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]

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]

[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] |
- Reflexive Property
- Diagonals of a parallelogram are congruent
- Given by the diagram
- 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]

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]

[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] |
- [math] Delta ABC ~= Delta ACD [/math]
- [math] Delta ABD ~= Delta CDB [/math]
- [math] Delta ABC ~= Delta CDA [/math]
- [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]

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]

[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] |
- Elimination Property of Equality
- Corresponding angles are congruent
- Definition of congruent terms
- 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).

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

[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] |
- [math] ang ABC ~= ang CDA[/math]
- [math] ang BAD ~= ang BCD [/math]
- [math] ang BAD ~= ang CDA [/math]
- [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.


[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?
- [math]Delta ABC ~= Delta DCB[/math]
- [math]Delta ABD ~= Delta DCA[/math]
- [math]Delta ADC ~= Delta ACB [/math]
- [math]Delta ABD ~= Delta DCB[/math]
B.
What is the missing reason in step 8?
- Parallel Postulate
- Supplementary angles
- Same side interior angles are supplementary
- Linear Pairs are supplementary
C.
What is the missing reason in step 10?
- Double Angle Identity
- Supplementary angles
- From the diagram
- Substitution Property of Equality
D.
What is the missing statement in step 14?
- [math]ang BAD " and " ang CDA " are right angles"[/math]
- [math]bar{BA} _|_ bar{AD}[/math]
- [math]bar{CD} _|_ bar{AD}[/math]
- [math]m ang BAD = m ang CDA[/math]
E.
What is the missing reason in step 15?
- Given
- Opposite sides of a parallelogram are parallel
- From the diagram
- Opposite sides of a rectangle are parallel
F.
What is the missing reason in step 20?
- Parallelogram with four right angles is a rectangle
- Image given is a rectangle
- Quadrilateral with opposite sides parallel and congruent is a rectangle
- QED
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