Unit 1: The Basics of Geometry
In this unit, we will explore the basic building blocks, vocabulary, and classifications of geometry. If you know how to identify and classify the shapes and properties of certain types of triangles, you can apply the appropriate rules and relationships to quickly and easily make predictive calculations.
Completing this unit should take you approximately 5 hours.
Upon successful completion of this unit, you will be able to:
- identify angles, rays, line segments, and points;
- determine the distance between two points;
- determine the lengths of a sections of a line segment;
- measure and classify angles;
- determine the measurement of unknown angles; and
- classify polygons by sides.
1.1: Language and Notation
In this section, we discuss the vocabulary you need to know to understand geometry. You should be comfortable with the meaning of the words used in geometry so you can use its language to understand and solve geometry problems.
Watch this video. Make sure you understand the definitions for line, line segment, ray, midpoint, collinear, and planar.
This video discusses the differences between geometric objects that may appear to be similar: lines, line segments, and rays.
1.2: The Distance between Two Points
Now that we have reviewed the basic language you need to know to understand geometry, we are ready to begin problem solving. First, let's explore how to determine the distance between two points on a line segment.
Read this article and watch the videos. Pay attention to the sections on the ruler postulate and the segment addition postulate. These two postulates help us perform certain measurements. The videos give examples. Read the examples near the end of the page closely to see step-by-step solutions.
Then, complete review questions 1, 2, 6, 7, 9, 10, 11, 16, and 17 and check your answers.
1.3: Midpoints and Segment Bisectors
A midpoint is the point on a line segment that divides it into two equal parts or the halfway point of a line segment. A segment bisector is a line, ray or segment that cuts another line segment into two equal parts.
Read this article and watch the video to learn about these concepts. Pay close attention to the midpoint formula, since we use this formula to determine the midpoint of a segment.
Then, complete guided practice questions 1–3 and check your answers.
1.4: Measuring Angles
Now that we have studied lines and line segments, we learn about angles between lines or line segments.
First, watch this video to learn the vocabulary of angles.
Once you feel comfortable with the vocabulary we use to describe angles, watch these videos to learn how to use a protractor, the tool we use to measure angles accurately. The videos also explain the units we use to define angles.
1.5: Congruent Angles and Angle Bisectors
In this section, we explore how to identify congruent angles and apply properties of an angle bisector.
Read this article and watch the videos. Pay attention to the section on investigation, which explains the step-by-step method for constructing an angle bisector. Carefully read the examples.
Then, complete practice questions 1, 5, 11, and 12 and check your answers.
1.6: Acute, Obtuse, and Right Angles
Acute angles are less than 90 degrees, right angles are 90 degrees, and obtuse angles are greater than 90 degrees.
Watch these videos to learn about ways to classify angles based on their degree.
1.7: Complementary and Supplementary Angles
A set of complementary angles add up to 90 degrees. A set of supplementary angles add up to 180 degrees.
Watch these videos. Pay close attention to the definition of complementary and supplementary used to describe sets of angles.
Then, complete this assessment and check your answers.
1.8: Linear Pairs
A linear pair of angles is formed when two lines intersect. Two angles are said to be linear if they are adjacent angles formed by two intersecting lines. The measure of a straight angle is 180 degrees, so a linear pair of angles must add up to 180 degrees. Two angles are adjacent when they have a common side and a common vertex (corner point) and do not overlap.
Read this article and watch the videos to learn about linear pairs, adjacent angles, and angles that form a linear pair. Pay attention to the examples of how to measure angles and identify linear pairs.
Then, complete review questions 6, 7, 12, and 13 and check your answers.
1.9: Vertical Angles
Vertical angles are nonadjacent angles formed by intersecting lines. We can determine the angles of vertical angles based on what we have already learned in this unit.
Watch this video to learn about vertical angles and how they relate to each other.
Then, watch this video for more advanced examples that use algebra.
Then, complete this assessment, which combines what you know about vertical angles with basic algebra to solve problems.
1.10: Polygon Classification
Now we can combine line segments with certain angles to form shapes called polygons. In this section, we classify polygons based on the types of angles (concave or convex) used to produce them.
Read this article. The table lists the common types of polygons we will encounter in geometry.