5.4 SSS Triangle Congruence

Transcription

1 OMMON OR Locker LSSON ommon ore Math Standards The student is expected to: OMMON OR G-O..8 xplain how the criteria for triangle congruence (... SSS) follow from the definition of congruence in terms of rigid motions. lso G-O..7, G-O..10, G-SRT..5 Mathematical Practices OMMON OR 5.4 SSS Triangle ongruence MP.7 Using Structure Language Objective Have small groups of students complete a triangle congruence chart. Name lass ate 5.4 SSS Triangle ongruence ssential Question: What does the SSS Triangle ongruence Theorem tell you about triangles? xplore onstructing Triangles Given Three Side Lengths Two triangles are congruent if and only if a rigid motion transformation maps one triangle onto the other triangle. Many theorems can also be used to identify congruent triangles. ollow these steps to construct a triangle with sides of length 5 in.,, and 3 in. Use a ruler, compass, and either tracing paper or a transparency. Use a ruler to draw a line segment of length 5 inches. Label the endpoints and. Open a compass to 4 inches. Place the point of the compass on, and draw an arc as shown. Now open the compass to 3 inches. Place the point of the compass on, and draw a second arc. Resource Locker NGG ssential Question: What does the SSS Triangle ongruence Theorem tell you about triangles? If three sides of one triangle are congruent to three sides of another triangle, you can conclude that the triangles are congruent. PRVIW: LSSON PRORMN TSK View the ngage section online. Point out that the structural beams mentioned in the Preview refer to the steel sides of the triangles. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing ompany 5 in. Next, find the intersection of the two arcs. Label the intersection. raw and. Label the side lengths on the figure. 3 in. 5 in. 3 in. 5 in. Repeat steps through to draw on a separate piece of tracing paper. The triangle should have sides with the same lengths as. Start with a segment that is long. Label the endpoints and as shown. 3 in. 5 in. Module Lesson 4 Name lass ate 5.4 SSS Triangle ongruence ssential Question: What does the SSS Triangle ongruence Theorem tell you about triangles? xplore onstructing Triangles Given Houghton Mifflin Harcourt Publishing ompany Three Side Lengths Two triangles are congruent if and only if a rigid motion transformation maps one triangle onto the other triangle. Many theorems can also be used to identify congruent triangles. ollow these steps to construct a triangle with sides of length 5 in.,, and 3 in. Use a ruler, compass, and either tracing paper or a transparency. G-O..8 xplain how the criteria for triangle congruence (... SSS) follow from the definition of congruence in terms of rigid motions. lso G-O..7, G-O..10, G-SRT..5 Use a ruler to draw a line segment of length 5 inches. Label the endpoints and. Open a compass to 4 inches. Place the point of the compass on, and draw an arc as shown. 5 in. Next, find the intersection of the two arcs. Label the intersection. raw and. Label the side lengths on the figure. Resource Now open the compass to 3 inches. Place the point of the compass on, and draw a second arc. 3 in. 5 in. Repeat steps through to draw on a separate piece of tracing paper. The triangle should have sides with the same lengths as. Start with a segment that is long. Label the endpoints and as shown. HROVR PGS Turn to these pages to find this lesson in the hardcover student edition. 3 in. 3 in. 5 in. 5 in. Module Lesson Lesson 5.4

2 ompare and. re they congruent? How do you know? Yes. Triangle can be mapped to by a translation that maps point to point, and then a rotation about point. Reflect 1. iscussion When you construct, how do you know that the intersection of the two arcs is a distance of 4 inches from and 3 inches from? The arcs are sections of circles sets of points, respectively, 3 inches and 4 inches from. The other circle is the set of points that is 3 inches away from. The intersection has the properties of both circles. 2. ompare your triangles to those made by other students. re they all congruent? xplain. Yes, because there is a sequence of rigid motions that maps any one of the triangles onto any of the others. XPLOR onstructing Triangles Given Three Side Lengths INTGRT THNOLOGY Students may use geometry software to explore the concept of constructing a triangle with given side lengths. xplain 1 Justifying SSS Triangle ongruence You can use rigid motions and the converse of the Perpendicular isector Theorem to justify this theorem. SSS Triangle ongruence Theorem If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. xample 1 In the triangles shown, let,, and. Use rigid motions to show that. Houghton Mifflin Harcourt Publishing ompany INTGRT MTHMTIL PRTIS ocus on Modeling MP.4 Give each student a piece of dry spaghetti. They should measure and break the spaghetti so that they have three pieces that are the same length as the sides of the triangle in the exercise. Instruct them to make a triangle with these pieces on top of the triangle they drew. Have them compare the triangles and explain why they are congruent. Then ask them to consider whether it is possible to create a triangle from the spaghetti pieces that is not congruent to the triangle they drew in the exercise. XPLIN 1 Justifying SSS Triangle ongruence Module Lesson 4 PROSSIONL VLOPMNT Math ackground The Side-Side-Side Triangle ongruence Theorem is often presented as the first of the Triangle ongruence Theorems because it is easy to demonstrate concretely. In this course, it is the third theorem presented because the justification requires students to apply the Perpendicular isector Theorem. Reinforce this justification throughout the lesson. INTGRT MTHMTIL PRTIS ocus on ommunication MP.3 Some justifications in this exercise may be difficult for some students to understand. Have students model each step with the triangles they have drawn on tracing paper. ncourage students to restate the steps in their own words. Students should fully understand each step before moving on to the next one. QUSTIONING STRTGIS raw two segments that share an endpoint. How many different segments could be drawn to turn this figure into a triangle? There is only one segment. SSS Triangle ongruence 256

3 Transform by a translation along followed by a rotation about point, so that and coincide. The segments coincide because they are the same length. oes a reflection across map point to Therefore, point lies on the perpendicular point? To show this, notice that =, bisector of by the converse of the which means that point is equidistant from perpendicular bisector theorem. ecause =, point and point. point also lies on the perpendicular bisector of. Since point and point both lie on the perpendicular bisector of and there is a unique line through any two points, is the perpendicular bisector of. y the definition of reflection, the image of point must be point. Therefore, is mapped onto by a translation, followed by a rotation, followed by a reflection, and the two triangles are congruent. Show that PQR. Q R Houghton Mifflin Harcourt Publishing ompany P Triangle is transformed by a sequence of rigid motions to form the figure shown below. Identify the sequence of rigid motions. (You will complete the proof on the following page.) R Q 1. Translation along P 2. Rotation about P so that PQ and coincide. P 3. Reflection across PQ. Module Lesson Lesson 5.4 OLLORTIV LRNING Peer-to-Peer ctivity Give each student five index cards. Students should prepare a card that shows two labeled triangles for each of the following situations: The triangles can be proved congruent by SSS. The triangles can be proved congruent by SS. The triangles can be proved congruent by S. The triangles can be proved noncongruent. There is not enough information to determine congruence. ollect and shuffle them. ivide students into pairs and give each pair ten cards. Have them sort their cards into the above categories.

4 omplete the explanation by filling in the blanks with the name of a point, line segment, or geometric theorem. ecause QR Q, point Q is equidistant from R (or ) and (or R). Therefore, by the converse of the Perpendicular isector Theorem, point Q lies on the of R. Similarly, perpendicular bisector PR P. So point P the perpendicular bisector of R. ecause two points determine a line, the line PQ is lies on the perpendicular bisector of R. y the definition of reflection, the image of point must be point R. Therefore, PQR because is mapped to PQR by a translation, a rotation, and a reflection. Reflect 3. an you conclude that two triangles are congruent if two pairs of corresponding sides are congruent? xplain your reasoning and include an example. No; you need a third piece of information to ensure a rigid motion maps one triangle to the other, such as congruent included angles or another pair of congruent sides. Your Turn 4. Use rigid motions and the converse of the perpendicular bisector theorem to explain why. and, so is equidistant from and, and is equidistant from and. y the converse of the Perpendicular isector Theorem, is the perpendicular bisector of. y the definition of a reflection, point is the image of point reflected across. The reflection also maps onto, so. Houghton Mifflin Harcourt Publishing ompany Module Lesson 4 IRNTIT INSTRUTION Manipulatives Have students create a triangle and a quadrilateral using strips of construction paper or tagboard and brass fasteners. Then have students attempt to change the angles in each without bending the strips of paper. Students should notice that the triangle always remains the same but that they can create many different quadrilaterals. iscuss how this activity illustrates the Side-Side-Side Triangle ongruence Theorem. SSS Triangle ongruence 258

