. Watch this lecture series and complete the interactive exercises.

### 2.6: Similarity and Proportional Measurements

The concept of similarity is often used in applications of geometry, such as in building models of large real-life objects.

In this lecture, the transformations you have learned about previously are used to define similar shapes. Watch this video and complete the interactive exercises.

Similar triangles don't only look very much alike, but they also have a unique relationship between the length of their sides. This lecture series describes the properties of similar triangles. Watch the videos and complete the interactive exercises.

Watch the examples in this lecture series to see how to use the similarity to find missing elements of a triangle. Complete the interactive exercises for practice.

Read this section and watch the videos to see the examples of applications of similar triangles in geometric and real-life problems.

Complete these exercises and check your answers.

### 2.7: Perimeter, Circumference, Area, and Volume

The study of geometry is concerned with the properties of various shapes and related measurements. In this section, you will explore concepts of perimeter, area, and volume. These measurements have many real-world applications. For example, one needs to know the perimeter of a garden to find out how much material is required to make a fence or the area of a wall to find out how much paint is needed to paint it.

Watch this lecture series introducing the perimeter, and complete the interactive exercises to practice calculating the perimeter of rectangles.

Area is the measurement indicating how much space on a plane is taken up by a two-dimensional shape. While perimeter is measured in units of length, area is measured in square units. We will discuss units of measurement in more detail in the next section. Watch this lecture series about calculating the area of rectangles, and complete the interactive exercises.

Watch this lecture series that describes how to find the area of a parallelogram given its height and base, and complete the interactive exercises.

You can think of any triangle as half of a parallelogram! This is why its area is half that of the parallelogram: half of height times base. Watch this lecture series and complete the interactive exercises to practice using the triangle area formula.

To calculate the circumference (which is like perimeter) and the area of a circle, you need a special irrational number: pi. Pi is the ratio of the circumference of a circle to its diameter. Watch this lecture series to explore the circumference and area of circles, and complete the interactive exercises.

The boundary of three-dimensional objects consists of several two-dimensional shapes. The total area of these shapes is called the surface area of the object. It gives some idea of how large an object is, but now how much space it takes up. This is measured by volume, which we will discuss next. Watch this lecture series and complete the interactive exercises to practice calculating the surface area of rectangular prisms.

Volume measures how much space is taken up by a three-dimensional object. The volume is measured in cubic units of length, such as cubic feet or cubic centimeters. Watch this lecture series and complete the interactive exercises to practice finding the volume of rectangular prisms and objects composed of several rectangular prisms.

This chapter gives an overview of volume formulas for more complicated three-dimensional objects: triangular prisms and pyramids, cylinders, cones, and spheres.

Complete these exercises and check your answers.

Complete these exercises and check your answers.

Complete these exercises and check your answers.

### 2.8: Units of Measurements

This section introduces units of measurement of basic geometric and physical quantities (distance, mass, and time) in both the metric and customary U.S system. You will learn how to express the same measurements in different units and determine appropriate units to use when measuring a property of an object.

This lecture series introduces the units of length (or distance) in the metric and U.S. customary systems. Watch the videos and complete the interactive exercises.

Watch this lecture series and complete the interactive exercises to get a feel for various units of volume in the metric and U.S. customary systems: liters, pints, gallons, and so on.

This lecture series discusses different units of mass in the metric and U.S. customary systems. Mass and weight are not the same thing, even though they are used interchangeably in everyday language.

Watch this lecture series and complete the interactive exercises to practice converting measurements of length to different units.

Watch this lecture series and complete the interactive exercises to practice converting measurements of time to different units.

Watch this lecture series and complete the interactive exercises to practice converting measurements of volume to different units.

Watch this lecture series and complete the interactive exercises to practice converting measurements of mass to different units.

This chapter explains a method for converting any units of measurement, including derived units such as square meters or miles per hour, given appropriate conversion factors. The list of common conversion factors is given. This chapter also provides real-world examples of when such conversions might need to be made. Read the chapter and work through the examples.

Complete this section to practice converting units. Then, watch the videos to see the solutions.

Complete these exercises and check your answers.

### Unit 2 Assessment

- Receive a grade
Take this assessment to see how well you understood this unit.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment