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CK-12 Geometry Page 3
CK-12 Geometry Read online
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Identify the midpoint of line segments.
Identify the bisector of a line segment.
Understand and identify congruent angles.
Understand and apply the Angle Bisector Postulate.
Introduction
Now that you have a better understanding of segments, angles, rays, and other basic geometric shapes, we can study the ways in which they can be divided. Any time you come across a segment or an angle, there are different ways to separate it into parts.
Congruent Line Segments
One of the most important words in geometry is congruent. This term refers to geometric objects that have exactly the same size and shape. Two segments are congruent if they have the same length.
Notation Notes:
When two things are congruent we use the symbol . For example if is congruent to , then we would write .
When we draw congruent segments, we use tic marks to show that two segments are congruent.
If there are multiple pairs of congruent segments (which are not congruent to each other) in the same picture, use two tic marks for the second set of congruent segments, three for the third set, and so on. See the two following illustrations.
Recall that the length of segment can be written in two ways: or simply . This might be a little confusing at first, but it will make sense as you use this notation more and more. Let’s say we used a ruler and measured and we saw that it had a length of . Then we could write , or .
If we know that , then we can write or simply .
You can prove two segments are congruent in a number of ways. You can measure them to find their lengths using any units of measurement—the units do not matter as long as you use the same units for both measurements. Or, if the segments are drawn in the plane, you can also find their lengths on the coordinate grid. Later in the course you will learn other ways to prove two segments are congruent.
Example 1
Henrietta drew a line segment on a coordinate grid as shown below.
She wants to draw another segment congruent to the first that begins at and travels straight up (that is, in the direction). What will be the coordinates of its second endpoint?
You will have to solve this problem in stages. The first step is to identify the length of the segment drawn onto the grid. It begins at and ends at . So, its length is .
The next step is to draw the second segment. Use a pencil to create the segment according to the specifications in the problem. You know that the segment needs to be congruent to the first, so it will be long. The problem also states that it travels straight up from the point . Draw in the point at and make a line segment long that travels straight up.
Now that you have drawn in the new segment, use the grid to identify the new endpoint. It has an coordinate of and a coordinate of . So, its coordinates are .
Segment Midpoints
Now that you understand congruent segments, there are a number of new terms and types of figures you can explore. A segment midpoint is a point on a line segment that divides the segment into two congruent segments. So, each segment between the midpoint and an endpoint will have the same length. In the diagram below, point is the midpoint of segment since is congruent to .
There is even a special postulate dedicated to midpoints.
Segment Midpoint Postulate: Any line segment will have exactly one midpoint—no more, and no less.
Example 2
Nandi and Arshad measure and find that their houses are apart. If they agree to meet at the midpoint between their two houses, how far will each of them travel?
The easiest way to find the distance to the midpoint of the imagined segment connecting their houses is to divide the length by .
So, each person will travel five miles to meet at the midpoint between Nandi’s and Arshad’s houses.
Segment Bisectors
Now that you know how to find midpoints of line segments, you can explore segment bisectors. A bisector is a line, segment, or ray that passes through a midpoint of another segment. You probably know that the prefix “bi” means two (think about the two wheels of a bicycle). So, a bisector cuts a line segment into two congruent parts.
Example 3
Use a ruler to draw a bisector of the segment below.
The first step in identifying a bisector is finding the midpoint. Measure the line segment to find that it is long. To find the midpoint, divide this distance by .
So, the midpoint will be from either endpoint on the segment. Measure from an endpoint and draw the midpoint.
To complete the problem, draw a line segment that passes through the midpoint. It doesn’t matter what angle this segment travels on. As long as it passes through the midpoint, it is a bisector.
Congruent Angles
You already know that congruent line segments have exactly the same length. You can also apply the concept of congruence to other geometric figures. When angles are congruent, they have exactly the same measure. They may point in different directions, have different side lengths, have different names or other attributes, but their measures will be equal.
Notation Notes:
When writing that two angles are congruent, we use the congruent symbol: . Alternatively, the symbol refers to the measure of , so we could write and that has the same meaning as . You may notice then, that numbers (such as measurements) are equal while objects (such as angles and segments) are congruent.
