Using Congruent Triangles

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2 To access a customizable version of this book, as well as other interactive content, visit AUTHORS CK12 Editor CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform. Copyright 2012 CK-12 Foundation, The names CK-12 and CK12 and associated logos and the terms FlexBook and FlexBook Platform (collectively CK-12 Marks ) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/Non- Commercial/Share Alike 3.0 Unported (CC BY-NC-SA) License ( as amended and updated by Creative Commons from time to time (the CC License ), which is incorporated herein by this reference. Complete terms can be found at Printed: July 10, 2012

3 1 CONCEPT 1 Using Congruent Triangles Learning Objectives Apply various triangles congruence postulates and theorems. Know the ways in which you can prove parts of a triangle congruent. Find distances using congruent triangles. Use construction techniques to create congruent triangles. Introduction As you can see, there are many different ways to prove that two triangles are congruent. It is important to know all of the different way that can prove congruence, and it is important to know which combinations of sides and angles do not prove congruence. When you prove properties of polygons in later chapters you will frequently use Congruence Theorem Review As you have studied in the previous lessons, there are five theorems and postulates that provide different ways in which you can prove two triangles congruent without checking all of the angles and all of the sides. It is important to know these five rules well so that you can use them in practical applications. TABLE 1.1: Name Corresponding congruent parts Does it prove congruence? SSS Three sides Yes SAS Two sides and the angle between Yes them ASA Two angles and the side between Yes them AAS Two angles and a side not between Yes them HL A hypotenuse and a leg in a right Yes triangle AAA Three angles No it will create a similar triangle, but not of the same size SSA Two sides and an angle not between them No this can create more than one distinct triangle When in doubt, think about the models we created. If you can construct only one possible triangle given the constraints, then you can prove congruence. If you can create more than one triangle within the given information, you cannot prove congruence. Example 1 Concept 1. Using Congruent Triangles

4 2 What rule can prove that the triangles below are congruent? A. SSS B. SSA C. ASA D. AAS The two triangles in the picture have two pairs of congruent angles and one pair of corresponding congruent sides. So, the triangle congruence postulate you choose must have two A s (for the angles) and one S (for the side). You can eliminate choices A and B for this reason. Now that you are deciding between choices C and D, you need to identify where the side is located in relation to the given angles. It is adjacent to one angle, but it is not in between them. Therefore, you can prove congruence using AAS. The correct answer is D. Proving Parts Congruent It is one thing to identify congruence when all of the important identifying information is provided, but sometimes you will have to identify congruent parts on your own. You have already practiced this in a few different ways. When you were testing SSS congruence, you used the distance formula to find the lengths of sides on a coordinate grid. As a review, the distance formula is shown below. distance = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 You can use the distance formula whenever you are examining shapes on a coordinate grid. When you were creating two-column and flow proofs, you also used the reflexive property of congruence. This property states that any segment or angle is congruent to itself. While this may sound obvious, it can be very helpful in proofs, as you saw in those examples. You may be tempted to use your ruler and protractor to check whether two triangles are congruent. However, this method does not necessarily work because not all pictures are drawn to scale. Example 2 How could you prove ABC = DECin the diagram below?

5 3 We can already see that BC = CE and AC = CD. We may be able to use SSS or SAS to show the triangles are congruent. However, to use SSS, we would need AB = DE and there is no obvious way to prove this. Can we show that two of the angles are congruent? Notice that BCA and ECD are vertical angles (nonadjacent angles made by the intersection of two lines i.e., angles on the opposite sides of the intersection). The Vertical Angle Theorem states that all vertical angles are also congruent. So, this tells us that BCA = ECD. Finally, by putting all the information together, you can confirm that ABC = DEC by the SAS Postulate. Finding Distances One way to use congruent triangles is to help you find distances in real life usually using a map or a diagram as a model. When using congruent triangles to identify distances, be sure you always match up corresponding sides. The most common error on this type of problem involves matching two sides that are not corresponding. Example 3 The map below shows five different towns. The town of Meridian was given its name because it lies exactly halfway between two pairs of cities: Camden and Grenata, and Lowell and Morsetown. Concept 1. Using Congruent Triangles

