OMMON OR Locker LSSON 5.3 SS Triangle ongruence Name lass ate 5.3 SS Triangle ongruence ssential Question: What does the SS Triangle ongruence Theorem tell you about triangles? ommon ore Math Standards The student is expected to: OMMON OR G-O..8 xplain how the criteria for triangle congruence ( SS...) 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 MP.3 Logic Language Objective Have students work in pairs to find an example in the lesson and write out a step-by-step explanation of how the SS Triangle ongruence Theorem works. NGG ssential Question: What does the SS Triangle ongruence Theorem tell you about triangles? If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. PRVIW: LSSON PRORMN TSK View the ngage section online. xplain that triangle congruence is important in the design of structures like pyramids. Then preview the Lesson Houghton Mifflin Harcourt Publishing ompany xplore 1 rawing Triangles Given Two Sides and an ngle You know that when all corresponding parts of two triangles are congruent, then the triangles are congruent. Sometimes you can determine that triangles are congruent based on less information. or this activity, cut two thin strips of paper, one long and the other long. On a sheet of paper use a straightedge to draw a horizontal line. rrange the strip to form a angle, as shown. Next, arrange the strip to complete the triangle. How many different triangles can you form? Support your answer with a diagram. 2 different triangles Now arrange the two strips of paper to form a angle so that the angle is included between the two consecutive sides, as shown. With this arrangement, can you construct more than one triangle? Why or why not? No, only one triangle is possible. Having the angle included between the sides fixes the position of the sides. Resource Locker Performance Task. 245 Module 5 245 Lesson 3 Name lass ate 5.3 SS Triangle ongruence ssential Question: What does the SS Triangle ongruence Theorem tell you about triangles? xplore 1 rawing Triangles Given Two Sides and an ngle You know that when all corresponding parts of two triangles are congruent, then the triangles are congruent. Sometimes you can determine that triangles are congruent based on less information. or this activity, cut two thin strips of paper, one long and the other long. Houghton Mifflin Harcourt Publishing ompany G-O..8 xplain how the criteria for triangle congruence ( SS ) follow from the definition of congruence in terms of rigid motions. lso G-O..7, G-O..10, G-SRT..5 On a sheet of paper use a straightedge to draw a horizontal line. rrange the strip to form a angle, as shown. Next, arrange the strip to complete the triangle. How many different triangles can you form? Support your answer with a diagram. 2 different triangles Now arrange the two strips of paper to form a angle so that the angle is included between the two consecutive sides, as shown. With this arrangement, can you construct more than one triangle? Why or why not? No, only one triangle is possible. Having the angle included between the sides fixes the position of the sides. Resource HROVR PGS 211 220 Turn to these pages to find this lesson in the hardcover student edition. Module 5 245 Lesson 3 245 Lesson 5.3
Reflect 1. iscussion If two triangles have two pairs of congruent corresponding sides and one pair of congruent corresponding angles, under what conditions can you conclude that the triangles must be congruent? xplain. The triangles must be congruent if the congruent corresponding angles are the angles included between the congruent corresponding sides. XPLOR 1 rawing Triangles Given Two Sides and an ngle xplore 2 Justifying SS Triangle ongruence You can explain the results of xplore 1 using transformations. onstruct by copying, side, and side. Let point correspond to point, point correspond to point, and point correspond to point, and place point on the segment shown. The diagram illustrates one step in a sequence of rigid motions that will map onto. escribe a complete sequence of rigid motions that will map onto. What can you conclude about the relationship between and? xplain your reasoning. because there is a sequence of rigid motions that maps one onto the other. Reflect 2. Is it possible to map onto using a single rigid motion? If so, describe the rigid motion. Yes; possible answer: reflect across