When drawing congruent angles, you use an arc in the middle of the angle to show that two angles are congruent. If two different pairs of angles are congruent, use one set of arcs for one pair, then two for the next pair and so on.
Use algebra to find a way to solve the problem below using this information.
Example 4
The two angles shown below are congruent.
What is the measure of each angle?
This problem combines issues of both algebra and geometry, so make sure you set up the problem correctly. It is given that the two angles are congruent, so they must have the same measurements. Therefore, you can set up an equation in which the expressions representing the angle measures are equal to each other.
Now that you have an equation with one variable, you can solve for the value of .
So, the value of is . You are not done, however. Use this value of to find the measure of one of the angles in the problem.
Finally, we know , so both of the angles measure .
Angle Bisectors
If a segment bisector divides a segment into two congruent parts, you can probably guess what an angle bisector is. An angle bisector divides an angle into two congruent angles, each having a measure exactly half of the original angle.
Angle Bisector Postulate: Every angle has exactly one bisector.
Example 5
The angle below measures .
If a bisector is drawn in this angle, what will be the measure of the new angles formed?
This is similar to the problem about the midpoint between the two houses. To find the measurements of the smaller angles once a bisector is drawn, divide the original angle measure by :
So, each of the newly formed angles would measure when the angle is bisected.
Lesson Summary
In this lesson, we explored segments and angles. Specifically, we have learned:
How to understand and identify congruent line segments.
How to identify the midpoint of line segments.
How to identify the bisector of a line segment.
How to understand and identify congruent angles.
How to understand and apply the Angle Bisector Postulate.
These skills are useful whenever performing measurements or calculations in diagrams. Make sure that you fully understand all concepts presented here before continuing in your study.
Review Questions
Copy the figure below and label it with the following information:
Sketch and label an angle bisector of below.
If we know tha
t , what is
Use the following diagram of rectangle for questions 4-10. (For these problems you can assume that opposite sides of a rectangle are congruent—later you will prove this is true.)
Given that is the midpoint of and , find the following lengths:
How many copies of can fit inside rectangle
Review Answers
Angle Pairs
Learning Objectives
Understand and identify complementary angles.
Understand and identify supplementary angles.
Understand and utilize the Linear Pair Postulate.
Understand and identify vertical angles.
Introduction
In this lesson you will learn about special angle pairs and prove the vertical angles theorem, one of the most useful theorems in geometry.
Complementary Angles
A pair of angles are Complementary angles if the sum of their measures is .
Complementary angles do not have to be congruent to each other. Rather, their only defining quality is that the sum of their measures is equal to the measure of a right angle: . If the outer rays of two adjacent angles form a right angle, then the angles are complementary.
Example 1
The two angles below are complementary. . What is the value of ?
Since you know that the two angles must sum to , you can create an equation. Then solve for the variable. In this case, the variable is .
Thus, the value of is .
Example 2
The two angles below are complementary. What is the measure of each angle?
This problem is a bit more complicated than the first example. However, the concepts are the same. If you add the two angles together, the sum will be . So, you can set up an algebraic equation with the values presented.
The best way to solve this problem is to solve the equation above for . Then, you must substitute the value for back into the original expressions to find the value of each angle.
The value of is . Now substitute this value back into the expressions to find the measures of the two angles in the diagram.
and . You can check to make sure these numbers are accurate by verifying if they are complementary.
Since these two angle measures sum to , they are complementary.
Supplementary Angles
Two angles are supplementary if their measures sum to .
Just like complementary angles, supplementary angles need not be congruent, or even touching. Their defining quality is that when their measures are added together, the sum is . You can use this information just as you did with complementary angles to solve different types of problems.
Example 3
The two angles below are supplementary. If , what is ?
This process is very straightforward. Since you know that the two angles must sum to , you can create an equation. Use a variable for the unknown angle measure and then solve for the variable. In this case, let's substitute for .
So, the measure of and thus .
Example 4
What is the measure of two congruent, supplementary angles?