5 XPLIN 2 Proving Triangles re ongruent Using SSS Triangle ongruence xplain 2 Proving Triangles re ongruent Using SSS Triangle ongruence You can apply the SSS Triangle ongruence Theorem to confirm that triangles are congruent. Remember, if any one pair of corresponding parts of two triangles is not congruent, then the triangles are not congruent. xample 2 Prove that the triangles are congruent or explain why they are not congruent. INTGRT MTHMTIL PRTIS ocus on Math onnections MP.1 lthough the example emphasizes the SSS Triangle ongruence Theorem, it is important for students to keep in mind that the triangles are congruent because one can be mapped onto the other by one or more rigid motions. Maintain this connection by asking which segments of one triangle map onto certain segments of another triangle. QUSTIONING STRTGIS When do you use the SSS Triangle ongruence Theorem instead of the S or SS Triangle ongruence Theorems to determine whether two triangles are congruent? When you know three pairs of corresponding congruent sides and no pairs of corresponding congruent angles, you cannot use a theorem that involves an angle. LNGUG SUPPORT Have students work in small groups. Have them complete a chart like the following, highlighting the sides and angles that are congruent in each pair of triangles. Houghton Mifflin Harcourt Publishing ompany = = 1.7 m, so. = = 2.4 m, so. = = 2.3 m, so. The three sides of are congruent to the three sides of. by the SSS Triangle ongruence Theorem. 1.7 m 2.3 m 2.4 m 1.7 m 2.3 m 2.4 m = G = 20 cm, so G. G 24 cm H = H = 12 cm, so H H. 12 cm H 20 cm H = GH = 24 cm, so H GH 20 cm. 12 cm 24 cm The three sides of H are congruent to the three sides of GH, so the two triangles are congruent by the SSS Triangle ongruence Theorem. Your Turn Prove that the triangles are congruent or explain why they are not congruent. 5. P 32 in. R Q 6. M 5 28 in. N 38 in. 38 in. S 5 The corresponding sides MN and QR are It is given that GK GL and JK JL, not congruent. Therefore, the triangles are not congruent. K J and GJ GJ by the Reflexive Property. G L Triangle ongruence Theorems Theorem efinition Picture Module Lesson 4 LNGUG SUPPORT onnect ontext The phrase Given Three Side Lengths in the title of the xplore section may confuse some nglish Learners. xplain that it indicates that you will be constructing triangles when you are given the lengths of the sides. Remind students that in math, they are often given partial information to help solve a problem, and that the information is called a given. 259 Lesson 5.4

6 xplain 3 pplying Triangle ongruence You can use the SSS Triangle ongruence Theorem and other triangle congruence theorems to solve many real-world problems that involve congruent triangles. xample 3 ind the value of x for which you can show the triangles are congruent. Lexi bought matching triangular pendants for herself and her mom in the shapes shown. or what value of x can you use a triangle congruence theorem to show that the pendants are congruent? Which triangle congruence theorem can you use? xplain. (4x - 6) cm 3 cm 3 cm 3 cm 3 cm J XPLIN 3 pplying Triangle ongruence VOI OMMON RRORS ecause students are concentrating on sides when using the SSS Triangle ongruence Theorem, they may not write corresponding angles in the correct order in the congruence statement. iscuss methods they can use to make sure they get the order right. K L (3x - 4) cm JK and JL, because they have the same measure. So, if KL, then JKL by the SSS Triangle ongruence Theorem. Write an equation setting the lengths equal and solve for x. 4x - 6 = 3x - 4; x = 2 deline made a design using triangular tiles as shown. or what value of x can you use a triangle congruence theorem to show that the tiles are congruent? Which triangle congruence theorem can you use? xplain. Notice that PQ MN and PR MO, because they have the same measure. If NO QR, then MNO PQR by the SSS Triangle ongruence Theorem. Write an equation setting the lengths equal and solve for x. 3x - 11 = 4, 3x = 15, x = 5 v Your Turn 7. raig made a mobile using geometric shapes including triangles shaped 60º V as shown. or what value of x and y 30º can you use a triangle congruence G theorem to show that the triangles are T (7y + 4)º 12 cm (8x - 12) cm congruent? Which triangle congruence theorem can you use? xplain. H U m H = 30 by the Triangle Sum Theorem, so H V. G U, they are right angles. If GH UV, then GH TUV by the S. x = 3; y = 8 M N O (3x - 11) in. R P Q Houghton Mifflin Harcourt Publishing ompany Image redits: MaximImages/lamy QUSTIONING STRTGIS How do you know whether angles in one triangle are congruent to the angles in the other triangle? You know the triangles are congruent by SSS, so you know that corresponding angles are also congruent because of PT. ONNT VOULRY Have students collaboratively write a definition for a triangle, providing supports. or example: triangle is a, It has sides. It has 3. right triangle has a. Module Lesson 4 SSS Triangle ongruence 260

