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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B 8. _ _ Given: AB is parallel to CD, ∠ACB ≅ ∠CAD. A B Prove: △ABC ≅ △CDA D 1. Given 2. ∠BAC ≅ ∠DCA 2. Alternate Interior Angles Theorem ¯ ≅ AC ¯ 3. AC 9. C Statements _ ¯ is parallel to CD. 1. AB Reasons 3. Reflexive Property of Congruence 4. ∠ACB ≅ ∠CAD 4. Given 5. △ABC ≅ △CDA 5. ASA Triangle Congruence Theorem _ Given: _ ∠H ≅ ∠J, G is the midpoint _ of HJ, FG is perpendicular to HJ. F Prove: △FGH ≅ △FGJ H Statements © Houghton Mifflin Harcourt Publishing Company 1. ∠H ≅ ∠J Reasons 1. Given ¯. 2. G is the midpoint of HJ 2. Given ¯ ≅ JG ¯ 3. HG 3. Definition of midpoint ¯ is perpendicular to HJ ¯. 4. FG 4. Given 5. ∠FGH and ∠FGJ are right angles. 5. Definition of perpendicular 6. ∠FGH ≅ ∠FGJ 6. All right angles are congruent. 7. △FGH ≅ △FGJ 7. ASA Triangle Congruence Theorem 10. The figure shows quadrilateral PQRS. What additional information do you need in order to conclude that △SPR ≅ △QRP by the ASA Triangle Congruence Theorem? Explain. ∠SRP ≅ ∠QPR; you need to have two pairs of congruent ¯ as the included side. corresponding angles with PR a. Describe a sequence of two rigid motions that maps △LMN to △WXY. ‹ › − ¯ to WX ¯, then reflect △LMN across WX Translate LM . b. How can you be sure that point N maps to point Y ? Since ∠L ≅ ∠W and ∠M ≅ ∠X, the images of ¯ LN and ¯ MN lie → → ‾ , respectively. The image of N must lie on both ‾ and XY on WY rays, so the image is the intersection point Y. GE_MNLESE385795_U2M05L2 239 239 Lesson 5.2 P 239 Q S _ _ 11. Communicate Mathematical Ideas In the figure, WX is parallel to LM. Module 5 J G R Y X W L M N Lesson 2 6/3/14 7:49 AM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B Write each proof. B _ 15. Given: ∠A ≅ ∠E, C is the midpoint of AE. ¯ ≅ ED ¯ Prove: AB C A E D Statements 1. ∠A ≅ ∠E Reasons 1. Given ¯. 2. C is the midpoint of AE 2. Given ¯ ≅ EC ¯ 3. AC 3. Definition of midpoint 4. ∠ACB ≅ ∠ECD 4. Vertical angles are congruent. 5. △ACB ≅ △ECD 5. ASA Triangle Congruence Theorem ¯ ≅ ED ¯ 6. AB 6. CPCTC 16. The figure shows △GHJ and △PQR on a coordinate plane. 4 a. Explain why the triangles are congruent using the ASA Triangle Congruence Theorem. ∠J ≅ ∠R, ∠H ≅ ∠Q (both are right angles), and ¯ (both are 4 units long). Two pairs of angles and ¯ ≅ QR HJ © Houghton Mifflin Harcourt Publishing Company their included sides are congruent, so the triangles are congruent by the ASA Triangle Congruence Theorem. -4 Q 0 -2 P J x R 4 -4 Possible answer: You can map △GHJ onto △PQR by a translation two units left followed by a reflection across the x-axis. By the definition of congruence in terms of rigid motions, the triangles must be congruent. GE_MNLESE385795_U2M05L2 241 Lesson 5.2 2 H b. Explain why the triangles are congruent using rigid motions. Module 5 241 y G 241 Lesson 2 6/3/14 7:50 AM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C %0/05&%*5$IBOHFTNVTUCFNBEFUISPVHI'JMFJOGP $PSSFDUJPO,FZ/-"$"" CRITICAL THINKING 8. _ _ Given that polygon ABCDEF is a regular hexagon, prove that AC ≅ AE . C B Draw non-collinear points M, O, and U on the board, connecting them to form an obtuse angle with vertex O. Ask 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. D A F E Statements 9. Reasons 1. ABCDEF is a regular hexagon. 