3.3 Corresponding Parts of Congruent Figures Are Congruent

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1 Name lass ate 3.3 orresponding arts of ongruent Figures re ongruent Essential Question: What can you conclude about two figures that are congruent? esource Locker Explore G.6. pply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles. lso G.3., G.3., G.3. Exploring ongruence of arts of Transformed Figures You will investigate some conclusions you can make when you know that two figures are congruent. Fold a sheet of paper in half. Use a straightedge to draw a triangle on the folded sheet. Then cut out the triangle, cutting through both layers of paper to produce two congruent triangles. Label them and EF, as shown. E F lace the triangles next to each other on a desktop. ince the triangles are congruent, there must be a sequence of rigid motions that maps to EF. escribe the sequence of rigid motions. The same sequence of rigid motions that maps to EF maps parts of to parts of EF. omplete the following. What does tep tell you about the corresponding parts of the two triangles? Why? Module Lesson 3

2 eflect 1. If you know that EF, what six congruence statements about segments and angles can you write? Why? 2. o your findings in this Explore apply to figures other than triangles? For instance, if you know that quadrilateral JKLM is congruent to quadrilateral Q, can you make any conclusions about corresponding parts? Why or why not? J K Q M L Explain 1 orresponding arts of ongruent Figures re ongruent The following true statement summarizes what you discovered in the Explore. orresponding arts of ongruent Figures re ongruent If two figures are congruent, then corresponding sides are congruent and corresponding angles are congruent. Example 1 EF. Find the given side length or angle measure. 2.6 cm 3.7 cm F 3.5 cm E Module Lesson 3

3 E tep 1 Find the side that corresponds to E. ince EF, E. tep 2 Find the unknown length. E =, and = 2.6 cm, so E = 2.6 cm. m tep 1 Find the angle that corresponds to. ince EF,. tep 2 Find the unknown angle measure. m = m, and m =, so m =. eflect 3. iscussion The triangles shown in the figure are congruent. an you conclude that JK Q? Explain. K Q J L Your Turn TU VWX. Find the given side length or angle measure. 32 ft 18 W 16 ft ft T 38 U V X 4. U 5. m Module Lesson 3

4 Explain 2 pplying the roperties of ongruence igid motions preserve length and angle measure. This means that congruent segments have the same length, so UV XY implies UV = XY and vice versa. In the same way, congruent angles have the same measure, so J K implies m J = m K and vice versa. roperties of ongruence eflexive roperty of ongruence ymmetric roperty of ongruence Transitive roperty of ongruence If, then. If and EF, then EF. Example 2 EF. Find the given side length or angle measure. (3x + 8) in. (5y + 11) 25 in. (6y + 2) (5x) in. 83 E F ince EF, E. Therefore, = E. Write an equation. ubtract 3x from each side. ivide each side by 2. 3x + 8 = 5x 8 = 2x 4 = x o, = 3x + 8 = 3 (4) + 8 = = 20 in. m ince EF,. Therefore, m = m. Write an equation. 5y + = + 2 ubtract 5y from each side. 11 = + 2 ubtract 2 from each side. = eflect o, m = (6y + 2) = (6 + 2) =. 6. an you determine any other side lengths of EF based on the fact that EF? Explain. Module Lesson 3

5 Your Turn Quadrilateral GHJK quadrilateral LMN. Find the given side length or angle measure. G (4x + 3) cm (9y + 17) H 18 cm (10y) L K (6x 13) cm J N (11y - 1) M 7. LM 8. m H Explain 3 Using ongruent orresponding arts in a roof Example 3 Write each proof. Given: rove: is the midpoint of. tatements Given orresponding parts of congruent figures are congruent. 3. is the midpoint of. 3. efinition of midpoint. Module Lesson 3

6 Given: Quadrilateral JKLM quadrilateral NQ; J K J K Q rove: J M L N tatements 1. Quadrilateral JKLM quadrilateral NQ J K K J 4. eflect 9. In art, how can you use corresponding parts of congruent figures to prove that a different angle of quadrilateral JKLM is congruent to an angle of quadrilateral NQ? Your Turn Write each proof. 10. Given: VT WT rove: T bisects VW. V tatements T W Module Lesson 3

7 11. Given: Quadrilateral quadrilateral EFGH; rove: GH E F H G tatements Elaborate 12. student claims that any two congruent triangles must have the same perimeter. o you agree? Explain. 13. If Q is a right triangle and Q XYZ, does XYZ have to be a right triangle? Why or why not? 14. Essential Question heck-in uppose you know that pentagon E pentagon FGHJK. How many additional congruence statements can you write using corresponding parts of the pentagons? Explain. Module Lesson 3