7 LORT QUSTIONING STRTGIS Two triangles appear to be congruent. You know that three pairs of corresponding parts are congruent, but you have no information about the other corresponding parts. How can you determine whether the triangles really are congruent? Sample answer: If the congruent parts are two pairs of corresponding angles and the included side, or two pairs of corresponding sides and the included angle, you can use the S or SS Triangle ongruence Theorem. If the congruent parts are three pairs of corresponding sides, you can use the SSS Triangle ongruence Theorem. Otherwise, check whether there is a sequence of rigid motions that map one triangle onto the other. laborate 8. n isosceles triangle has two sides of equal length. If we ask everyone in class to construct an isosceles triangle that has one side of length 8 cm and another side of length 12 cm, how many sets of congruent triangles might the class make? The class could make at most 2 sets of congruent triangles. One set would have sides of length 8 cm, 8 cm, and 12 cm. The second set would have sides of length 12 cm, 12 cm, and 8 cm. 9. ssential Question heck-in How do you explain the SSS Triangle ongruence Theorem? Possible answer: Two triangles are congruent if a series of rigid motion transformations maps one triangle onto the other. or two triangles that have pairs of congruent sides, begin by translating and then rotating one triangle so that it shares one side with the other triangle. Then apply the converse of the Perpendicular isector Theorem to show that a reflection completes the mapping. valuate: Homework and Practice Use a compass and a straightedge to complete the drawing of so that it is congruent to. 1. Online Homework Hints and Help xtra Practice SUMMRIZ TH LSSON When do you use the SSS Triangle ongruence Theorem? What information do you need in order to use this theorem? You use the SSS Triangle ongruence Theorem to prove that two triangles are congruent by using three pairs of congruent sides. You need to know that all three pairs of corresponding sides are congruent to use it. Houghton Mifflin Harcourt Publishing ompany On a separate piece of paper, use a compass and a ruler to construct two congruent triangles with the given side lengths. Label the lengths of the sides in., 3.5 in., 3. 3 cm, 11 cm, 12 cm 3 in. 3.5 in. 3 in. 3.5 in. 11 cm 11 cm 3 cm 12 cm 12 cm 3 cm Module Lesson Lesson 5.4