1. Given 2. AB = AF and BC = FE _ _ _ _ 3. AB ≅ AF and BC ≅ FE 2. Definition of regular polygon 4. m∠B = m∠F 4. Definition of regular polygon 3. Definition of congruence in terms of rigid motion 5. ∠B ≅ ∠F 5. Definition of congruence in terms of rigid motion 6. △ABC ≅ △AFE _ _ 7. AC ≅ AE 6. SAS Triangle Congruence Theorem 7. CPCTC A product designer is designing an easel braces_ as _ with _ extra_ shown in the diagram. Prove _ that if BD ≅ FD and CD ≅ ED, _ then the braces BE and FC are also congruent. ª)PVHIUPO.JGGMJO)BSDPVSU1VCMJTIJOH$PNQBOZt*NBHF$SFEJUT ª"OESFZVVJ4UPDL1IPUPDPN D C A B GE_MNLESE385795_U2M05L3 251 Lesson 5.3 F _ _ _ _ You are given that BD ≅ FD and CD ≅ ED. You also know that ∠D ≅ ∠D by the reflexive property. Two sides and the included angle of △BDE are congruent to two sides and the included angle of △FDC. The triangles are congruent by the SAS Triangle Congruence Theorem. So, by CPCTC, the ¯ and are also congruent. ¯ and FC braces BE Module 5 251 E 251 Lesson 3 22/03/14 9:49 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-C;CA-C 10. An artist is framing a large picture and wants to put metal poles across the back to strengthen the frame as shown in the diagram. metal poles _If the _ _ are _both the same length and they bisect each other, prove that AB ≅ CD and AD ≅ CB. A AVOID COMMON ERRORS Students may choose the wrong angle when SAS is used to prove triangles congruent. Explain that the angle must be formed by the sides. The included angle is named by the letter the segments share. B E D C _ _ Because BD and_ AC bisect _ each other, AE = CE and BE = DE, _ _ so AE ≅ CE and BE ≅ DE by the definition of congruence. By the Vertical Angle Theorem, you also know that ∠AEB ≅ ∠CED. Two sides and the included angle of △AEB are congruent to two sides and the included angle of △CED. The triangles are congruent _ _ by the SAS Triangle Congruence Theorem. _ _ By CPCTC, AB ≅ CD. You can use similar reasoning to show that AD ≅ CB 11. The figure shows a side panel of a skateboard ramp. Kalim wants to confirm that the right triangles in the panel are congruent. C A D B a. What measurements should Kalim take if he wants to confirm that the triangles are congruent by SAS? Explain. _ _ Measure AB and DB; so _ he can_ confirm that two pairs of sides and their included _ _ angles are congruent. (AB ≅ DB, CB ≅ CB, and ∠ABC ≅ ∠DBC) Measure ∠ACB and ∠DCB; so he can confirm that two pairs and their included _of angles _ sides are congruent. (∠ACB ≅ ∠DCB, ∠ABC ≅ ∠DBC, and CB ≅ CB) Module 5 GE_MNLESE385795_U2M05L3.indd 252 252 © Houghton Mifflin Harcourt Publishing Company b. What measurements should Kalim take if he wants to confirm that the triangles are congruent by ASA? Explain. Lesson 3 6/9/15 3:08 AM SAS Triangle Congruence 252 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-C;CA-C INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2"TLTUVEFOUTUPDPOTJEFSUIFDIBOHFTJO H.O.T. Focus on Higher Order Thinking 26. Explain the Error Ava wants to know the distance JK across a pond. She locates points as shown. She says that the distance across the pond must be 160 ft by the SSS Triangle Congruence Theorem. Explain her error. L 210 ft N 160 ft 190 ft M The distance is 160 ft, but the justification should be 190 ft Given: ∠BFC ≅ ∠ECF, ∠BCF ≅ ∠EFC _ _ _ _ AB ≅ DE, AF ≅ DC Prove: △ABF ≅ △DEC K A B 1. ∠BFC ≅ ∠ECF, ∠BCF ≅ ∠EFC ― C F Statements ― E 1. Given 2. Reflexive Property of Congruence 3. △BFC ≅ △ECF 3. ASA Triangle Congruence Theorem 4. FB ≅ CE 4. CPCTC ― ― ―― ― D Reasons 2. FC ≅ FC ― Mike’s bench if it were to transform from the original shape to complete collapse. Which properties of the bench would change and how would they change? Which properties would not change? Sample answer: Change: During collapse, the area inside the parallelogram would decrease continually; the measures of the angles would change, with the measures of one pair of opposite angles increasing continually and the measures of the other pair decreasing continually. 