CK-12 Geometry Read online

  Defining Triangles

  The first shape to examine is the triangle. Though you have probably heard of triangles before, it is helpful to review the formal definition. A triangle is any closed figure made by three line segments intersecting at their endpoints. Every triangle has three vertices (points at which the segments meet), three sides (the segments themselves), and three interior angles (formed at each vertex). All of the following shapes are triangles.

  You may have learned in the past that the sum of the interior angles in a triangle is always . Later we will prove this property, but for now you can use this fact to find missing angles. Other important properties of triangles will be explored in later chapters.

  Example 1

  Which of the figures below are not triangles?

  To solve this problem, you must carefully analyze the four shapes in the answer choices. Remember that a triangle has three sides, three vertices, and three interior angles. Choice A fits this description, so it is a triangle. Choice B has one curved side, so its sides are not exclusively line segments. Choice C is also a triangle. Choice D, however, is not a closed shape. Therefore, it is not a triangle. Choices B and D are not triangles.

  Example 2

  How many triangles are in the diagram below?

  To solve this problem, you must carefully count the triangles of different size. Begin with the smallest triangles. There are small triangles.

  Now count the triangles that are formed by four of the smaller triangles, like the one below.

  There are a total of seven triangles of this size, if you remember to count the inverted one in the center of the diagram.

  Next, count the triangles that are formed by nine of the smaller triangles. There are three of these triangles. And finally, there is one triangle formed by smaller triangles.

  Now, add these numbers together.

  So, there are a total of triangles in the figure shown.

  Classifications by Angles

  Earlier in this chapter, you learned how to classify angles as acute, obtuse, or right. Now that you know how to identify triangles, we can separate them into classifications as well. One way to classify a triangle is by the measure of its angles. In any triangle, two of the angles will always be acute. This is necessary to keep the total sum of the interior angles at . The third angle, however, can be acute, obtuse, or right.

  This is how triangles are classified. If a triangle has one right angle, it is called a right triangle.

  If a triangle has one obtuse angle, it is called an obtuse triangle.

  If all of the angles are acute, it is called an acute triangle.

  The last type of triangle classifications by angles occurs when all angles are congruent. This triangle is called an equiangular triangle.

  Example 3

  Which term best describes below?

  The triangle in the diagram has two acute angles. But, so is an obtuse angle. If the angle measure were not given, you could check this using the corner of a piece of notebook paper or by measuring the angle with a protractor. An obtuse angle will be greater than (the square corner of a paper) and less than (a straight line). Since one angle in the triangle above is obtuse, it is an obtuse triangle.

  Classifying by Side Lengths

  There are more types of triangle classes that are not based on angle measure. Instead, these classifications have to do with the sides of the triangle and their relationships to each other. When a triangle has all sides of different length, it is called a scalene triangle.

  When at least two sides of a triangle are congruent, the triangle is said to be an isosceles triangle.

  Finally, when a triangle has sides that are all congruent, it is called an equilateral triangle. Note that by the definitions, an equilateral triangle is also an isosceles triangle.

  Example 4

  Which term best describes the triangle below?

  A. scalene

  B. isosceles

  C. equilateral

  To classify the triangle by side lengths, you have to examine the relationships between the sides. Two of the sides in this triangle are congruent, so it is an isosceles triangle. The correct answer is B.

  Lesson Summary

  In this lesson, we explored triangles and their classifications. Specifically, we have learned:

  How to define triangles.

  How to classify triangles as acute, right, obtuse, or equiangular.

  How to classify triangles as scalene, isosceles, or equilateral.

  These terms or concepts are important in many different types of geometric practice. It is important to have these concepts solidified in your mind as you explore other topics of geometry and mathematics.

  Review Questions

  Exercises 1-5: Classify each triangle by its sides and by its angles. If you do not have enough information to make a classification, write “not enough information.”

  Sketch an equiangular triangle. What must be true about the sides?

  Sketch an obtuse isosceles triangle.

  True or false: A right triangle can be scalene.

  True or false: An obtuse triangle can have more than one obtuse angle.

  One of the answers in 8 or 9 is false. Sketch an illustration to show why it is false, and change the false statement to make it true.

  Review Answers

  A is an acute scalene triangle.

  B is an equilateral triangle.

  C is a right isosceles triangle.

  D is a scalene triangle. Since we don’t know anything about the angles, we cannot assume it is a right triangle, even though one of the angles looks like it may be .

  E is an obtuse scalene triangle.

  If a triangle is equiangular then it is also equilateral, so the sides are all congruent.

  Sketch below:

  True.

  False.

  9 is false since the three sides would not make a triangle. To make the statement true, it should say: “An obtuse triangle has exactly one obtuse angle.”

  Classifying Polygons

  Learning Objectives

  Define polygons.

  Understand the difference between convex and concave polygons.

  Classify polygons by number of sides.

  Use the distance formula to find side lengths on a coordinate grid.

  Introduction

  As you progress in your studies of geometry, you can examine different types of shapes. In the last lesson, you studied the triangle, and different ways to classify triangles. This lesson presents other shapes, called polygons. There are many different ways to classify and analyze these shapes. Practice these classification procedures frequently and they will get easier and easier.

  Defining Polygons

  Now that you know what a triangle is, you can learn about other types of shapes. Triangles belong to a larger group of shapes called polygons. A polygon is any closed planar figure that is made entirely of line segments that intersect at their endpoints. Polygons can have any number of sides and angles, but the sides can never be curved.

  The segments are called the sides of the polygons, and the points where the segments intersect are called vertices. Note that the singular of vertices is vertex.

  The easiest way to identify a polygon is to look for a closed figure with no curved sides. If there is any curvature in a shape, it cannot be a polygon. Also, the points of a polygon must all lie within the same plane (or it wouldn’t be two-dimensional).

  Example 1

  Which of the figures below is a polygon?

  The easiest way to identify the polygon is to identify which shapes are not polygons. Choices B and C each have at least one curved side. So they cannot be polygons. Choice D has all straight sides, but one of the vertices is not at the endpoints of the two adjacent sides, so it is not a polygon. Choice A is composed entirely of line segments that intersect at their endpoints. So, it is a polygon. The correct answer is A.

  Example 2

  Which of the figures below is n
ot a polygon?

  All four of the shapes are composed of line segments, so you cannot eliminate any choices based on that criteria alone. Notice that choices A, B, and D have points that all lie within the same plane. Choice C is a three-dimensional shape, so it does not lie within one plane. So it is not a polygon. The correct answer is C.

  Convex and Concave Polygons

  Now that you know how to identify polygons, you can begin to practice classifying them. The first type of classification to learn is whether a polygon is convex or concave. Think of the term concave as referring to a cave, or an interior space. A concave polygon has a section that “points inward” toward the middle of the shape. In any concave polygon, there are at least two vertices that can be connected without passing through the interior of the shape. The polygon below is concave and demonstrates this property.

  A convex polygon does not share this property. Any time you connect the vertices of a convex polygon, the segments between nonadjacent vertices will travel through the interior of the shape. Lines segments that connect to vertices traveling only on the interior of the shape are called diagonals.

  Example 3

  Identify whether the shapes below are convex or concave.

  To solve this problem, connect the vertices to see if the segments pass through the interior or exterior of the shape.

  A. The segments go through the interior.

  Therefore, the polygon is convex.

  B. The segments go through the exterior.

  Therefore, the polygon is concave.

  C. One of the segments goes through the exterior.

  Thus, the polygon is concave.

  Classifying Polygons

  The most common way to classify a polygon is by the number of sides. Regardless of whether the polygon is convex or concave, it can be named by the number of sides. The prefix in each name reveals the number of sides. The chart below shows names and samples of polygons.

  Polygon Name Number of Sides Sample Drawings

  Triangle

  Quadrilateral

  Pentagon

  Hexagon

  Heptagon

  Octagon

  Nonagon

  Decagon

  Undecagon or hendecagon (there is some debate!)

  Dodecagon

  gon (where )

  Practice using these polygon names with the appropriate prefixes. The more you practice, the more you will remember.

  Example 4

  Name the three polygons below by their number of sides.

  A. This shape has seven sides, so it is a heptagon.

  B. This shape has five sides, so it is a pentagon.

  C. This shape has ten sides, so it is a decagon.

  Using the Distance Formula on Polygons

  You can use the distance formula to find the lengths of sides of polygons if they are on a coordinate grid. Remember to carefully assign the values to the variables to ensure accuracy. Recall from algebra that you can find the distance between points and using the following formula.

  Example 5

  A quadrilateral has been drawn on the coordinate grid below.

  What is the length of segment ?

  Use the distance formula to solve this problem. The endpoints of are and . Substitute for , for , for , and for . Then we have:

  So the distance between points and is , or about .

  Lesson Summary

  In this lesson, we explored polygons. Specifically, we have learned:

  How to define polygons.

  How to understand the difference between convex and concave polygons.

  How to classify polygons by number of sides.

  How to use the distance formula to find side lengths on a coordinate grid.

  Polygons are important geometric shapes, and there are many different types of questions that involve them. Polygons are important aspects of architecture and design and appear constantly in nature. Notice the polygons you see every day when you look at buildings, chopped vegetables, and even bookshelves. Make sure you practice the classifications of different polygons so that you can name them easily.

  Review Questions

  For exercises 1-5, name each polygon in as much detail as possible.

  Explain why the following figures are NOT polygons:

  How many diagonals can you draw from one vertex of a pentagon? Draw a sketch of your answer.

  How many diagonals can you draw from one vertex of an octagon? Draw a sketch of your answer.

  How many diagonals can you draw from one vertex of a dodecagon?

  Use your answers to 7, 8, and 9 and try more examples if necessary to answer the question: How many diagonals can you draw from one vertex of an gon?

  Review Answers

  This is a convex pentagon.

  Concave octagon.

  Concave gon (note that the number of sides is equal to the number of vertices, so it may be easier to count the points [vertices] instead of the sides).

  Concave decagon.

  Convex quadrilateral.

  A is not a polygon since the two sides do not meet at a vertex; B is not a polygon since one side is curved; C is not a polygon since it is not enclosed.

  The answer is .

  The answer is .

  A dodecagon has twelve sides, so you can draw nine diagonals from one vertex.

  Use this table to answer question 10.

  Sides Diagonals from One Vertex

  To see the pattern, try adding a “process” column that takes you from the left column to the right side.

  Sides Process Diagonals from One Vertex

  Notice that we subtract

  from each number on the left to arrive at the number in the right column. So, if the number in the left column is

  (standing for some unknown number), then the number in the right column is

  .

  Problem Solving in Geometry

  Learning Objectives

  Read and understand given problem situations.

  Use multiple representations to restate problem situations.

  Identify problem-solving plans.

  Solve real-world problems using planning strategies.

  Introduction

  One of the most important things we hope you will learn in school is how to solve problems. In real life, problem solving is not usually as clear as it is in school. Often, performing a calculation or measurement can be a simple task. Knowing what to measure or solve for can be the greatest challenge in solving problems. This lesson helps you develop the skills needed to become a good problem solver.

  Understanding Problem Situations

  The first step whenever you approach a complicated problem is to simplify the problem. That means identifying the necessary information, and finding the desired value. Begin by asking yourself the simple question: What is this problem asking for?

  If the problem had to ask you only one question, what would it be? This helps you identify how you should respond in the end.

  Next, you have to find the information you need to solve the problem. Ask yourself another question: What do I need to know to find the answer?

  This question will help you sift through information that may be helpful with this problem.

  Use these basic questions to simplify the following problem. Don’t try to solve it yet, just begin this process with questioning.

  Example 1

  Ehab drew a rectangle on the chalkboard. was and was . If Ehab draws in the diagonal , what will be its length?

  Begin to understand this problem by asking yourself two questions:

  1. What is the problem asking for?

  The question asks for the length of diagonal .

  2. What do I need to know to find the answer?

  You need to know three things:

  The angles of a rectangle are all equal to .

  The lengths of the sides of the rectangle are and .

  The Pythagorean Theorem can be used to find the third side of a right triangle.


  Answering these questions is the first step to success with this problem.

  Drawing Representations

  Up to this point, the analysis of the sample problem has dealt with words alone. It is important to distill the basic information from the problem, but there are different ways to proceed from here. Often, visual representations can be very helpful in understanding problems. Make a simple drawing that represents what is being discussed. For example, a tray with six cookies could be represented by the diagram below.