CK-12 Geometry
Page 14
The missing exterior angle measures You can use this information to find the value of , because the interior and exterior angles form a linear pair and therefore they must sum to
Exterior Angles in a Triangle Theorem
In a triangle, the measure of an exterior angle is equal to the sum of the remote interior angles.
We won’t prove this theorem with a two-column proof (that will be an exercise), but we will use the example above to illustrate it. Look at the diagram from the previous example for a moment. If we look at the exterior angle at , then the interior angles at and are called “remote interior angles.”
Notice that the exterior angle at point measured At the same time, the interior angle at point measured and the interior angle at measured The sum of interior angles . Notice the measures of the remote interior angles sum to the measure of the exterior angle at . This relationship is always true, and it is a result of the linear pair postulate and the triangle sum theorem. Your job will be to show how this works.
Lesson Summary
In this lesson, we explored triangle sums. Specifically, we have learned:
How to identify interior and exterior angles in a triangle.
How to understand and apply the Triangle Sum Theorem
How to utilize the complementary relationship of acute angles in a right triangle.
How to identify the relationship of the exterior angles in a triangle.
These skills will help you understand triangles and their unique qualities. Always look for triangles in diagrams, maps, and other mathematical representations.
Points to Consider
Now that you understand the internal qualities of triangles, it is time to explore the basic concepts of triangle congruence.
Review Questions
Questions 1 and 2 use the following diagram:
Find in the triangle above.
What is in the triangle above?
Questions 3-6 use the following diagram:
What is
What is
What is
What is the relationship between and ? Write one or two sentences to explain how you know this is the relationship.
Find in the diagram below:
Use the diagram below for questions 8-13. (Note )
_____. Why?
_____. Why?
_____. Why?
_____. Why?
_____. Why?
_____. Why?
Prove the Remote Exterior Angle Theorem: The measure of an exterior angle in a triangle equals the sum of the measures of the remote interior angles. To get started, you may use the following: Given triangle as in the diagram below, prove .
Review Answers
and are complementary. Since the measures of the three angles of the triangle must add up to we can use the fact that is a right angle to conclude that
and add up to
. is an alternate interior angle with
. is an alternate interior angle with the labeled
. is a linear pair with
. Use the triangle sum theorem with and solve for
. Use the triangle sum theorem with
We will prove this using a two-column proof.
Statement Reason
1. 1. Given
2. 2. Triangle Sum Theorem
3. 3. Linear Pair Postulate
4. 4. Substitution
5. 5. Subtraction property of equality (subtracted on both sides)
Congruent Figures
Learning Objectives
Define congruence in triangles.
Create accurate congruence statements.
Understand that if two angles of a triangle are congruent to two angles of another triangle, the remaining angles will also be congruent.
Explore properties of triangle congruence.
Introduction
Triangles are important in geometry because every other polygon can be turned into triangles by cutting them up (formally we call this adding auxiliary lines). Think of a square: If you add an auxiliary line such as a diagonal, then it is two right triangles. If we understand triangles well, then we can take what we know about triangles and apply that knowledge to all other polygons. In this chapter you will learn about congruent triangles, and in subsequent chapters you will use what you know about triangles to prove things about all kinds of shapes and figures.
Defining Congruence in Triangles
Two figures are congruent if they have exactly the same size and shape. Another way of saying this is that the two figures can be perfectly aligned when one is placed on top of the other—but you may need to rotate or flip the figures over to make them line up. When that alignment is done, the angles that are matched are called corresponding angles, and the sides that are matched are called corresponding sides.
In the diagram above, sides and have the same length, as shown by the tic marks. If two sides have the same number of tic marks, it means that they have the same length. Since and each have one tic mark, they have the same length. Once we have established that , we need to examine the other sides of the triangles. and each have two tic marks, showing that they are also congruent. Finally, as you can see, because they each have three tic marks. Each of these pairs corresponds because they are congruent to each other. Notice that the three sides of each triangle do not need to be congruent to each other, as long as they are congruent to their corresponding side on the other triangle.
When two triangles are congruent, the three pairs of corresponding angles are also congruent. Notice the tic marks in the triangles below.