8 Identify a sequence of rigid motions that maps one side of onto one side of VLUT Possible answer: translation along, and then a clockwise rotation about so that coincides with Possible answer: translation along, and then a clockwise rotation about so that coincides with. In each figure, identify the perpendicular bisector and the line segment it bisects, and explain how to use the information to show that the two triangles are congruent is the perpendicular bisector of. This shows that maps to by a reflection across. Possible answer: translation along, and then a counterclockwise rotation about so that coincides with. Possible answer: translation along, and then a counterclockwise rotation about so that coincides with. G is the perpendicular bisector of G. This shows that maps to G by a reflection across. Houghton Mifflin Harcourt Publishing ompany SSIGNMNT GUI oncepts and Skills xplore onstructing Triangles Given Side Lengths xample 1 Justifying SSS Triangle ongruence xample 2 Proving Triangles re ongruent Using SSS Triangle ongruence xample 3 pplying Triangle ongruence VOI OMMON RRORS Practice xercises 1 3 xercises 4 9 xercises xercises Some students may have trouble applying SSS to adjacent triangles. djacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent sides. Module Lesson 4 xercise epth of Knowledge (.O.K.) OMMON OR Mathematical Practices Skills/oncepts MP.4 Modeling Skills/oncepts MP.1 Problem Solving Skills/oncepts MP.2 Reasoning Strategic Thinking MP.1 Problem Solving Skills/oncepts MP.3 Logic SSS Triangle ongruence 262

9 OMMUNIT MTH Introduce the word criterion and its plural, criteria. sk students if they can define criterion and give an example of its use. or instance, you might discuss a college s criteria for accepting students (a completed application, a minimum grade-point average, a minimum ST score, and so on). Tell students that they have developed three criteria for determining whether two triangles are congruent, the S, SS, and SSS Triangle ongruence Theorems. iscuss how they can determine whether two given triangles meet any of these criteria. Prove that the triangles are congruent or explain why this is not possible. 10. S 9 m T 11. R 6 m 9 m Q 6 m ongruent, by SSS congruence; RS TQ because they have the same measure, RT RT by the reflexive property, and ST QR because they have the same measure cm K cm J W X 12 cm 8 cm Z J 50º X 50º Y M 50º N 51º Not congruent, because only one triangle has an interior angle of 51 so there is no rigid motion that will map one to the other. P 7 cm K W 8 cm X 9 cm 10 cm L Y Possibly congruent, depending on the lengths of the unlabeled sides. L Y Not congruent. The diagrams show a total of 4 different side lengths, but two congruent triangles have only 3 different side lengths. 14. arol bought two chairs with triangular backs. or what value of x can you use a triangle congruence theorem to show that the triangles are congruent? Which triangle congruence theorem can you use? xplain. Houghton Mifflin Harcourt Publishing ompany G T 25 in. 25 in. 25 in. 25 in. H (9x - 21) in. I U 15 in. V X = 4; GH TU and GI TV, because they have the same measure. So, if HI UV, then GHI TUV by the SSS Triangle ongruence Theorem. Module Lesson Lesson 5.4

10 15. or what values of x and y can you use a triangle congruence theorem to show that the triangles are congruent? Which triangle congruence theorem can you use? xplain. x = 4, y = 4;, G, and G, S ind all possible solutions for x such that is congruent to. One or more of the problems may have no solution. 30º (14x - 47) cm 9 cm 60º (15y)º 30º G INTGRT THNOLOGY Students may find it useful to use geometry software to model exercises they are struggling with. 16. : sides of length 6, 8, and x. : sides of length 6, 9, and x - 1. x = : sides of length 3, x + 1, and 14. : sides of length 13, x - 9, and 2x - 6 no solution 18. : sides of length 17, 17, and 2x + 1. : sides of length 17, 17, and 3x : sides of length 19, 25, and 5x - 2. : sides of length 25, 28, and 4 - y x = 10 x = 6, y = : sides of length 8, x - y, and x + y 21. : sides of length 9, x, and 2x - y : sides of length 8, 15, and 17 : sides of length 8, 9, and 2y - x x = 16, y = 1; x = 16, y = -1 Possible solution: x = 8, y = These statements are part of an explanation for the SSS Triangle ongruence Theorem. Write the numbers 1 to 6 to place these strategies in a logical order. The statements refer to triangles and shown here Rotate the image of about, so that the image of coincides with. pply the definition of reflection to show is the reflection of across. onclude that because a sequence of rigid motions maps one triangle onto the other. Translate along. Houghton Mifflin Harcourt Publishing ompany 4 efine as the perpendicular bisector of the line connecting and the image of. 3 Identify, and then, as equidistant from and the image of. Module Lesson 4 SSS Triangle ongruence 264