There is no diagram to help you visualize this scenario, so you’ll have to imagine the angles (or even better, draw it yourself by translating the words into a picture!). Two supplementary angles must sum to . Congruent angles must have the same measure. So, you need to find two congruent angles that are supplementary. You can divide by two to find the value of each angle.
Each congruent, supplementary angle will measure . In other words, they will be right angles.
Linear Pairs
Before we talk about a special pair of angles called linear pairs, we need to define adjacent angles. Two angles are adjacent if they share the same vertex and one side, but they do not overlap. In the diagram below, and are adjacent.
However, and are not adjacent since they overlap (i.e. they share common points in the interior of the angle).
Now we are ready to talk about linear pairs. A linear pair is two angles that are adjacent and whose non-common sides form a straight line. In the diagram below, and are a linear pair. Note that is a line.
Linear pairs are so important in geometry that they have their own postulate.
Linear Pair Postulate: If two angles are a linear pair, then they are supplementary.
Example 5
The two angles below form a linear pair. What is the value of each angle?
If you add the two angles, the sum will be . So, you can set up an algebraic equation with the values presented.
The best way to solve this problem is to solve the equation above for . Then, you must plug the value for back into the original expressions to find the value of each angle.
The value of is . Now substitute this value back into the expressions to determine the measures of the two angles in the diagram.
The two angles in the diagram measure and . You can check to make sure these numbers are accurate by verifying if they are supplementary.
Vertical Angles
Now that you understand supplementary and complementary angles, you can examine more complicated situations. Special angle relationships are formed when two lines intersect, and you can use your knowledge of linear pairs of angles to explore each angle further.
Vertical angles are defined as two non-adjacent angles formed by intersecting lines. In the diagram below, and are vertical angles. Also, and are vertical angles.
Suppose that you know . You can use that information to find the measurement of all the other angles. For example, and must be supplementary since they are a linear pair. So, to find , subtract from .
So measures . Knowing that angles and are also supplementary means that , since the sum of and is . If angle measures , then the measure of angle must be , since and are also supplementary. Notice that angles and are congruent and and are congruent .
The Vertical Angles Theorem states that if two angles are vertical angles then they are congruent.
We can prove the vertical angles theorem using a process just like the one we used above. There was nothing special about the given measure of . Here is proof that vertical angles will always be congruent: Since and form a linear pair, we know that they are supplementary: . For the same reason, and are supplementary: . Using a substitution, we can write . Finally, subtracting on both sides yields . Or, by the definition of congruent angles, .
Use your knowledge of vertical angles to solve the following problem.
Example 6
What is in the diagram below?
Using your knowledge of intersecting lines, you can identify that is vertical to the angle marked . Since vertical angles are congruent, they will have the same measure. So, is also equal to .
Lesson Summary
In this lesson, we explored angle pairs. Specifically, we have learned:
How to understand and identify complementary angles.
How to understand and identify supplementary angles.
How to understand and utilize the Linear Pair Postulate.
How to understand and identify vertical angles.
The relationships between different angles are used in almost every type of geometric application. Make sure that these concepts are retained as you progress in your studies.
Review Questions
Find the measure of the angle complementary to if
Find the measure of the angle supplementary to if
Find and .
Given , Find .
Use the diagram below for exercises 5 and 6. Note that .
Identify each of the following (there may be more than one correct answer for some of these questions). Name one pair of vertical angles.
Name one linear pair of angles.
Name two complementary angles.
Nam two supplementary angles.
Given that , find .
.
.
.
Review Answers
,
and (or and also works);
and (o
r and also works);
and ;
same as (b) and (or and also works).
Classifying Triangles
Learning Objectives
Define triangles.
Classify triangles as acute, right, obtuse, or equiangular.
Classify triangles as scalene, isosceles, or equilateral.
Introduction
By this point, you should be able to readily identify many different types of geometric objects. You have learned about lines, segments, rays, planes, as well as basic relationships between many of these figures. Everything you have learned up to this point is necessary to explore the classifications and properties of different types of shapes. The next two sections focus on two-dimensional shapes—shapes that lie in one plane. As you learn about polygons, use what you know about measurement and angle relationships in these sections.