210 ft the SAS Triangle Congruence Theorem. 27. Analyze Relationships Write a proof. J 5. AB ≅ DE, AF ≅ DC 5. Given 6. △ABF ≅ △DEC 6. SSS Triangle Congruence Theorem Not change: The lengths of the sides and the perimeter of the parallelogram would not change. AVOID COMMON ERRORS Students may argue that Mike’s bench may be solidly constructed with screws, nails, and glue, so that it would not collapse. Stress that the bench is a real-world object introduced here to model an abstract geometrical principle, and that it is not a perfect model. The important conclusion to draw is the one relating to quadrilaterals, not to benches. Lesson Performance Task Mike Michelle © Houghton Mifflin Harcourt Publishing Company Mike and Michelle each hope to get a contract with the city to build benches for commuters to sit on while waiting for buses. The benches must be stable so that they don’t collapse, and they must be attractive. Their designs are shown. Judge the two benches on stability and attractiveness. Explain your reasoning. Answers will vary. While Mike's bench will probably be chosen as the more attractive, students should note that Michelle's is more stable. There are many quadrilaterals with the same side lengths as Mike's bench. But, by the SSS Congruence Theorem, the triangles in Michelle's design cannot change shape. Module 5 266 Lesson 4 EXTENSION ACTIVITY GE_MNLESE385795_U2M05L4.indd 266 Have students construct models of Mike and Michelle’s benches from sturdy pieces of cardboard or photo-print paper cut into long, narrow strips and attached at the corners with brads. Students can explore further by making other polygons, checking their stability, and explaining why, for example, there is no SSSSS Pentagon Congruence Theorem. A 5-sided polygon is not stable. This means that there are many different shapes that can be constructed from the same 5 sides of a pentagon. 6/9/15 2:29 AM 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. SSS Triangle Congruence 266 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B Reflect EXPLAIN 2 3. Using Angle-Angle-Side Congruence Discussion The Third Angles Theorem says “If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent.” How could using this theorem simplify the proof of the AAS Congruence Theorem? Steps 2–8 could be simplified to one step, ∠B ≅ ∠E, making the whole proof only three steps. QUESTIONING STRATEGIES 4. How does the AAS Congruence Theorem differ from the ASA Congruence Theorem? While both the ASA and AAS Congruence Theorems require you to know that two pairs of corresponding angles are congruent between two triangles, the ASA Congruence Theorem requires you to also know that a pair of corresponding included sides are congruent. The AAS Congruence Theorem requires you to know also that a pair of corresponding non-included sides are congruent. Could the AAS Congruence Theorem be used in the proof? Explain. No, that is the theorem being proved, so it cannot be used until it has been proven. Explain 2 Example 2 Using Angle-Angle-Side Congruence Use the AAS Theorem to prove the given triangles are congruent. _ _ Given: AC ≅ EC and m‖n D E Prove: △ABC ≅ △EDC m C A B n AC ≅ EC If AAS is used as the method of proof, can the triangles also be proved congruent using ASA? Yes; you can use the Third Angles Theorem to show the third pair of angles is congruent, and then you can use ASA. Given mǁn Given ∠E ≅ ∠A △ABC ≅ △EDC Alt. Int ! Thm. AAS Cong. Thm. ∠B ≅ ∠D © Houghton Mifflin Harcourt Publishing Company Alt. Int ! Thm. _ _ _ _ _ _ Given: CB ‖ ED, AB ‖ CD, and CB ≅ ED. A Prove: △ABC ≅ △CDE CB ≅ ED C B E D