a vertical line halfway between points and. Possible answer: Translate so that point maps to point. Then rotate 180 counterclockwise about point. Point will map to point because =. Then reflect across. Point will map to point because and =. Module 5 246 Lesson 3 Houghton Mifflin Harcourt Publishing ompany INTGRT THNOLOGY Have students use geometry software to explore included angles. QUSTIONING STRTGIS If I hold up a compass and increase the angle, what happens to the distance between the tips? If I decrease the angle, what happens to the distance between the tips? If I keep the angle the same, what happens to the distance between the tips? The angle increases; it decreases; it stays the same. XPLOR 2 Justifying SS Triangle ongruence INTGRT MTHMTIL PRTIS ocus on ritical Thinking MP.3 ach time the students perform a transformation, have them note the effect of the transformation on the angles and sides. They should notice that they are transformed in the same way and that their measures stay the same. PROSSIONL VLOPMNT Integrate Mathematical Practices This lesson provides an opportunity to address Mathematical Practice MP.3, which calls for students to construct viable arguments and critique the reasoning of others. s students explore congruent triangles, ask them to share their observations and conclusions with the class. s they share their findings, ask if anyone got different results. iscuss the differences. Promoting this type of dialogue in the classroom is an essential aspect of the standard. QUSTIONING STRTGIS oes it matter on which side of the angle you place each segment? No; they will make the same triangle, with the only difference being a reflection. SS Triangle ongruence 246
XPLIN 1 eciding Whether Triangles re ongruent Using SS Triangle ongruence xplain 1 eciding Whether Triangles are ongruent Using SS Triangle ongruence What you explored in the previous two activities can be summarized in a theorem. You can use this theorem and the definition of congruence in terms of rigid motions to determine whether two triangles are congruent. SS Triangle ongruence Theorem If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. QUSTIONING STRTGIS How do you know that two sides of a triangle are congruent? Two sides are congruent if they have the same length. INTGRT MTHMTIL PRTIS ocus on ritical Thinking MP.3 Remind students that they know how to find the measure of an angle of a triangle when they know the measures of the other two angles. This makes it possible to apply the S Triangle ongruence Theorem to many sets of triangles. Tell them to suppose that they know the lengths of two sides of a triangle. Is it possible to use that information to find the length of the third side? It is possible only if the triangle is a right triangle; then the length of the third side can be found using the Pythagorean Theorem. Houghton Mifflin Harcourt Publishing ompany xample 1 etermine whether the triangles are congruent. xplain your reasoning. Look for congruent corresponding parts. Sides and do not correspond to side 20 cm, 43 because they are not 15 cm long. corresponds to 19 cm 15 cm, because = = 20 cm. corresponds to, because = = 19 cm. and are corresponding angles because they are included between pairs of corresponding sides, but they don t have the same measure. The triangles are not congruent, because there is no sequence of rigid motions that maps onto. 74 in. J 37 46 in. L K N P 46 in. 37 M 74 in. Look for congruent corresponding parts. JL corresponds to MP, because JL = MP = 46 in. JK corresponds to MN, because JK = MN = 74 in. J corresponds to M, because m J = m M = 37. Two sides and the included angle of JKL are congruent to two sides and the included angle of MNP. JKL MNP SS Triangle ongruence Theorem by the. 20 cm 19 cm Module 5 247 Lesson 3 OLLORTIV LRNING Small Group ctivity Instruct students to illustrate the difference between the S and SS Triangle ongruence Theorems. They may make a poster, write an essay, create a model, or use another technique to convey the information. Have students share their work in small groups. Then have each group choose one project to present to the class. 247 Lesson 5.3
