Proofs Involving Geometric Figures (Grade 10)

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Proofs Involving Geometric Figures

1. 
For the following proof, choose the correct missing statements and reasons in the questions below.

Given that [math]Delta AFB ~= Delta EDC[/math], [math]bar{BC} " || " bar{FD}[/math], [math]m ang BFE = 90°[/math], and points [math]A,F,E[/math] are collinear, prove that quadrilateral [math]BCDF[/math] is a rectangle.

Trapezoid ABCDEF

[math] \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{"Statement"} \ \ \ \ \ \ \ \ \ \ \ \ [/math][math]
\mathbf{"Reason"}[/math]
[math]1. m ang BFE = 90° [/math][math]1. "Given"[/math]
[math]2. A,F,E " are collinear"[/math][math]2. "Given"[/math]
[math]3. m ang AFB + m ang BFE = 180°[/math][math]3. ""[/math]
[math]4. [/math][math]4. "Substitution Property of Equality"[/math]
[math]5. m ang AFB = 180° - 90°[/math][math]5. "Subtraction Property of Equality"[/math]
[math]6. m ang AFB = 90°[/math][math]6. "Algebra (subtract)"[/math]
[math]7. Delta AFB ~= Delta EDC[/math][math]7. "Given"[/math]
[math]8. [/math][math]8. "Corr. angles of congruent triangles congruent"[/math]
[math]9. m ang AFB = m ang EDC[/math][math]9. "Definition congruent angles" [/math]
[math]10. 90° = m ang EDC [/math][math]10. "Substitution Property of Equality"[/math]
[math]11. bar{BC} " || " bar{DF} [/math][math]11. "Given"[/math]
[math]12. m ang EDC + m ang BCD = 180°[/math][math]12. ""[/math]
[math]13. 90° + m ang BCD = 180°[/math][math]13. "Substitution Property of Equality"[/math]
[math]14. m ang BCD = 180° - 90°[/math][math]14. "Subtraction Property of Equality"[/math]
[math]15. m ang BCD = 90°[/math][math]15. "Algebra (subtract)"[/math]
[math]16. m ang BFE + m ang EDC + m ang BCD + [/math] [math] \ \ \ \ m ang CBF = 360°[/math][math]16. ""[/math]
[math]17. 90° + 90° + 90° + m ang CBF = 360°[/math][math]17. "Substitution Property of Equality"[/math]
[math]18. 270° + m ang CFB = 360°[/math][math]18. "Algebra (add)"[/math]
[math]19. m ang CFB = 360° - 270°[/math][math]19. "Subtraction Property of Equality"[/math]
[math]20. m ang CFB = 90°[/math][math]20. "Algebra (subtract)"[/math]
[math]21. ang BFE, \ ang EDC, \ ang BCD, \ ang CBF [/math] [math] \ \ \ " are right angles" [/math][math]21. "Definition of right angles"[/math]
[math]22. "Quad. " BCDF " is a rectangle" [/math][math]22. "Quad. with 4 right angles is a rectangle"[/math]
A. 
What is the missing reason in step 3?
  1. The sum of the angles on a straight line is 180°
  2. [math]ang ABF[/math] and [math]ang BFE[/math] are complimentary
  3. Right angles are congruent
  4. [math]ang AFB ~= ang BFE[/math]
B. 
What is the missing statement in step 4?
  1. [math]m ang AFB + m ang BFE = m ang AFE[/math]
  2. [math]90° + 90° = 180°[/math]
  3. [math]m ang AFB + 90° = 180°[/math]
  4. [math]90° + m ang BFE = 180°[/math]
C. 
What is the missing statement in step 8?
  1. [math]ang AFB ~= ang BFE[/math]
  2. [math]ang AFB ~= ang EDC[/math]
  3. [math]ang BAF ~= ang EDC[/math]
  4. [math]ang BAF ~= ang ECD[/math]
D. 
What is the missing reason in step 12?
  1. If 2 parallel lines are cut by a transversal, alternate angles are supplementary
  2. If 2 parallel lines are cut by a transversal, consecutive exterior angles are supplementary
  3. If 2 parallel lines are cut by a transversal, alternate interior angles are congruent
  4. If 2 parallel lines are cut by a transversal, consecutive interior angles are supplementary
E. 
What is the missing reason in step 16?
  1. The sum of four non-obtuse angles is 360°
  2. The sum of the interior angles of any two dimensional figure is 360°
  3. Four angles make a full rotation, which is 360°
  4. The sum of the interior angles in a quadrilateral is 360°
2. 
For the following proof, choose the correct missing statement and reasons in the questions below.