Given CB ǁ ED ∠ACB ≅ ∠CED △ABC ≅ △CDE Given Corr. ! Thm. AAS Cong. Thm. AB ǁ CD ∠CAB ≅ ∠ECD Given Corr. ! Thm. Module 6 285 Lesson 2 COLLABORATIVE LEARNING GE_MNLESE385795_U2M06L2 285 Small Group Activity Students may benefit from exploring conditions that do not guarantee congruent triangles. Have students work in small groups and use a protractor to draw triangles with angles that measure 35°, 50°, and 95°. Then then compare their triangles within each group. Students will see that all the triangles have the same shape, but not necessarily the same size. Since the triangles are not all congruent, the activity shows that there is no AAA Congruence Criteria. 285 Lesson 6.2 5/22/14 9:04 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-C;CA-C Your Turn 5. EXPLAIN 3 _ _ _ _ Given: ∠ABC ≅ ∠DEF, BC ∥ EF, AC ≅ DF. Use the AAS Theorem to prove the triangles are congruent. Applying Angle-Angle-Side Congruence Write a paragraph proof. E B AVOID COMMON ERRORS C A F D The real-world situation may distract some students. Make sure they are certain what they are solving for before they begin. ¯ is parallel to EF ¯, this means that ∠ACB is congruent to ∠DFE, Because BC using the Corresponding Angles Theorem. Since ∠ABC is congruent to ¯ is congruent to DF ¯, then one pair of corresponding angles ∠DEF and AC and two pairs of non-included corresponding sides are congruent. This means that △ABC is congruent to △DEF using the AAS Triangle Congruence Theorem. Applying Angle-Angle-Side Congruence Explain 3 The triangular regions represent plots of land. Use the AAS Theorem to explain why the same amount of fencing will surround either plot. Example 3 Given: ∠A ≅ ∠D It is given that ∠A ≅ ∠D. Also, ∠B ≅ ∠E because both are right angles. Compare AC and DF using the Distance Formula. 1 2 2 1 y D E 2 -2 0 2 F 4 x -2 -4 2 © Houghton Mifflin Harcourt Publishing Company 2 A -4 2 2 4 2 ―――――――― AC = √(x - x ) + (y - y ) ――――――――― = √(-1-(-4)) + (4 - 0) ――― = √3 + 4 ― = √25 2 C B =5 ―――――――― ――――――― = √(4 - 0) + (1 - 4) ―――― = 4 + (-3) ― = √25 DF = √(x 2 - x 1) 2 + (y 2 - y 1) 2 2 2 2 2 =5 Because two pairs of angles and a pair of non-included sides are congruent, △ABC ≅ △DEF by AAS. Therefore the triangles have the same perimeter and the same amount of fencing is needed. Module 6 286 Lesson 2 DIFFERENTIATE INSTRUCTION GE_MNLESE385795_U2M06L2 286 Auditory Cues 6/16/15 10:23 AM As students work through the lesson, have them identify orally whether the triangles are congruent by ASA, AAS, SAS, or SSS. Then have them work with a partner to identify the congruent parts and to determine if the congruent pairs of angles or sides are corresponding parts. AAS Triangle Congruence 286 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B 4. QUESTIONING STRATEGIES △GKJ and △JHG G H 5. Yes. You cannot use the HL Triangle K J Congruence Theorem since it is not known whether the triangles are right triangles, but you can use the SSS Triangle Congruence Theorem. What type of triangle must be given to use HL as a method of proof? It must be a right triangle. △EFG and △SQR F R E S G Q Yes. You can use Pythagorean Theorem _ _ the to show that FG ≅ QR and then use the HL Triangle Congruence Theorem, or you can use the SAS Triangle Congruence Theorem with the given information. Write a two-column proof, using the HL Congruence Theorem, to prove that the triangles are congruent. 6. ― ― Given: ∠A and ∠B are right angles. AB ≅ DC Prove: △ABC ≅ △DCB A D B C Statements Reasons 1. ∠A and ∠B are right angles. _ _ 2. AB ≅ DC _ _ 3. BC ≅ BC 4. △ABC ≅ △DCB 7. 