5.2 ASA Triangle Congruence

Transcription

1 Name lass ate 5.2 S Triangle ongruence ssential question: What does the S Triangle ongruence Theorem tell you about triangles? xplore 1 rawing Triangles Given Two ngles and a Side You have seen that two triangles are congruent if they have six pairs of congruent corresponding parts. However, it is not always possible to check all three pairs of corresponding sides and all three pairs of corresponding angles. ortunately, there are shortcuts for determining whether two triangles are congruent. Resource Locker raw a segment that is 4 inches long. Label the endpoints and. Use a protractor to draw a 30 angle so that one side is _ and its vertex is point. Use a protractor to draw a 40 angle so that one side is _ and its vertex is point. Label the point where the sides of the angles intersect as point in. 40 ut your triangle and a classmate s triangle beside each other. Is there a sequence of rigid motions that maps one to the other? What does this tell you about the triangles? Reflect Houghton Mifflin Harcourt ublishing ompany 1. In a polygon, the side that connects two consecutive angles is the included side of those two angles. escribe the triangle you drew using the term included side. e as precise as possible. 2. iscussion ased on your results, how can you decide whether two triangles are congruent without checking that all six pairs of corresponding sides and corresponding angles are congruent? Module Lesson 2

2 xplore 2 Justifying S Triangle ongruence xplain the results of xplore 1 using transformations. Use tracing paper to make two copies of the triangle from xplore 1 as shown. Identify the corresponding parts you know to be congruent and mark these congruent parts on the figure. _ What can you do to show that these triangles are congruent? translate so that point maps to point. What translation vector did you use? Use a rotation to map point to point. What is the center of the rotation? What is the angle of the rotation? How do you know the image of point is point? What rigid motion do you think will map point to point? G H To show that the image of point is point, notice that is reflected across, so the measure of the angle is preserved. Since you can conclude that the image of _ lies on. In particular, the image of point must lie on. y similar reasoning, the image of _ lies on and the image of point must lie on. The only point that lies on both _ and _ is. escribe the sequence of rigid motions used to map to. Reflect 3. iscussion rturo said the argument in the activity works for any triangles with two pairs of congruent corresponding angles, and it is not necessary for the included sides to be congruent. o you agree? xplain. Houghton Mifflin Harcourt ublishing ompany Module Lesson 2

3 xplain 1 eciding Whether Triangles re ongruent Using S Triangle ongruence You can state your findings about triangle congruence as a theorem. this theorem can help you decide whether two triangles are congruent. S Triangle ongruence Theorem If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. xample 1 etermine whether the triangles are congruent. xplain your reasoning. Step 1 ind m. m + m + m = m = 180 m = 180 m = cm 2.3 cm Step 2 ompare the angle measures and side lengths. m = m = 45, = = 2.3 cm, and m = m = 61 So,, _ _, and. and include side _, and and include side _. So, by the S triangle ongruence theorem. Step 1 ind m. m M + m N + m = m = 180 J in. 110 K Houghton Mifflin Harcourt ublishing ompany + m = 180 m = Step 2 ompare the angle measures and side lengths. N None of the angles in MN has a measure of. therefore, there is/is not a sequence of rigid motions that maps MN onto JKL, and MN is/is not congruent to JKL. Reflect 4. In art, do you need to find m K? Why or why not? L in. 31 M Module Lesson 2

4 Your Turn etermine whether the triangles are congruent. xplain your reasoning S Q R in. T 1 in. U xplain 2 roving Triangles re ongruent Using S Triangle ongruence the S triangle ongruence theorem may be used as a reason in a proof. xample 2 Write each proof. M Given: MQ NQ, MQ NQ rove: MQ NQ Q N 1. MQ NQ 1. Given 2. MQ NQ 2. Given 3. _ Q _ Q 3. Reflexive roperty of ongruence 4. MQ NQ 4. S Triangle ongruence Theorem Houghton Mifflin Harcourt ublishing ompany Module Lesson 2

5 Given:, is the midpoint of _. rove: is the midpoint of. _ _ _ Reflect 7. In art, suppose the length of _ is 8.2 centimeters. an you determine the length of any other segments in the figure? xplain. Your Turn Write each proof. 8. Given: JLM KML, JML KLM rove: JML KLM J K L M Houghton Mifflin Harcourt ublishing ompany Module Lesson 2

6 9. Given: S and U are right angles, _ RV bisects _ SU. rove: RST VUT R S T U V laborate 10. iscussion Suppose you and a classmate both draw triangles with a 30 angle, a 70 angle, and a side that is 3 inches long. How will they compare? xplain your reasoning. 11. iscussion How can a diagram show you that corresponding parts of two triangles are congruent without providing specific angle measures or side lengths? 12. ssential Question heck-in What must be true in order for you to use the S Triangle ongruence Theorem to prove that triangles are congruent? Houghton Mifflin Harcourt ublishing ompany Module Lesson 2