In the following figure, [math]bar{AB} " || " bar{CE}[/math], [math]ang BFE[/math] and [math]ang CDE[/math] are supplementary, and points [math]A,F,E[/math] are collinear. Also, [math]ang BCD ~= ang CDE[/math] and [math]bar{AF} ~= bar{ED}[/math]. If [math]BC = 2CD[/math] and [math]CD = 2 ED[/math], prove that [math]BD*BF = AB*BC[/math] ([math]bar{BD}[/math] is not drawn).

Trapezoid ABCDEF

[math] \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{"Statement"} \ \ \ \ \ \ \ \ \ \ \ \ [/math][math]
\mathbf{"Reason"}[/math]
[math]1. ang BCD ~= ang CDE [/math][math]1. "Given"[/math]
[math]2. BC = 2CD[/math][math]2. "Given"[/math]
[math]3. (BC)/(CD) = 2[/math][math]3. "Division Property of Equality"[/math]
[math]4. CD = 2ED [/math][math]4. "Given"[/math]
[math]5. (CD)/(ED) = 2[/math][math]5. "Division Property of Equality"[/math]
[math]6. (BC)/(CD) = (CD)/(ED)[/math][math]6. "Transitive Property of Equality"[/math]
[math]7. [/math][math]7. "SAS Similarity Theorem"[/math]
[math]8. (BD)/(CE) = (BC)/(CD)[/math][math]8. "Ratios corr. sides of similar triangles are equal" [/math]
[math]9. bar{AF} ~= bar{ED} [/math][math]9. "Given"[/math]
[math]10. bar{AB} " || " bar{CE} [/math][math]10. "Given"[/math]
[math]11. ang BAF ~= ang CED[/math][math]11. ""[/math]
[math]12. ang BFE " and " ang CDE " are supplementary"[/math][math]12. "Given"[/math]
[math]13. m ang BFE + m ang CDE = 180°[/math][math]13. ""[/math]
[math]14. m ang BFE = 180° - m ang CDE[/math][math]14. "Subtraction Property of Equality"[/math]
[math]15. A, F, E " are collinear"[/math][math]15. "Given"[/math]
[math]16. m ang AFB + m ang BFE = 180°[/math][math]16. ""[/math]
[math]17. m ang BFE = 180° - m ang AFB[/math][math]17. "Subtraction Property of Equality"[/math]
[math]18. 180° - m ang CDE = 180° - m ang ABF[/math][math]18. "Transitive Property of Equality"[/math]
[math]19. - m ang CDE = - m ang AFB[/math][math]19. "Subtraction Property of Equality"[/math]
[math]20. m ang CDE = m ang AFB[/math][math]20. "Multiplication Property of Equality"[/math]
[math]21. ang CDE ~= ang AFB [/math][math]21. "Definition of Congruent Angles"[/math]
[math]22. Delta AFB ~= Delta EDC [/math][math]22. ""[/math]
[math]23. bar{AB} ~= bar{CE} [/math][math]23. "Corr. sides of congruent triangles congruent"[/math]
[math]24. AB = CE [/math][math]24. "Definition of Congruent Segments"[/math]
[math]25. bar{BF} ~= bar{CD} [/math][math]25. "Corr. sides of congruent triangles congruent"[/math]
[math]26. BF = CD [/math][math]26. "Definition of Congruent Segments"[/math]
[math]27. (BD)/(CE) = (BC)/(CD)[/math][math]27. "Earlier result" [/math]
[math]28. (BD)/(AB) = (BC)/(BF) [/math][math]28. "Substitution Property of Equality"[/math]
[math]29. BD*BF = AB*BC [/math][math]29. "Multiplication Property of Equality"[/math]
A. 
What is the missing statement in step 7?
  1. [math]Delta BCD \ ~ \ Delta CDE[/math]
  2. [math]Delta BCD \ ~ \ Delta CED[/math]
  3. [math]Delta BCD \ ~ \ Delta DEC[/math]
  4. [math]Delta BCD \ ~ \ Delta DCE[/math]
B. 
What is the missing reason in step 11?
  1. If two parallel lines are cut by a transversal, alternate exterior angles are congruent
  2. If two parallel lines are cut by a transversal, alternate interior angles are congruent
  3. If two parallel lines are cut by a transversal, alternate angles are congruent
  4. If two parallel lines are cut by a transversal, corresponding angles are congruent
C. 
What is the missing reason in step 13?
  1. Given
  2. Definition of supplementary angles
  3. The sum of two right angles is 180°
  4. The sum of any two angles of a rectangle is 180°
D. 
What is the missing reason in step 16?
  1. The sum of the exterior angle and its corresponding interior angle is 180°
  2. The sum of adjacent angles which are also equal is 180°
  3. The sum of the angles on a straight line is 180°
  4. The sum of two right angles is 180°
E. 
What is the missing reason in step 22?
  1. SAS Congruence Theorem
  2. AAS Congruence Theorem
  3. HL Congruence Theorem
  4. ASA Congruence Theorem

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