5.4 SSS Triangle Congruence

Transcription

1 Locker LSSON 5.4 SSS Triangle ongruence Name lass ate 5.4 SSS Triangle ongruence ssential uestion: What does the SSS Triangle ongruence Theorem tell you about triangles? Texas Math Standards The student is expected to:.6. rove two triangles are congruent by applying the Side-ngle-Side, ngle-side-ngle, Side-Side-Side, ngle-ngle-side, and Hypotenuse- Leg congruence conditions. xplore.6. rove two triangles are congruent by... Side-Side-Side... ;.5. Use... constructions... to make conjectures.... lso.3.,.5. onstructing Triangles iven 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,, and 3 in. Use a ruler, compass, and either tracing paper or a transparency. esource Locker.5. Use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships. lso.3.,.5. Mathematical rocesses Use a ruler to draw a line segment of length 5 inches. Label the endpoints and. Open a compass to 4 inches. lace the point of the compass on, and draw an arc as shown..1. nalyze mathematical relationships to connect and communicate mathematical ideas. Language Objective 1., 1., 2..4, 2..3 Have small groups of students complete a triangle congruence chart. N ssential uestion: 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. Houghton Mifflin Harcourt ublishing ompany Now open the compass to 3 inches. lace the point of the compass on, and draw a second arc. Next, find the intersection of the two arcs. Label the intersection. raw and. Label the side lengths on the figure. VIW: LSSON OMN TSK View the ngage section online. oint out that the structural beams mentioned in the review refer to the steel sides of the triangles. Then preview the Lesson erformance Task. Module Lesson 4 Name lass ate 5.4 SSS Triangle ongruence ssential uestion: What does the SSS Triangle ongruence Theorem tell you about triangles?.6. rove two triangles are congruent by... Side-Side-Side... ;.5. Use... constructions... to make conjectures.... lso.3.,.5. xplore onstructing Triangles iven Houghton Mifflin Harcourt ublishing 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,, 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. lace the point of the compass on, and draw an arc as shown. Now open the compass to 3 inches. lace the point of the compass on, and draw a second arc. esource HOV S Turn to these pages to find this lesson in the hardcover student edition. Next, find the intersection of the two arcs. Label the intersection. raw and. Label the side lengths on the figure. Module Lesson Lesson 5.4

2 epeat 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. 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. or each side in, write its congruent side in. eflect 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. One circle is the set of points that are 4 inches away 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 all the triangles congruent? How do you know? Yes, because there is a sequence of rigid motions that maps any one of the triangles onto any of the others. xplain 1 Justifying SSS Triangle ongruence You can use rigid motions and the converse of the erpendicular 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. Houghton Mifflin Harcourt ublishing ompany XLO onstructing Triangles iven Three Side Lengths INTT THNOLOY Students may use geometry software to explore the concept of constructing a triangle with given side lengths. INTT MTHMTIL OSSS ocus on Modeling ive 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. XLIN 1 Justifying SSS Triangle ongruence Module Lesson 4 OSSIONL VLOMNT 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 erpendicular isector Theorem. einforce this justification throughout the lesson. INTT MTHMTIL OSSS ocus on ommunication 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. USTIONIN STTIS 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 288

3 xample 1 In the triangles shown, let,, and. Use rigid motions to show that. 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 point? To show this, notice that =, which means that point is equidistant from point and point. Houghton Mifflin Harcourt ublishing ompany Therefore, point lies on the perpendicular bisector of by the converse of the perpendicular bisector theorem. ecause =, 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. Module Lesson Lesson 5.4 OLLOTIV LNIN eer-to-eer ctivity ive 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 Show that. Triangle is transformed by a sequence of rigid motions to form the figure shown below. Identify the sequence of rigid motions. 1. Translation along 2. otation about so that and coincide. 3. eflection across. omplete the explanation by filling in the blanks with the name of a point, line segment, or geometric theorem. ecause, point is equidistant from (or ) and (or ). Therefore, erpendicular isector by the converse of the Theorem, point lies on the of. Similarly, perpendicular bisector. So point lies on the perpendicular bisector of. ecause two points determine a line, the line is the perpendicular bisector of. y the definition of reflection, the image of point must be point. Therefore, because is mapped to by a translation, a rotation, and a reflection. eflect 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. Houghton Mifflin Harcourt ublishing ompany Module Lesson 4 INTIT INSTUTION 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 290

5 XLIN 2 roving Triangles re ongruent Using SSS Triangle ongruence INTT MTHMTIL OSSS ocus on Math onnections 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. 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 erpendicular 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. xplain 2 roving Triangles re ongruent Using SSS Triangle ongruence You can apply the SSS Triangle ongruence Theorem to confirm that triangles are congruent. emember, if any one pair of corresponding parts of two triangles is not congruent, then the triangles are not congruent. xample 2 rove that the triangles are congruent or explain why they are not congruent. USTIONIN STTIS 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. LNU SUOT 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. Triangle ongruence Theorems Theorem efinition icture Houghton Mifflin Harcourt ublishing 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 = = 20 cm, so. 24 cm H = H = 12 cm, so H H. 12 cm H 20 cm 20 cm H = H = 24 cm, so H H. 12 cm 24 cm The three sides of H are congruent to the three sides of H, so the two triangles are congruent by the SSS Triangle ongruence Theorem. Module Lesson 4 LNU SUOT onnect ontext The phrase iven 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. emind students that in math, they are often given partial information to help solve a problem, and that the information is called a given. 291 Lesson 5.4

6 Your Turn rove that the triangles are congruent or explain why they are not congruent. XLIN The corresponding sides MN and are not congruent. Therefore, the triangles are not congruent. It is given by the figure that K L and JK JL, and J J by the eflexive roperty. M K 5 28 in. N S 38 in. 38 in. 32 in. 5 pplying Triangle ongruence VOI OMMON OS 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. xplain 3 (4x - 6) cm 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. 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. 3 cm 3 cm 3 cm 3 cm J 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 - 6 = -4 x = 2 J L Houghton Mifflin Harcourt ublishing ompany Image redits: MaximImages/lamy USTIONIN STTIS 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 T. ONNT VOULY 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 292

7 LOT USTIONIN STTIS 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. 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 MN and M N O (3x - 11) in. MO, because they have the same measure. If NO SSS, then MNO by the Triangle ongruence Theorem. Write an equation setting the lengths equal and solve for x. 3x - 11 = x = 5 x = v Your Turn 7. raig made a mobile using geometric V shapes including triangles shaped 60º as shown. or what value of x and y 30º can you use a triangle congruence T (7y + 4)º 12 cm theorem to show that the triangles are (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. U, they are right angles. If H UV, then H TUV by the S. x = 3; y = 8 SUMMIZ 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 ublishing ompany laborate 8. xplain how the SSS ongruence Theorem is useful to anyone who is asked to construct a triangle, such as an engineer, a graphic designer, or a draftsman. The SSS ongruence Theorem shows that all triangles made with the same three lengths of sides are congruent. n engineer or other worker can specify the shape and size of a triangle merely by identifying the three lengths of its sides. 9. sequence of rigid motions maps JL onto T, as shown. How can you use the converse of the erpendicular isector Theorem to confirm that a reflection across T will map point K to point S? J and T are both equidistant from S and K. So T is the perpendicular bisector of SK, and a reflection across T maps point K onto point S. T L S K Module Lesson Lesson 5.4