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.

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.

[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?
- The sum of the angles on a straight line is 180°
- [math]ang ABF[/math] and [math]ang BFE[/math] are complimentary
- Right angles are congruent
- [math]ang AFB ~= ang BFE[/math]
B.
What is the missing statement in step 4?
- [math]m ang AFB + m ang BFE = m ang AFE[/math]
- [math]90° + 90° = 180°[/math]
- [math]m ang AFB + 90° = 180°[/math]
- [math]90° + m ang BFE = 180°[/math]
C.
What is the missing statement in step 8?
- [math]ang AFB ~= ang BFE[/math]
- [math]ang AFB ~= ang EDC[/math]
- [math]ang BAF ~= ang EDC[/math]
- [math]ang BAF ~= ang ECD[/math]
D.
What is the missing reason in step 12?
- If 2 parallel lines are cut by a transversal, alternate angles are supplementary
- If 2 parallel lines are cut by a transversal, consecutive exterior angles are supplementary
- If 2 parallel lines are cut by a transversal, alternate interior angles are congruent
- If 2 parallel lines are cut by a transversal, consecutive interior angles are supplementary
E.
What is the missing reason in step 16?
- The sum of four non-obtuse angles is 360°
- The sum of the interior angles of any two dimensional figure is 360°
- Four angles make a full rotation, which is 360°
- 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).

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).

[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?
- [math]Delta BCD \ ~ \ Delta CDE[/math]
- [math]Delta BCD \ ~ \ Delta CED[/math]
- [math]Delta BCD \ ~ \ Delta DEC[/math]
- [math]Delta BCD \ ~ \ Delta DCE[/math]
B.
What is the missing reason in step 11?
- If two parallel lines are cut by a transversal, alternate exterior angles are congruent
- If two parallel lines are cut by a transversal, alternate interior angles are congruent
- If two parallel lines are cut by a transversal, alternate angles are congruent
- If two parallel lines are cut by a transversal, corresponding angles are congruent
C.
What is the missing reason in step 13?
- Given
- Definition of supplementary angles
- The sum of two right angles is 180°
- The sum of any two angles of a rectangle is 180°
D.
What is the missing reason in step 16?
- The sum of the exterior angle and its corresponding interior angle is 180°
- The sum of adjacent angles which are also equal is 180°
- The sum of the angles on a straight line is 180°
- The sum of two right angles is 180°
E.
What is the missing reason in step 22?
- SAS Congruence Theorem
- AAS Congruence Theorem
- HL Congruence Theorem
- ASA Congruence Theorem
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