Parallelogram Related Proofs (Grade 10)

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Parallelogram Related Proofs

1. 
What is the missing reason in step 4 of the following proof?

Given: Quadrilateral [math]ABCD[/math] where [math]bar{AD}[/math] || [math]bar{BC}[/math] and [math]bar{AD} ~= bar{BC}[/math]
Prove: [math]bar{AB} " || " bar{CD}[/math]

Parallelogram ABCD v3

[math] \ \ \ \ \ \ \ \ \ \ \ \ " Statement " \ \ \ \ \ \ \ \ \ \ \ \ [/math][math] " Reason "[/math]
[math]1. bar{AD} " || " bar{BC} [/math][math]1. "Given"[/math]
[math]2. ang CBD ~= ang ADB [/math][math]2. "Alternate interior angles are congruent"[/math]
[math]3. bar{BC} ~= bar{AD}[/math][math]3. "Given"[/math]
[math]4. bar{BD} ~= bar{BD} [/math][math]4. ""[/math]
[math]5. Delta ABD ~= Delta CDB [/math][math]5. "Side-Angle-Side Postulate"[/math]
[math]6. ang ABD ~= ang CDB [/math][math]6. "Corresponding angles of congruent triangles are congruent"[/math]
[math]7. bar{AB} " || " bar{CD}[/math][math]7. "If two lines are cut by a transversal, and alternate interior"[/math] [math] \ \ \ \ \ " angles are congruent, the lines are parallel"[/math]
  1. Reflexive Property
  2. Diagonals of a parallelogram are congruent
  3. Given by the diagram
  4. Corresponding sides of congruent triangles are congruent
2. 
What is the missing statement in step 6 of the following proof?

Given: Quadrilateral [math]ABCD[/math] with diagonal [math]bar{BD}[/math] where [math]bar{AB}[/math] || [math]bar{CD}[/math] and [math]bar{BC}[/math] || [math]bar{AD}[/math]
Prove: [math]bar{AB} ~= bar{CD}[/math]

Parallelogram ABCD v4

[math] \ \ \ \ \ \ \ \ \ \ \ \ " Statement " \ \ \ \ \ \ \ \ \ \ \ \ [/math][math] " Reason "[/math]
[math]1. bar{AB} [/math] || [math] bar{CD} [/math][math]1. "Given"[/math]
[math]2. ang ABD ~= ang CDB[/math][math]2. "Alternate interior angles are congruent"[/math]
[math]3. bar{BC} [/math] || [math] bar{AD}[/math][math]3. "Given"[/math]
[math]4. ang ADB ~= ang CBD [/math][math]4. "Alternate interior angles are congruent"[/math]
[math]5. bar{BD} ~= bar{BD} [/math][math]5. "Reflexive Property"[/math]
[math]6. [/math][math]6. "Angle-Side-Angle Postulate"[/math]
[math]7. bar{AB} ~= bar{CD} [/math][math]7. "Corresponding sides of congruent triangles are congruent"[/math]
  1. [math] Delta ABC ~= Delta ACD [/math]
  2. [math] Delta ABD ~= Delta CDB [/math]
  3. [math] Delta ABC ~= Delta CDA [/math]
  4. [math] Delta ABD ~= Delta DCB [/math]
3. 
What is the missing reason in step 8 of the following proof?

Given: Quadrilateral [math]ABCD[/math] with diagonal [math]bar{BD}[/math] where [math]bar{AB}[/math] || [math]bar{CD}[/math] and [math]bar{BC}[/math] || [math]bar{AD}[/math]
Prove: [math]ang BAD ~= ang DCB[/math]

Parallelogram ABCD v3

[math] \ \ \ \ \ \ \ \ \ \ \ \ " Statement " \ \ \ \ \ \ \ \ \ \ \ \ [/math][math] " Reason "[/math]
[math]1. bar{AB} [/math] || [math] bar{CD} [/math][math]1. "Given"[/math]
[math]2. ang ABD ~= ang CDB[/math][math]2. "Alternate interior angles are congruent"[/math]
[math]3. bar{BC} [/math] || [math] bar{AD}[/math][math]3. "Given"[/math]
[math]4. ang ADB ~= ang CBD [/math][math]4. "Alternate interior angles are congruent"[/math]
[math]5. m ang ABD = m ang CDB, \ \ m ang ADB = m ang CBD [/math][math]5. "Definition congruent angles"[/math]
[math]6. m ang ABD + m ang ADB + m ang BAD = 180°,[/math] [math] \ \ \ \ \ \ \ \ \ m ang CDB + m ang DCB + m ang CBD = 180° [/math][math]6. "Sum of angles in a triangle is 180°"[/math]
[math]7. m ang CDB + m ang DCB + m ang CBD = [/math] [math] \ \ \ \ \ \ \ \ \ \ m ang ABD + m ang ADB + m ang BAD [/math][math]7. "Substitution Property of Equality"[/math]
[math]8. m ang DCB = m ang BAD [/math][math]8. ""[/math]
[math]9. ang BAD ~=ang DCB [/math][math]9. "Definition of congruent angles"[/math]
  1. Elimination Property of Equality
  2. Corresponding angles are congruent
  3. Definition of congruent terms
  4. Subtraction Property of Equality
4. 
What is the missing statement in step 4 in the following proof?

Given that [math]ABCD[/math] is a rectangle, prove that [math]bar{AC}~=bar{BD} \ \ [/math] ([math]bar{AC}, bar{BD}[/math] not pictured).