We use arcs inside the angle to show congruence in angles just as tic marks show congruence in sides. From the markings in the angles we can see and .
By definition, if two triangles are congruent, then you know that all pairs of corresponding sides are congruent and all pairs of corresponding angles are congruent. This is sometimes called CPCTC: Corresponding parts of congruent triangles are congruent.
Example 1
Are the two triangles below congruent?
The question asks whether the two triangles in the diagram are congruent. To identify whether or not the triangles are congruent, each pair of corresponding sides and angles must be congruent.
Begin by examining the sides. and both have one tic mark, so they are congruent. and both have two tic marks, so they are congruent as well. and have three tic marks each, so each pair of sides is congruent.
Next you must check each angle. and both have one arc, so they are congruent. because they each have two arcs. Finally, because they have three arcs.
We can check that each angle in the first triangle matches with its corresponding angle in the second triangle by examining the sides. corresponds with because they are formed by the sides with two and three tic marks. Since all pairs of corresponding sides and angles are congruent in these two triangles, we conclude that the two triangles are congruent.
Creating Congruence Statements
We have already been using the congruence sign when talking about congruent sides and congruent angles.
For example, if you wanted to say that was congruent to , you could write the following statement.
In Chapter 1 you learned that the line above with no arrows means that is a segment (and not a line or a ray). If you were to read this statement out loud, you could say “Segment is congruent to segment .”
When dealing with congruence statements involving angles or triangles, you can use other symbols. Whereas the symbol means “segment ,” the symbol means “angle .” Similarly, the symbol means “triangle .”
When you are creating a congruence statement of two triangles, the order of the letters is very important. Corresponding parts must be written in order. That is, the angle at first letter of the first triangle corresponds with the angle at the first letter of the second triangle, the angles at the second letter correspond, and so on.
In the diagram above, if you were to name each triangle individually, they could be and . Those n
ames seem the most appropriate because the letters are in alphabetical order. However, if you are writing a congruence statement, you could NOT say that . If you look at , it does not correspond to . corresponds to instead (indicated by the two arcs in the angles). corresponds to , and corresponds to . Remember, you must compose the congruent statement so that the vertices are lined up for congruence. The statement below is correct.
This form may look strange at first, but this is how you must create congruence statements in any situation. Using this standard form allows your work to be easily understood by others, a crucial element of mathematics.
Example 2
Compose a congruence statement for the two triangles below.
To write an accurate congruence statement, you must be able to identify the corresponding pairs in the triangles above. Notice that and each have one arc mark. Similarly, and each have two arcs, and and have three arcs. Additionally, (or ), and
So, the two triangles are congruent, and to make the most accurate statement, this should be expressed by matching corresponding vertices. You can spell the first triangle in alphabetical order and then align the second triangle to that standard.
Notice in example 2 that you don’t need to write the angles in alphabetical order, as long as corresponding parts match up. If you’re feeling adventurous, you could also express this statement as shown below.
Both of these congruence statements are accurate because corresponding sides and angles are aligned within the statement.
The Third Angle Theorem
Previously, you studied the triangle sum theorem, which states that the sum of the measures of the interior angles in a triangle will always be equal to This information is useful when showing congruence. As you practiced, if you know the measures of two angles within a triangle, there is only one possible measurement of the third angle. Thus, if you can prove two corresponding angle pairs congruent, the third pair is also guaranteed to be congruent.
Third Angle Theorem
If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles are also congruent.
This may seem like an odd statement, but use the exercise below to understand it more fully.
Example 3
Identify whether or not the missing angles in the triangles below are congruent.
To identify whether or not the third angles are congruent, you must first find their measures. Start with the triangle on the left. Since you know two of the angles in the triangle, you can use the triangle sum theorem to find the missing angle. In we know
The missing angle of the triangle on the left measures Repeat this process for the triangle on the right.
So, . Remember that you could also identify this without using the triangle sum theorem. If two pairs of angles in two triangles are congruent, then the remaining pair of angles also must be congruent.
Congruence Properties
In earlier mathematics courses, you have learned concepts like the reflexive or commutative properties. These concepts help you solve many types of mathematics problems. There are a few properties relating to congruence that will help you solve geometry problems as well.