Your Turn 3. etermine whether the triangles are congruent. xplain your reasoning. GH, GJ, and G, and and G are included by congruent corresponding sides. HGJ by the SS Triangle ongruence Theorem. H 2.5 cm 2.5 cm 2.9 cm G 1.7 cm 1.7 cm XPLIN 2 Proving Triangles re ongruent Using SS Triangle ongruence xplain 2 Proving Triangles re ongruent Using SS Triangle ongruence Theorems about congruent triangles can be used to show that triangles in real-world objects are congruent. xample 2 Write each proof. J VOI OMMON RRORS Remind students that they should not assume information from a figure unless it is marked or stated in the given information. Write a proof to show that the two halves of a triangular window are congruent if the vertical post is the perpendicular bisector of the base. Given: is the perpendicular bisector of. Prove: It is given that is the perpendicular bisector of. y the definition of a perpendicular bisector, =, which means, and, which means and are congruent right angles. In addition, by the reflexive property of congruence. So two sides and the included angle of are congruent to two sides and the included angle of. The triangles are congruent by the SS Triangle ongruence Theorem. Given: bisects and bisects Prove: It is given that bisects and bisects. So by the definition = of a bisector, = and, which makes and. because they are vertical angles included. So two sides and the angle of are congruent to two sides and the included angle of. The SS Triangle ongruence Theorem triangles are congruent by the. Houghton Mifflin Harcourt Publishing ompany Image redits: Ulrich Niehoff/Imagebroker/age fotostock QUSTIONING STRTGIS To use SS, is it essential that the congruent angles be included between the pairs of congruent sides? Yes, because it is possible for an acute triangle and an obtuse triangle to have two pairs of corresponding congruent sides and a pair of corresponding congruent nonincluded angles. There is no Side-Side-ngle (SS) Theorem. Module 5 248 Lesson 3 IRNTIT INSTRUTION Kinesthetic xperience Have students place two pencils on their desks so that the points intersect and the pencils model an angle. Have students measure the distance between the erasers. Have the students rotate one pencil to change the angle. Have them measure the distance between the erasers again. fter they have experimented with different angles, discuss whether or not it is possible to change the distance between the erasers without changing the angle. SS Triangle ongruence 248
LORT QUSTIONING STRTGIS The solution to an exercise is JKL MNP. Suppose that ud concludes that JLK MPN and Kim concludes that KJL MPN. an both students be correct? xplain. ud s answer is correct because the order of the vertices lines up congruent angles. Kim s is not because the order of the vertices does not line up congruent angles. Your Turn 4. Given: and 1 2 Prove: Possible answer: You are given that and 1 2. You also know that by the reflexive property. Two sides and the included angle of are congruent to two sides and the included angle of. The triangles are congruent by the SS Triangle ongruence Theorem laborate 5. xplain why the corresponding angles must be included angles in order to use the SS Triangle ongruence Theorem. Possible answer: If the corresponding angles are not included angles, then there is more than one possible angle between the congruent corresponding sides. 1 2 SUMMRIZ TH LSSON Why would you use the SS Triangle ongruence Theorem? What do you need to know to use it? You would use the SS Triangle ongruence Theorem to prove that two triangles are congruent by using only three pairs of congruent parts. You need to know that two pairs of corresponding sides are congruent and the angles included between those sides are also congruent. Houghton Mifflin Harcourt Publishing ompany 6. Jeffrey draws PQR and TUV. He uses a translation to map point P to point T and point R to point V as shown. What should be his next step in showing the triangles are congruent? Why? Reflect PQR across TV ; this will map point Q to point U and show that there is a sequence of rigid motions that maps PQR to TUV. 7. ssential Question heck-in If two triangles share a common side, what else must be true for the SS Triangle ongruence Theorem to apply? second side and an included angle must be congruent. valuate: Homework and Practice 1. Sarah performs rigid motions mapping point to point and point to point, as shown. oes she have enough information to confirm that the triangles are congruent? xplain your reasoning. No; she can map to by a reflection across, but will map to only if =. Q P T U R V Online Homework Hints and Help xtra Practice Module 5 249 Lesson 3 LNGUG SUPPORT onnect Vocabulary Open and shut a door and talk about the function of a hinge. ompare the concept of an included angle to a hinge. raw a triangle on the board, labeling the vertices. Have students identify the angle that is included between each pair of sides. 249 Lesson 5.3