6 4 Using the information in the map, what is the distance between Camden and Lowell? The first step in this problem is to identify whether or not the marked triangles are congruent. Since you know that the distance from Camden to Meridian is the same as Meridian to Grenata, those two sides are congruent. Similarly, since the distance from Lowell to Meridian is the same as Meridian to Morsetown, those two sides are also a congruent pair. The angles between these lines are also congruent because they are vertical angles. So, by the SAS postulate, these two triangles are congruent. This allows us to find the distance between Camden and Lowell by identifying its corresponding side on the other triangle. Because they are both opposite the vertical angle, the side connecting Camden and Lowell corresponds to the side connecting Morsetown and Grenata. Since

7 5 the triangles are congruent, these corresponding sides will also be congruent to each other. Therefore, the distance between Camden and Lowell is five miles. This use of the definition of congruent triangles is one of the most powerful tools you will use in geometry class. It is often abbreviated as CPCTC, meaning Corresponding Parts of Congruent Triangles are Congruent. Constructions Another important part of geometry is creating geometric figures through construction. A construction is a drawing that is made using only a straightedge and a compass you can think of construction as a special game in geometry in which we make figures using only these tools. You may be surprised how many shapes can be made this way. Example 4 Use a compass and straightedge to construct the perpendicular bisector of the segment below. Begin by using your compass to create an arc with the same distance from a point as the segment. Repeat this process on the opposite side. Concept 1. Using Congruent Triangles

8 6 Now draw a line through the two points of intersections. This forms the perpendicular bisector. Draw segments connecting the points on the bisector to the original endpoints. Knowing that the center point is the midpoint of both line segments and that all angles formed around point M are right angles, you can prove that all four triangles created are congruent by the SAS rule. Lesson Summary In this lesson, we explored applications triangle congruence. Specifically, we have learned to: Identify various triangles congruence postulates and theorems.

9 7 Use the fact that corresponding parts of congruent triangles are congruent. Find distances using congruent triangles. Use construction techniques to create congruent triangles. 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 You now know all the different ways in which you can prove two triangles congruent. In the next chapter you ll learn more about isosceles and equilateral triangles. Review Questions Use the following diagram for exercises Find AB in the diagram above. 2. Find BC in the diagram above. 3. What is m ABC? How do you know? 4. What postulate can you use to show ABC = JKL? 5. Use the distance formula to find AC. How can use this to find JL? 6-8: For each pair of triangles, complete the triangle congruence statement, or write no congruence statement possible. Name the triangle congruence postulate you use, or write a sentence to explain why you can t write a triangle congruence statement. 6. PAL =. Concept 1. Using Congruent Triangles

10 BIN =. 8. BOW =. 9. In the following diagram, Midtown is exactly halfway between Uptown and Downtown. What is the distance between Downtown and Lower East Side? How do you know? Write a few sentences to convince a reader your answer is correct. 10. Given: R is the midpoint of PN and PA LN Prove: PA = LN Review Answers 1. AB = 4

11 BC = Since AB is horizontal (parallel to the x axis) and BC is vertical (parallel to the y axis), we can conclude that they intersect at a right angle. 4. SAS (other answers are possible) 5. AC = ( 1 ( 5)) 2 + ( 2 3) 2 = = 41. Since the triangles are congruent, we can conclude JL = PAL = BUD. SAS triangle congruence postulate 7. BIN = RAT. SSS triangle congruence postulate 8. No congruence statement is possible; we don t have enough information km. Since Midtown is the midpoint of the line connecting Uptown and Downtown, we can use the vertical angle theorem for the angles made by the two lines that meet at Midtown, and then we can conclude that the triangles are congruent using AAS. If the triangles are congruent then all corresponding parts are also congruent TABLE 1.2: Statement Reason 1. R is the midpoint of PN 1. Given 2. PR = RN 2. Definition of midpoint 3. PA LN 3. Given 4. RNL = RPA 4. Alternate interior angles theorem 5. PAR = NLR 5. Alternate interior angles theorem 6. PAR = NLR 6. AAS triangle congruence postulate 7. PA = LN 7. Definition of congruent triangles (corresponding parts are congruent) Concept 1. Using Congruent Triangles

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Triangle Sum Theorem. Bill Zahner Lori Jordan. Say Thanks to the Authors Click (No sign in required)

Triangle Sum Theorem. Bill Zahner Lori Jordan. Say Thanks to the Authors Click (No sign in required) Triangle Sum Theorem Bill Zahner Lori Jordan Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive