* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Document
Survey
Document related concepts
Rational trigonometry wikipedia , lookup
Integer triangle wikipedia , lookup
Trigonometric functions wikipedia , lookup
Atiyah–Singer index theorem wikipedia , lookup
Line (geometry) wikipedia , lookup
Euler angles wikipedia , lookup
History of geometry wikipedia , lookup
Noether's theorem wikipedia , lookup
Four color theorem wikipedia , lookup
History of trigonometry wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Transcript
Answers for the lesson “Show that a Quadrilateral is a Parallelogram” LESSON 8.3 12. Skill Practice y B 1. The definition of a parallelogram is that it is a quadrilateral with opposite pairs of parallel sides. Since } AB, } CD, and } AD, } BC are opposite pairs of parallel sides, quadrilateral ABCD is a parallelogram. 1 21 A C 2. Yes; both pairs of opposite sides D are congruent. Sample answer: AB} 5 CD 5 4 and BC 5 DA 5 Ï61 3. The congruent sides must be opposite one another. x 13. y B C 4. Theorem 8.8 5. Theorem 8.7 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 6. Theorem 8.10 A 7. Since both pairs of opposite sides of JKLM always remain congruent, JKLM is always a parallelogram and } JK remains } parallel to ML. 8. 4 2 10. } 3 9. 8 11. D 1 22 x Sample answer: AB} 5 CD 5 5 and BC 5 DA 5 Ï65 14. B y y C B 1 A 2 A 22 21 C x D x Sample answer: AB 5 CD 5 5 and BC 5 DA 5 8 D } Sample answer: AB 5 CD 5 Ï 41 and BC 5 DA 5 5 Geometry Answer Transparencies for Checking Homework 233 15. Sample answer: Show nADB > nCBD using the SAS Congruence Postulate. This makes } AD > } CB and } BA > } CD using corresponding parts of congruent triangles are congruent. 16. Sample answer: Show nADB > nCBD using the ASA Congruence Theorem. This makes } > CD } and AD } > CB } using AB corresponding parts of congruent triangles are congruent. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 17. Sample answer: Show } AB i } DC by the Alternate Interior Angles Converse, and show } AD i } BC by the Corresponding Angles Converse. 18. A 19. 114 20. 45 21. 50 22. A quadrilateral is a parallelogram if and only if both pairs of opposite sides are congruent. 23. A quadrilateral is a parallelogram if and only if both pairs of opposite angles are congruent. 25. (23, 2); since } DA must be parallel and congruent to } BC, use } the slope and length of BC to find point D by starting at point A. 26. (3, 1); since } AB must be parallel and congruent to } CD, use } the slope and length of AB to find point D by starting at point C. 27. (25, 23); since } DA must be parallel and congruent to } BC, use } the slope and length of BC to find point D by starting at point A. 28. (7, 22); since } AB must be parallel and congruent to } CD, use } the slope and length of AB to find point D starting at point C. 29. Sample answer: Draw a line passing through points A and B. ]› At points A and B, construct AP ]› such that the angle each and BQ ray makes with the line is the same. Mark off congruent segments starting at points A and ]› and BQ ]›, respectively. B along AP Draw the line segment joining these two endpoints. 24. Joining the endpoints of the two line segments that bisect one another forms a quadrilateral. Using Theorem 8.10 you know that it is a parallelogram. Q P A B Geometry Answer Transparencies for Checking Homework 234 30. 8; since ABCD is a parallelogram you know that } AD > } CB and } } AD i CB. ADE > CBF using 34. the Alternate Interior Angles Congruence Theorem. It was The point of intersection of given that } BF > } DE, therefore the diagonals is not necessarily nADE > nCBF by the SAS their midpoint. Congruence Theorem. Using corresponding parts of congruent 35. triangles are congruent, you know that CF 5 EA 5 8. Problem Solving 31. a. EFJK, FGHJ, EGHK; in each The opposite sides that are not marked in the given diagram are not necessarily the same length. case opposite pairs of sides are congruent. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. b. Since EGHK is a parallelogram, opposite sides are parallel. 36. 8 8 32. AEFD and EBCF are parallelograms by Theorem 8.8 so } AD and } BC both remain parallel } } to EF; AE and } DF, } BE and } CF. The sides of length 8 are not necessarily parallel. 33. 