About This Chapter
Parallel Lines and Polygons - Chapter Summary and Learning Objectives
In this geometry chapter, instructors will help you learn about parallel lines, those straight objects on a plane that never intersect and are always the same distance away from each other. Using easy-to-understand explanations and drawings, the experienced instructors who lead these lessons will introduce you to the parallel postulate or hypothesis and the concept of indirect proof. You'll also learn about congruent angles, or those that are the same shape and size, as well as how to identify and measure angles and polygons. When you complete this chapter, you should be able to:
- Identify the angles formed by parallel lines and a transversal
- Use geometrical theories to prove that lines are parallel
- Indentify and measure the area, diagonal and perimeter of a polygon
- Measure the angles of polygons and triangles
|The Parallel Postulate and Indirect Proof||Define and provide examples of indirect proof, and explain how it relates to the parallel postulate.|
|Angles Formed by a Transversal||Define transversal. Identify angles formed by parallel lines and a transversal, and provide examples.|
|Parallel Lines: How to Prove Lines Are Parallel||Discuss how to prove lines are parallel, including those intersected by a transversal and pairs of congruent alternate exterior angles, congruent corresponding angles, congruent alternate exterior angles or supplementary interior angles on the same side of the transversal.|
|Using Converse Statements to Prove Lines Are Parallel||Apply the opposite of a statement to prove that: 1) When two parallel lines are intersected by a transversal, all pairs of alternate interior angles are congruent. 2) When two lines are parallel and a third line is perpendicular to one of them, it is also perpendicular to the other. 3) When two parallel lines are intersected by a transversal, all pairs of corresponding angles are congruent. 4) When two parallel lines are intersected by a transversal, all pairs of alternate exterior angles are congruent. 5) When two parallel lines are cut by a transversal, all pairs of interior angles on the same side of the transversal are supplementary.|
|What Are Polygons? - Definition and Measurements||Define a polygon, and describe how to measure its diagonal and perimeter. Determine properties and provide examples of a polygon's angles by constructing a line parallel to a given line that passes through a point not on the given line.|
|Measuring the Area of Regular Polygons||Describe the apothem of a regular polygon. Explain how to apply the area formula, and provide examples.|
|Measuring the Angles of Triangles: 180 Degrees||Establish that the sum of angles of a triangle is 180°.|
|Measuring the Angles of a Polygon||Show how to find the sum of measures of angles in a polygon.|
1. Angles Formed by a Transversal
When you have a pair of parallel lines and a transversal, something very interesting happens to the angles that are formed. You can see this happen in real life at street intersections and such. Watch this video lesson to learn about all of this.
2. Parallel Lines: How to Prove Lines Are Parallel
Watch this video lesson to learn how you can prove that two lines are parallel just by matching up pairs of angles. Learn which angles to pair up and what to look for.
3. Using Converse Statements to Prove Lines Are Parallel
Because a pair of parallel lines produces unique angle characteristics, we can use this information to our advantage. Watch this video lesson to see how we turn this advantage into converse statements to help us prove parallel lines.
4. Constructing a Parallel Line Using a Point Not on the Given Line
Watch this video lesson, and you will learn how to draw parallel lines with just a compass and a straightedge. Also, learn why you would want to be able to do this in real life.
5. The Parallel Postulate: Definition & Examples
In this lesson, you will learn about an important postulate in Euclidean geometry, called the Parallel Postulate. It sounds kind of hard, but this lesson explains it in simple terms and provides several examples of it as well.
6. What Are Polygons? - Definition and Examples
Watch this video lesson to see how the shapes that you grew up with are all related. Learn why the shapes of bricks, stars, and street blocks are considered polygons while the sun, moon, and rolling hills are not.
7. Regular Polygons: Definition & Parts
What makes a polygon a polygon? And what shapes are considered polygons? In this lesson, find out the answers to these questions and more as we learn all about polygons and their parts.
8. How to Find the Number of Diagonals in a Polygon
At first, you might think that the diagonals of a polygon are pretty useless, but watch this video lesson to learn how diagonals are used in real life. You will learn the formula to find how many diagonals a polygon has as well as how to use it.
9. Finding the Perimeter of Polygons
In this video lesson, you will learn the process you need to take to find the perimeter of polygons. You will also learn that there is a shortcut if your polygon is a regular polygon.
10. Measuring the Area of Regular Polygons: Formula & Examples
Watch this video lesson to learn why a regular polygon makes your life easier when it comes to finding the area inside one. Learn the one formula for area that will work for any type of regular polygon.
11. Measuring the Angles of Triangles: 180 Degrees
Watch this video lesson to see why a triangle's angles always add up to 180 degrees. Also, learn how you can use this unique fact about triangles to find an unknown angle in a triangle.
12. How to Measure the Angles of a Polygon & Find the Sum
Watch this video lesson to learn the one formula that lets you find the measure of angles in any regular polygon. Also, learn how you can tell if you are working with a regular polygon or not.
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Other chapters within the Geometry: High School course
- High School Geometry: Foundations of Geometry
- High School Geometry: Logic in Mathematics
- High School Geometry: Introduction to Geometric Figures
- High School Geometry: Properties of Triangles
- High School Geometry: Triangles, Theorems and Proofs
- High School Geometry: Similar Polygons
- High School Geometry: Quadrilaterals
- High School Geometry: Circular Arcs and Circles
- High School Geometry: Conic Sections
- High School Geometry: Geometric Solids
- High School Geometry: Analytical Geometry
- High School Geometry: Probability
- High School Geometry: Introduction to Trigonometry