8 Evaluate: Homework and ractice 1. anielle finds that she can use a translation and a reflection to make quadrilateral fit perfectly on top of quadrilateral WXYZ. What congruence statements can anielle write using the sides and angles of the quadrilaterals? Why? Online Homework Hints and Help Extra ractice Y Z X W EF GHJ. Find the given side length or angle measure. 19 ft E 42 ft F J H 31 ft G 2. JH 3. m KLMN Q. Find the given side length or angle measure. K L 2.1 cm N 2.9 cm M Q 4. m 5. Module Lesson 3

9 TUV. Find the given side length or angle measure. (5x + 7) cm V U (4y - 18) (6x - 1) cm (3y + 2) (4y) (6x + 2) cm T m U Quadrilateral EFG quadrilateral KLMN. Find the given side length or angle measure. G N (4y - 29) in. (20x + 12) E 68 F (25x - 8) (2y + 3)in. K (y + 9) in. L M 8. FG 9. m Module Lesson 3

10 Write each proof. 10. Given: Quadrilateral QTU quadrilateral QT rove: QT bisects. Q U T tatements 11. Given: rove: bisects and bisects. tatements 12. Given: MN T; MN M rove: MN T tatements N M T Module Lesson 3

11 13. Given: entagon E pentagon FGHJK; E rove: K G H F E J K tatements GHJ Q and Q TU. omplete the following using a side or angle of TU. Justify your answers. 14. GH 15. J 16. HJ 17. H 18. GJ = 19. m G = Module Lesson 3

12 EF. Find the given side length or angle measure. J E 62 M 71 F 20. m 21. m 22. The figure shows the dimensions of two city parks, where T XYZ and YX YZ. city employee wants to order new fences to surround both parks. What is the total length of the fences required to surround the parks? X 320 ft T 210 ft Z Y tower crane is used to lift steel, concrete, and building materials at construction sites. The figure shows part of the horizontal beam of a tower crane. G H HG G 59 H 23. Is it possible to determine m GH? If so, how? If not, why not? member of the construction crew claims that is twice as long as. o you agree? Explain. Image redits: Ken rown/e+/getty Images Module Lesson 3

13 25. Multi-tep company installs triangular pools at hotels. ll of the pools are congruent and JKL MN in the figure. What is the perimeter of each pool? J 41 ft M (4x - 4) ft L K (5x + 1) ft N (20x - 15) (15x + 15) 26. Kendall and va lay out the course shown below for their radio-controlled trucks. In the figure,. The trucks travel at a constant speed of 15 feet per second. How long does it take a truck to travel on the course from to to to? ound to the nearest tenth of a second. 30 ft 50 ft Image redits: (t) Image ollective/lamy; (b) Oleksiy Maksymenko hotography/lamy 40 ft Module Lesson 3

14 27. MN Q. etermine whether each statement about the triangles is true or false. elect the correct answer for each lettered part. M N 52 mm Q 52 mm (x + 19) 34 mm (2x - 33) a. Q is isosceles. True False b. M is longer than MN. True False c. m = 52 True False d. The perimeter of Q is 120 mm. True False e. M Q True False H.O.T. Focus on Higher Order Thinking 28. Justify easoning Given that EF, = 2.7 ft, and = 3.4 ft, is it possible to determine the length of EF? If so, find the length and justify your steps. If not, explain why not. Module Lesson 3

15 29. Explain the Error student was told that GHJ T and was asked to find GH. The student s work is shown below. Explain the error and find the correct answer. tudent's Work 5x - 2 = 6x = x - 5 G (5x - 2) m J H T (6x - 5) m (4x + 3) m 3 = x GH = 5x - 2 = 5 (3) - 2 = 13 m 30. ritical Thinking In, m = 55, m = 50, and m = 75. In EF, m E = 50, and m F = 65. Is it possible for the triangles to be congruent? Explain. 31. nalyze elationships Q Q and T. student said that point appears to be the midpoint of T. Is it possible to prove this? If so, write the proof. If not, explain why not. Q T tatements Module Lesson 3

16 Lesson erformance Task The illustration shows a Yankee uzzle quilt. a. Use the idea of congruent shapes to describe the design of the quilt. b. Explain how the triangle with base can be transformed to the position of the triangle with base. c. Explain how you know that =. Module Lesson 3

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