1. Given 2. Given 3. Reflexive Property of Congruence 4. HL Triangle Congruence Theorem Given: ∠FGH and ∠JHK are right _ _ _ angles. H is the midpoint of GK. FH ≅ JK Prove: △FGH ≅ △JHK G © Houghton Mifflin Harcourt Publishing Company Statements 2. H is the midpoint of GK. _ _ 3. GH ≅ HK _ _ 4. FH ≅ JK 3. Definition of midpoint 4. Given 5. △FGH ≅ △JHK 5. HL Triangle Congruence Theorem ― 2. Given ― to QR. Given: MP is perpendicular ―_ ― N is the midpoint of MP. QP ≅ RM Prove: △MNR ≅ △PNQ M _ _ MP ⊥ QR so ∠QNP and ∠MNR are right angles (definition of perpendicular). N is the midpoint _ _ PN (definition midpoint). of MP, so MN ≅ ¯ _ of _ Then, since it is given that QP ≅ RM , △MNR ≅ △PNQ by the HL Triangle Congruence Theorem. Exercise GE_MNLESE385795_U2M06L3.indd 299 Lesson 6.3 Reasons 1. Given Module 6 299 K H 1. ∠FGH and ∠JHK are right angles. ― 8. J F Q N P R 299 Lesson 3 Depth of Knowledge (D.O.K.) Mathematical Practices 23/05/14 7:18 PM 22 3 Strategic Thinking MP.2 Reasoning 23 3 Strategic Thinking MP.2 Reasoning DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A 20° ladd er The house is perpendicular to the ground, so the other remote interior angle is 90°. 20 + 90 = 110, so the measure of the indicated exterior angle is 110°. house 18. A ladder propped up against a house makes a 20° angle with the wall. What would be the ladder's angle measure with the ground facing away from the house? ? ground 19. Photography The aperture of a camera is made by overlapping blades that form a regular decagon. a. What is the sum of the measures of the interior angles of the decagon? (10 - 2)180° = (8)180° = 1440° b. What would be the measure of each interior angle? each exterior angle? 1440° ÷ 10 = 144°; 180° - 144° = 36° c. Find the sum of all ten exterior angles. 36°(10) = 360° 20. Determine the measure of ∠UXW in the diagram. m∠WUX = 90° Y V © Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: ©neyro2008/iStockPhoto.com U m∠UXW = 36° W Z 21. Determine the measures of angles x, y, and z. x = 180 - (100 + 60) = 20° 80° y = 180 - (80 + 55) = 45° 100° 55° GE_MNLESE385795_U2M07L1.indd 323 Lesson 7.1 78° 54° X Module 7 323 180° = 54° + 90° + m∠UXW x° z° y° 60° z = 180 - (20 + 45) = 115° 323 Lesson 1 20/03/14 12:28 PM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B _ 19. Critical Thinking Prove ∠B ≅ ∠C, given point M is the midpoint of BC. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Remind students that an isosceles triangle has A B Statements _ 1. M is the midpoint of BC. ― ― C M at least two congruent sides, and its properties can be used to prove that it also has at least two congruent angles. Reasons 1. Given 2. BM ≅ CM 2. Definition of midpoint ― ― 4. AM ≅ AM 4. Reflexive Property of Congruence 5. △AMB ≅ △AMC 5. SSS Triangle Congruence Theorem 6. ∠B ≅ ∠C 6. CPCTC ― ― 3. AB ≅ AC 3. Given _ _ 20. Given that △ABC is an isosceles triangle and AD and CD are angle bisectors, what is m∠ADC? B m∠BAC = m∠BCA = 70°, so m∠DAC = m∠DCA = 35°. Then, m∠ADC = 180° − (35° + 35°) = 110°. 40° D A C H.O.T. Focus on Higher Order Thinking The triangles are congruent isosceles triangles; the bisected right angle results in two 45° angles, and the perpendicular segments result in two right angles, so angles A and C must also measure 45°. Since ¯ BD ≅ ¯ BD by the Reflexive Property, triangles ABD and CBD are congruent by ASA. A D B Module 7 GE_MNLESE385795_U2M07L2 338 © Houghton Mifflin Harcourt Publishing Company 21. Analyze Isosceles right triangle_ ABC has _ _Relationships _ angle at B and _ _ a right AB ≅ CB. BD bisects angle B, and point D is on AC. If BD ⟘ AC, describe triangles ABD and CBD. Explain. HINT: Draw a diagram. C 338 Lesson 2 5/22/14 4:55 PM Isosceles and Equilateral Triangles 338