7 valuate: Homework and ractice 1. Natasha draws a segment _ Q that is 6 centimeters long. She uses a protractor to draw a 60 angle so that one side is _ Q and its vertex is point. then she uses a protractor to draw an 35 angle so that one side is _ Q and its vertex is point Q. a. raw a triangle following the instructions that Natasha used. Label the vertices and the known side and angle measures. Online Homework Hints and Help xtra ractice b. Will there be a sequence of rigid motions that will map your triangle onto Natasha s triangle? xplain. 2. tomas drew two triangles, as shown, so that, _ _, and. escribe a sequence of one or more rigid motions tomas can use to show that. Houghton Mifflin Harcourt ublishing ompany etermine whether the triangles are congruent. xplain your reasoning K J cm L 9.3 cm in M in. N Module Lesson 2

8 etermine whether the triangles are congruent. xplain your reasoning m Q T 4.1 m S 34 J 58 M 17 L R U K 73 Write each proof. _ 7. Given: bisects and. rove: 1. _ bisects and efinition of bisector efinition of bisector Reflexive roperty of ongruence Houghton Mifflin Harcourt ublishing ompany Module Lesson 2

9 _ 8. Given: is parallel to, _. rove: 9. Given: H J, G is the midpoint of HJ, G is perpendicular to _ HJ. rove: GH GJ H G J Houghton Mifflin Harcourt ublishing ompany 10. The figure shows quadrilateral QRS. What additional information do you need in order to conclude that SR QR by the S Triangle ongruence Theorem? xplain. 11. ommunicate Mathematical Ideas In the figure, _ WX is parallel to _ LM. a. escribe a sequence of two rigid motions that maps LMN to WXY. b. How can you be sure that point N maps to point Y? S W Y L N X R Q M Module Lesson 2

10 Use a compass and straightedge and the S Triangle ongruence Theorem to construct a triangle that is congruent to Multi-Step or what values of the variables is QR congruent to SR? In this case, what is m Q? (2x +1) (x +18) Q S (8y - 4) R (4y + 28) Houghton Mifflin Harcourt ublishing ompany Module Lesson 2

11 Write each proof. 15. Given:, is the midpoint of _. rove: 16. The figure shows GHJ and QR on a coordinate plane. a. xplain why the triangles are congruent using the S Triangle ongruence Theorem. 4 y G 2 H J x -4 Q 0 R Houghton Mifflin Harcourt ublishing ompany b. xplain why the triangles are congruent using rigid motions. Module Lesson 2

12 17. Justify Reasoning factory makes triangular traffic signs. ach sign is an equilateral triangle with three 60 angles. xplain why two signs that each have a side 36 inches long must be congruent. 18. Represent Real-World roblems Rob is making the kite shown in the figure. a. an Rob conclude that? Why or why not? b. Rob says that = and =. o you agree? xplain. c. Given that = x + 15 cm and = x cm, write an expression for the distance around the kite in centimeters. 19. In order to find the distance across a canyon, Mariela sites a tree across the canyon (point ) and locates points on her side of the canyon as shown. xplain how she can use this information to find the distance across the canyon. 500 ft 500 ft Houghton Mifflin Harcourt ublishing ompany Image redits: (t) TheYok/iStockhoto.com; (b) View Stock/Getty Images Module Lesson 2

13 20. etermine whether each of the following provides enough information to prove that SQ SQR. Select the correct answer for each lettered part. Q R a. Q is the midpoint of _ R. Yes No b. R Yes No c. SQ is a right angle, SQ RSQ Yes No d. SQ is a right angle, m = 32, m RSQ = 58. Yes No e. R, SQ RSQ Yes No S H.O.T. ocus on Higher Order Thinking 21. ounterexamples Jasmine said that the S Triangle ongruence Theorem works for quadrilaterals. That is, if two angles and the included side of one quadrilateral are congruent to two angles and the included side of another quadrilateral, then the quadrilaterals are congruent. Sketch and mark a figure of two quadrilaterals as a counterexample to show that Jasmine is incorrect. Houghton Mifflin Harcourt ublishing ompany 22. ritique Reasoning and are both right triangles and both triangles contain a 30 angle. oth triangles have a side that is 9.5 mm long. Yoshio claims that he can use the S Triangle ongruence Theorem to show that the triangles are congruent. o you agree? xplain. Module Lesson 2

14 23. raw onclusions o you think there is an SS ongruence Theorem for quadrilaterals? Suppose two quadrilaterals have a pair of congruent consecutive angles with a pair of congruent included sides and an additional pair of congruent corresponding sides. Must the quadrilaterals be congruent? Justify your response. Lesson erformance Task The flag of the ongo Republic consists of green and red right triangles separated by a yellow parallelogram. onstruct an argument to prove that. Houghton Mifflin Harcourt ublishing ompany Module Lesson 2