The reflexive property of congruence states that any shape is congruent to itself. This may seem obvious, but in a geometric proof, you need to identify every possibility to help you solve a problem. If two triangles share a line segment, you can prove congruence by the reflexive property.
In the diagram above, you can say that the shared side of the triangles is congruent because of the reflexive property. Or in other words, .
The symmetric property of congruence states that congruence works frontwards and backwards, or in symbols, if then .
The transitive property of congruence states that if two shapes are congruent to a third, they are also congruent to each other. In other words, if , and , then . This property is very important in identifying congruence between different shapes.
Example 4
Which property can be used to prove the statement below?
If and , then .
A. reflexive property of congruence
B. identity property of congruence
C. transitive property of congruence
D. symmetric property of congruence
The transitive property is the one that allows you to transfer congruence to different shapes. As this states that two triangles are congruent to a third, they must be congruent to each other by the transitive property. The correct answer is C.
Lesson Summary
In this lesson, we explored congruent figures. Specifically, we have learned:
How to define congruence in triangles.
How to create accurate congruence statements.
To understand that if two angles of a triangle are congruent to two angles of another triangle, the remaining angles will also be congruent.
How to employ properties of triangle congruence.
These skills will help you understand issues of congruence involving triangles. Always look for triangles in diagrams, maps, and other mathematical representations.
Points to Consider
Now that you understand the issues inherent in triangle congruence, you will create your first congruence proof.
Review Questions
Use the diagram below for problem 1.
Write a congruence statement for the two triangles above.
Exercises 2-3 use the following diagram.
Suppose the two triangles above are congruent. Write a congruence statement for these two triangles.
Explain how we know that if the two triangles are congruent, then .
Use the diagram below for exercises 4-5.
Explain how we know .
Are these two triangles congruent? Explain why (note, “looks” are not enough of a reason!).
If you want to know the measure of all three angles in a triangle, how many angles do you need to measure with your protractor? Why?
Use the following diagram for exercises 7-10.
What is the relationship between and ? How do you know?
What is ? How do you know?
What property tells us ?
Write a congruence statement for these triangles.
Review Answers
(Note the order of the letters is important!)
If the two triangles are congruent, then corresponds with and therefore they are congruent to each other by the definition of congruence.
The third angle theorem states that if two pairs of angles are congruent in two triangles, then the third pair of angles must also be congruent
No. corresponds with but they are not the same length
You only need to measure two angles. The triangle sum theorem will help you find the measure of the third angle
and are supplementary since they are a linear pair
The reflexive property of congruence
Triangle Congruence using SSS
Learning Objectives
Use the distance formula to analyze triangles on a coordinate grid.
Understand and apply the SSS postulate of triangle congruence.
Introduction
In the last section you learned that if two triangles are congruent then the three pairs of corresponding sides are congruent and the three pairs of corresponding angles are congruent. In symbols, means , and .
Wow, that’s a lot of information—in fact, one triangle congruence statement contains six different congruence statements! In this section we show that proving two triangles are congruent does not necessarily require showing all six congruence statements are true. Lucky for us, there are shortcuts for showing two triangles are congruent—this section and the next explore some of these shortcuts.
Triangles on a Coordinate Grid
To begin looking at rules of triangle congruence, we can use a coordinate grid. The following grid shows two triangles.
The first step in finding out if these triangles are congruent is to identify the
lengths of the sides. In algebra, you learned the distance formula, shown below.
You can use this formula to find the distances on the grid.
Example 1
Find the distances of all the line segments on the coordinate grid above using the distance formula.
Begin with . First write the coordinates.
is
is
is
Now use the coordinates to find the lengths of each segment in the triangle.
So, the lengths are as follows.
, , and
Next, find the lengths in triangle . First write the coordinates.
is
is
is
Now use the coordinates to find the lengths of each segment in the triangle.
So, the lengths are as follows:
, , and
Using the distance formula, we showed that the corresponding sides of the two triangles have the same length. We don’t have the tools to find the measures of the angles in these triangles, so we can show congruence in a different way.