Rectangle ABCD

[math] \ \ \ \ \ \ \ \ \ \ \ \ " Statement " \ \ \ \ \ \ \ \ \ \ \ \ [/math][math] " Reason "[/math]
[math]1. ABCD " is a rectangle"[/math][math]1. "Given"[/math]
[math]2. bar{AB} ~= bar{CD} [/math][math]2. "Opposite sides of a rectangle are congruent"[/math]
[math]3. ang ABC, ang BCD " are right angles"[/math][math]3. "Interior angles of rectangles are all right angles"[/math]
[math]4. [/math][math]4. "All right angles are congruent"[/math]
[math]5. bar{BC} ~= bar{BC} [/math][math]5. "Reflexive Property"[/math]
[math]6. Delta ABC ~= Delta DCB[/math][math]6. "Side-Angle-Side Postulate"[/math]
[math]7. bar{AC} ~= bar{BD}[/math][math]7. "Corresponding sides of congruent triangles are congruent"[/math]
  1. [math] ang ABC ~= ang CDA[/math]
  2. [math] ang BAD ~= ang BCD [/math]
  3. [math] ang BAD ~= ang CDA [/math]
  4. [math] ang ABC ~= ang BCD [/math]
5. 
Given that [math]ABCD[/math] is a parallelogram and [math]bar{BD}~=bar{AC}[/math] (line segments not pictured) prove that [math]ABCD[/math] is a rectangle.

Rectangle ABCD

[math] \ \ \ \ \ \ \ \ \ \ \ \ " Statement " \ \ \ \ \ \ \ \ \ \ \ \ [/math][math] " Reason "[/math]
[math]1. bar{AC} ~= bar{BD} [/math][math]1. "Given"[/math]
[math]2. ABCD " is a parallelogram"[/math][math]2. "Given"[/math]
[math]3. bar{AB} ~= bar{DC} [/math][math]3. "Opposite sides of a parallelogram are congruent"[/math]
[math]4. bar{AD} ~= bar{AD} [/math][math]4. "Reflexive Property"[/math]
[math]5. [/math][math]5. "Side-Side-Side Postulate"[/math]
[math]6. ang BAD ~= ang CDA[/math][math]6. "Corresponding angles of congruent" [/math] [math] \ \ \ \ \ \ \ " triangles are congruent"[/math]
[math]7. bar{AB} " || " bar{CD} [/math][math]7. "Opposite sides of a parallelogram are parallel"[/math]
[math]8. m ang BAD + m ang CDA = 180°[/math][math]8. ""[/math]
[math]9. m ang BAD = m ang CDA [/math][math]9. "Definition of congruent angles"[/math]
[math]10. m ang BAD + m ang BAD = 180° [/math][math]10. ""[/math]
[math]11. 2m ang BAD = 180° [/math][math]11. "Distributive Property"[/math]
[math]12. m ang BAD = 90° [/math][math]12. "Division Property of Equality"[/math]
[math]13. m ang CDA = 90° [/math][math]13. m ang CDA = m ang BAD[/math]
[math]14. [/math][math]14. "Definition of right angles"[/math]
[math]15. bar{BC} " || " bar{AD} [/math][math]15. ""[/math]
[math]16. m ang BAD + m ang ABC = 180°, [/math] [math] \ \ \ \ \ m ang ADC + m ang DCB = 180° [/math][math]16. "Same side interior angles are supplementary"[/math]
[math]17. 90° + m ang ABC = 180°, [/math] [math] \ \ \ \ \ 90° + m ang DCB = 180° [/math][math]17. "Substitution Property of Equality"[/math]
[math]18. m ang ABC = 90°, [/math] [math] \ \ \ \ \ m ang DCB = 90° [/math][math]18. "Subtraction Property of Equality"[/math]
[math]19. ang ABC, ang DCB " are right angles" [/math][math]19. "Definition of right angles"[/math]
[math]20. ABCD " is a rectangle" [/math][math]20. ""[/math]
A. 
What is the missing statement in step 5?
  1. [math]Delta ABC ~= Delta DCB[/math]
  2. [math]Delta ABD ~= Delta DCA[/math]
  3. [math]Delta ADC ~= Delta ACB [/math]
  4. [math]Delta ABD ~= Delta DCB[/math]
B. 
What is the missing reason in step 8?
  1. Parallel Postulate
  2. Supplementary angles
  3. Same side interior angles are supplementary
  4. Linear Pairs are supplementary
C. 
What is the missing reason in step 10?
  1. Double Angle Identity
  2. Supplementary angles
  3. From the diagram
  4. Substitution Property of Equality
D. 
What is the missing statement in step 14?
  1. [math]ang BAD " and " ang CDA " are right angles"[/math]
  2. [math]bar{BA} _|_ bar{AD}[/math]
  3. [math]bar{CD} _|_ bar{AD}[/math]
  4. [math]m ang BAD = m ang CDA[/math]
E. 
What is the missing reason in step 15?
  1. Given
  2. Opposite sides of a parallelogram are parallel
  3. From the diagram
  4. Opposite sides of a rectangle are parallel
F. 
What is the missing reason in step 20?
  1. Parallelogram with four right angles is a rectangle
  2. Image given is a rectangle
  3. Quadrilateral with opposite sides parallel and congruent is a rectangle
  4. QED

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