etermine whether the triangles are congruent. xplain your reasoning. 2. 3. Q P 2 in. S 2 in. PS RQ, PR PR, SPR QRP, and SPR and QRP are included by congruent corresponding sides. SPR QRP by SS. 4. 5. H 30 mm 40 mm 30 mm 95 40 mm G J 52 mm GH and HJ, but included angles and H are not congruent. The triangles are not congruent, because there is no sequence of rigid motions that maps onto GHJ. ind the value of the variable that results in congruent triangles. xplain. 6. 7. 9 in 30 (2x) in. R 9 in 2x = x + 4; x = 4; by SS when x is 4. 30 (x + 4) in. 30 mm 30 mm,, and, and and are included by congruent corresponding sides. by SS. 1.3 m 1.3 m,, and and are included by congruent corresponding sides. by SS. (2x + 14) 13.5 in. 12 in. 13.5 in. (4x) 12 in. 2x+ 14 = 4x; x = 7; by SS when x is 7. Houghton Mifflin Harcourt Publishing ompany VLUT SSIGNMNT GUI oncepts and Skills xplore 2 Justifying SS Triangle ongruence xample 1 eciding Whether Triangles re ongruent Using SS Triangle ongruence xample 2 Proving Triangles re ongruent Using SS Triangle ongruence Practice xercises 1 6 xercises 7 11 xercises 12 20 INTGRT MTHMTIL PRTIS ocus on ommunication MP.3 Write S and SS on the board. sk students to do each of the following in their Math Journals: Tell what each stands for in terms of triangle congruence. raw and label a diagram illustrating each. Tell how the two theorems are the same. Tell how the two theorems are different. Module 5 250 Lesson 3 xercise epth of Knowledge (.O.K.) OMMON OR Mathematical Practices 1 6 2 Skills/oncepts MP.3 Logic 8 2 Skills/oncepts MP.1 Problem Solving 9 10 2 Skills/oncepts MP.4 Modeling 11 12 2 Skills/oncepts MP.3 Logic 13 3 Strategic Thinking MP.4 Modeling 14 3 Strategic Thinking MP.2 Reasoning 15 3 Strategic Thinking MP.6 Precision SS Triangle ongruence 250
RITIL THINKING raw non-collinear points M, O, and U on the board, connecting them to form an obtuse angle with vertex O. sk students to visualize a translation, a rotation, and a reflection of the figure shown. In each case, have them describe the effect on the segment that connects point M with point U. Sample response: The segment connecting points M and U will follow the same movements as the rest of the figure. Its length will remain the same no matter what rigid-motion transformation is used. 8. Given that polygon is a regular hexagon, prove that. Statements Reasons 1. is a regular hexagon. 1. Given 2. = and = 2. efinition of regular polygon 3. and 3. efinition of congruence in terms of rigid motion 4. m = m 4. efinition of regular polygon 5. 5. efinition of congruence in terms of rigid motion 6. 6. SS Triangle ongruence Theorem 7. 7. PT 9. product designer is designing an easel with extra braces as shown in the diagram. Prove that if and, then the braces and are also congruent. Houghton Mifflin Harcourt Publishing ompany Image redits: ndreyuu/istockphoto.com You are given that and. You also know that by the reflexive property. Two sides and the included angle of are congruent to two sides and the included angle of. The triangles are congruent by the SS Triangle ongruence Theorem. So, by PT, the braces and and are also congruent. Module 5 251 Lesson 3 251 Lesson 5.3
10. n artist is framing a large picture and wants to put metal poles across the back to strengthen the frame as shown in the diagram. If the metal poles are both the same length and they bisect each other, prove that and. VOI OMMON RRORS Students may choose the wrong angle when SS is used to prove triangles congruent. xplain that the angle must be formed by the sides. The included angle is named by the letter the segments share. ecause and bisect each other, = and =, so and by the definition of congruence. y the Vertical ngle Theorem, you also know that. Two sides and the included angle of are congruent to two sides and the included angle of. The triangles are congruent by the SS