1st column: Alternate Interior Angles Congruence Theorem; Reflexive Property of Segment Congruence; Given 2nd column: SAS; Corr. Parts of s are >; Theorem 8.7 >n Geometry Answer Transparencies for Checking Homework 235 37. In a quadrilateral, if consecutive angles are supplementary, then the quadrilateral is a parallelogram. In ABCD you are given A and B are supplementary, and C and B are supplementary, which gives you mA 5 mC. Also B and C are supplementary, and C and D are supplementary which gives you mB 5 mD. So ABCD is a parallelogram by Theorem 8.8. B Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. A C D 38. It is given that A > C and B > D. Let mA 5 mC 5 x and mB 5 mD 5 y. Since ABCD is a quadrilateral, you know that 2x 1 2y 5 360 using the Polygon Interior Angles Theorem, so x 1 y 5 180. Using the definition of supplementary angles, A and B, B and C, C and D, and D and A are supplementary. Using Theorem 8.5, ABCD is a parallelogram. } > MP } and 39. It is given that KP } > LP } by definition of segment JP bisector. KPL > MPJ and KPJ > MPL since they are vertical angles. nKPL > nMPJ and nKPJ > nMPL by the SAS Congruence Postulate. Using corresponding parts of congruent triangles are congruent, } } and JM } > LK }. Using KJ > ML Theorem 8.7, JKLM is a parallelogram. 40. It is given that DEBF is a parallelogram and AE 5 CF. Since DEBF is a parallelogram, you know that FD 5 EB, BFD > DEB, and ED 5 FB. AE 1 EB 5 CF 1 FD which implies that AB 5 CD, which } > CD }. BFC and implies that AB BFD, and DEB and DEA form linear pairs, thus making them supplementary. Using the Congruent Supplements Theorem, BFC > DEA making nAED > nCFB using SAS. Using corresponding parts of congruent triangles are congruent, } }. Theorem 8.7 tells you AD > CB that ABCD is a parallelogram. Geometry Answer Transparencies for Checking Homework 236 41. F B E C 2. The sum of the interior angles of G 3. x 5 4, y 5 4; the diagonals of a A H D Sample answer: Consider the } is the midsegment diagram. FG of nCBD and therefore is parallel } and half of its length. EH } to BD is the midsegment of nABD and } therefore is parallel to BD and half of its length. This makes } both parallel and } and FG EH congruent. Using Theorem 8.9, EFGH is a parallelogram. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. ABCDE is 5408. To find mA and mC, subtract 2708 from 5408 and divide by 2. parallelogram bisect each other. Set 12x 1 1 5 49 and 8y 1 4 5 36 and solve for x and y. 4. 85; 8 5 } is the midsegment of nAED 42. FJ } and and therefore is parallel to AD } is the half of its length. GH midsegment of nBEC and } and therefore is parallel to BC half of its length. Together this } and FJ }. } > GH } i GH gives you FJ Using Theorem 8.9, FGHJ is a parallelogram. Mixed Review of Problem Solving 1. a. 5; pentagon 5. In a parallelogram consecutive interior angles are supplementary. Solve x 1 3x 2 12 5 180 for x. Use the value of x to find the degree measure of the two consecutive angles. In a parallelogram the angle opposite each of these known angles has the same measure. b. 5408 c. 3608 Geometry Answer Transparencies for Checking Homework 237 6. a. EFGH is a parallelogram by Theorem 8.9. As EFGH changes the shape, E and G remain congruent, and H and F remain congruent, keeping } FG i } EH. b. mE and mG decrease from Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 558 to 508. mH and mF change from 1258 to 1308. Since EFGH is always a parallelogram, mF and mH are always the same and mE and mG are always the same. Since mE is always supplementary to the mF, their sum must be 1808. Alternate Interior Angles Congruence Theorem, } > CD } since BAX > DCY. AB opposite sides of a parallelogram are congruent. nBXA > nDYC by the AAS Congruence Theorem, making } } using corresponding BX > DY parts of congruent triangles are congruent. Use Theorem 8.9 to show XBYD is a parallelogram. 7. a. Since the slope of } MN and } PQ 3 is } and the slope of } NP and 11 5 } QM is 2}4 , then it is a parallelogram by definition. } b. MN 5 PQ 5 Ï130 making } and NP 5 QM 5 } > PQ MN } }. Using } > QM Ï41 making NP the Theorem 8.3, MNPQ is a parallelogram. } i DY } using the Lines 8. BX Perpendicular to a Transversal } and } > AC Theorem. Since BX }, then BXA and DYC }> AC DY are right angles making BXA > DYC. Using the Geometry Answer Transparencies for Checking Homework 238