11 JOURNL Have students compare and contrast the SSS and SS theorems and support their answers with a sketch that illustrates each theorem. 23. etermine whether the given information is sufficient to guarantee that two triangles are congruent. Select the correct answer for each lettered part.. The triangles have three pairs of congruent corresponding angles. sufficient not sufficient. The triangles have three pairs of congruent corresponding sides. sufficient not sufficient. The triangles have two pairs of congruent corresponding sides and one pair of congruent corresponding angles. sufficient not sufficient. The triangles have two pairs of congruent corresponding angles and one pair of congruent corresponding sides. sufficient not sufficient. Two angles and the included side of one triangle are congruent to two angles and the included side of the other triangle. sufficient not sufficient. Two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. sufficient not sufficient. the triangles have the same shape but not necessarily the same size.. SSS Triangle ongruence Theorem. These conditions do not produce a unique triangle.. The Triangle Sum Theorem will show that the third pair of angles is congruent, so the triangles are congruent by the S Triangle ongruence Theorem.. S Triangle ongruence Theorem. SS Triangle ongruence Theorem 24. Make a onjecture oes a version of SSS congruence apply to quadrilaterals? Provide an example to support your answer. No; Possible example: a variety of non-congruent rhombuses, including Houghton Mifflin Harcourt Publishing ompany a square, may have 4 sides of the same length. 25. re two triangles congruent if all pairs of corresponding angles are congruent? Support your answer with an example. No; t least one pair of sides must be congruent. or example, two triangles that have three 60 degree angles may not have at least one pair of congruent corresponding sides, such as a triangle with side lengths of 4 inches and another triangle with side lengths of 8 inches. Module Lesson Lesson 5.4

12 H.O.T. ocus on Higher Order Thinking 26. xplain the rror va wants to know the distance JK across a pond. She locates points as shown. She says that the distance across the pond must be 160 ft by the SSS Triangle ongruence Theorem. xplain her error. The distance is 160 ft, but the justification should be the SS Triangle ongruence Theorem. 27. nalyze Relationships Write a proof. Given:,, Prove: Statements 1., , 1. Given L 160 ft M Reasons 210 ft 190 ft N 190 ft 210 ft 2. Reflexive Property of ongruence 3. S Triangle ongruence Theorem 4. PT 5. Given J K INTGRT MTHMTIL PRTIS ocus on Reasoning MP.2 sk students to consider the changes in Mike s bench if it were to transform from the original shape to complete collapse. Which properties of the bench would change and how would they change? Which properties would not change? Sample answer: hange: uring collapse, the area inside the parallelogram would decrease continually; the measures of the angles would change, with the measures of one pair of opposite angles increasing continually and the measures of the other pair decreasing continually. Not change: The lengths of the sides and the perimeter of the parallelogram would not change. 6. Lesson Performance Task Mike and Michelle each hope to get a contract with the city to build benches for commuters to sit on while waiting for buses. The benches must be stable so that they don t collapse, and they must be attractive. Their designs are shown. Judge the two benches on stability and attractiveness. xplain your reasoning. Mike 6. SSS Triangle ongruence Theorem Michelle nswers will vary. While Mike's bench will probably be chosen as the more attractive, students should note that Michelle's is more stable. There are many quadrilaterals with the same side lengths as Mike's bench. ut, by the SSS ongruence Theorem, the triangles in Michelle's design cannot change shape. Houghton Mifflin Harcourt Publishing ompany VOI OMMON RRORS Students may argue that Mike s bench may be solidly constructed with screws, nails, and glue, so that it would not collapse. Stress that the bench is a real-world object introduced here to model an abstract geometrical principle, and that it is not a perfect model. The important conclusion to draw is the one relating to quadrilaterals, not to benches. Module Lesson 4 XTNSION TIVITY Have students construct models of Mike and Michelle s benches from sturdy pieces of cardboard or photo-print paper cut into long, narrow strips and attached at the corners with brads. Students can explore further by making other polygons, checking their stability, and explaining why, for example, there is no SSSSS Pentagon ongruence Theorem. 5-sided polygon is not stable. This means that there are many different shapes that can be constructed from the same 5 sides of a pentagon. Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. SSS Triangle ongruence 266