Triangle ongruence Theorem. y PT,. You can use similar reasoning to show that 11. The figure shows a side panel of a skateboard ramp. Kalim wants to confirm that the right triangles in the panel are congruent. a. What measurements should Kalim take if he wants to confirm that the triangles are congruent by SS? xplain. Measure and ; so he can confirm that two pairs of sides and their included angles are congruent. (,, and ) b. What measurements should Kalim take if he wants to confirm that the triangles are congruent by S? xplain. Measure and ; so he can confirm that two pairs of angles and their included sides are congruent. (,, and ) Houghton Mifflin Harcourt Publishing ompany Module 5 252 Lesson 3 SS Triangle ongruence 252
JOURNL Have students describe how they can use color coding to help them recognize the SS theorem. Have them support their descriptions with colored sketches. 12. Which of the following are reasons that justify why the triangles are congruent? Select all that apply.. SS Triangle ongruence Theorem. SS Triangle ongruence Theorem. S Triangle ongruence Theorem. onverse of PT. PT. SS is not a valid congruence theorem.. You do not know that all of the corresponding parts are congruent.. PT is a property of congruent triangles, not a justification for congruence. H.O.T. ocus on Higher Order Thinking 13. Multi-Step Refer to the following diagram to answer each question. Houghton Mifflin Harcourt Publishing ompany a. Use a triangle congruence theorem to explain why these triangles are congruent. ach triangle has side lengths of 2 and 6 and an included right angle. y SS they are congruent. b. escribe a sequence of rigid motions to map the top triangle onto the bottom triangle to confirm that they are congruent. Possible answer: Reflect the triangle across the y-axis. Next translate it 1 unit to the left. Then translate it 6 units down. Module 5 253 Lesson 3 253 Lesson 5.3
14. xplain the rror Mark says that the diagram confirms that a given angle and two given side lengths determine a unique triangle even if the angle is not an included angle. xplain Mark s error. 15. Justify Reasoning The opposite sides of a rectangle are congruent. an you conclude that a diagonal of a rectangle divides the rectangle into two congruent triangles? Justify your response. Lesson Performance Task J The diagram of the Great Pyramid at Giza gives the approximate lengths of edge and slant height. The slant height is the perpendicular bisector of. ind the perimeter of. xplain how you found the answer. ecause is the perpendicular bisector of, = and m = m = 90. lso, = so by the SS Triangle ongruence Theorem. Therefore, = 720 ft by PT. To find, use the Pythagorean Theorem: () 2 = 720 2-600 2 = 158,400; = 158,400 398; =, so = 398 perimeter of = + + + 720 + 398 + 398 + 720 = 2,236 ft. L M Possible answer: circle with its center at M and radius MJ = MK will intersect JL and KL at two other points closer to L. The triangles formed by each of these two points and the points L and M will be different than the original triangles, even though they are formed by the same given angle and two given side lengths. Yes; since the opposite sides of a rectangle are congruent and the included angles between the sides are right angles, the two triangles are congruent by the SS Theorem. K edge 720 ft slant height 600 ft Houghton Mifflin Harcourt Publishing ompany INTGRT MTHMTIL PRTIS ocus on Reasoning MP.2 sk students to identify the single congruence they would need to establish, in addition to the given information, to enable them to prove by the S Triangle ongruence Theorem. INTGRT MTHMTIL PRTIS ocus on Math onnections MP.1 The base of the Great Pyramid is square. What is the area of the base in acres? (1 acre = 43,560 ft 2 ) about 14.5 acres Module 5 254 Lesson 3 XTNSION TIVITY Some authorities claim that there is a relationship between the dimensions of the Great Pyramid at Giza and π, the ratio of the circumference of a circle to the radius. Have students research this claim and offer evidence as to its truth or falsity. The claim is that the ratio of the perimeter of the base of the Great Pyramid to its height equals 2π. Some sources give 923 m as the perimeter of the base and 146.3 m as the height. 923 6.309; 2π 6.283 146.3 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. SS Triangle ongruence 254