SAS Triangle Congruence

Transcription

1 Locker LSSON 5.3 SS Triangle ongruence Texas Math Standards The student is expected to: G.6. Prove two triangles are congruent by applying the Side-ngle-Side, ngle-side-ngle, Side-Side-Side, ngle-ngle-side, and Hypotenuse- Leg congruence conditions. lso G.3., G.5., G.5. Mathematical Processes G.1.G isplay, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. Language Objective 1.,.3, 2..4, 2..1 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. Name lass ate 5.3 SS Triangle ongruence ssential Question: What does the SS Triangle ongruence Theorem tell you about triangles? G.6. Prove two triangles are congruent by applying the... Side-ngle-Side... congruence conditions. lso G.3., G.5., G.5. 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. Use paper and pencil or real-world objects to create two side lengths of a triangle, one long and the other long. On a sheet of paper use a straightedge to draw a horizontal line. rrange the side to form a angle, as shown. Next, arrange the side to complete the triangle. How many different triangles can you form? Support your answer with a diagram. Resource Locker 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. Houghton Mifflin Harcourt Publishing ompany 2 different triangles 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 Performance Task. Module Lesson 3 Name lass ate 5.3 SS Triangle ongruence ssential Question: What does the SS Triangle ongruence Theorem tell you about triangles? G.6. Prove two triangles are congruent by applying the... Side-ngle-Side... congruence conditions. lso G.3., G.5., G.5. xplore 1 rawing Triangles Given Two Sides and an ngle Resource 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. Use paper and pencil or real-world objects to create two side lengths of a triangle, one long and the other long. On a sheet of paper use a straightedge to draw a horizontal line. rrange the side to form a angle, as shown. Next, arrange the side to complete the triangle. How many different triangles can you form? Support your answer with a diagram. 2 different triangles HROVR PGS Turn to these pages to find this lesson in the hardcover student edition. Houghton Mifflin Harcourt Publishing ompany Module Lesson Lesson 5.3

2 Now arrange the two sides 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? XPLOR 1 rawing Triangles Given Two Sides and an ngle Reflect No, only one triangle is possible. Having the angle included between the sides fixes the position of the sides. 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. 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. Module Lesson 3 PROSSIONL VLOPMNT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TKS G.1.G, which calls for students to display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. 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. 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 PROSSS ocus on ritical Thinking 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. 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 276

3 XPLIN 1 eciding Whether Triangles re ongruent Using SS Triangle ongruence 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. QUSTIONING STRTGIS How do you know that two sides of a triangle are congruent? Two sides are congruent if they have the same length. 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 =. INTGRT MTHMTIL PROSSS ocus on ritical Thinking 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 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. 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. Module 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. 277 Lesson 5.3

4 xample 1 etermine whether the triangles are congruent. xplain your reasoning. 20 cm cm 15 cm 20 cm 19 cm Look for congruent corresponding parts. Sides and do not correspond to side, because they are not 15 cm long. must correspond to, because = = 20 cm. must correspond to, because = = 19 cm. and must be corresponding angles, 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 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 MNP and of. JKL MNP by the SS Triangle ongruence Theorem. Your Turn 3. etermine whether the triangles are congruent. xplain your reasoning. H 2.5 cm 2.5 cm 2.9 cm G 1.7 cm 1.7 cm Houghton Mifflin Harcourt Publishing ompany J GH, GJ, G, because corresponding parts have the same measure. and G are included by congruent corresponding sides. HGJ by the SS Triangle ongruence Theorem. Module 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 278

5 XPLIN 2 Proving Triangles re ongruent Using SS Triangle ongruence VOI OMMON RRORS Remind students that they should not assume information from a figure unless it is marked or stated in the given information. 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. 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: 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. Houghton Mifflin Harcourt Publishing ompany Image redits: Ulrich Niehoff/Imagebroker/age fotostock 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 SS Triangle ongruence Theorem angle of. The triangles are congruent by the. Module 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. 279 Lesson 5.3

6 Your Turn 4. Given: and 1 2 Prove: laborate Q P T U 1 2 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 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. 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. R V Houghton Mifflin Harcourt Publishing ompany 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. 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. Module Lesson 3 SS Triangle ongruence 280