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In the diagram, a∥b and e∥f . Sylvia writes a proof to prove that opposite angles, ∠6 and ∠11 , are congruent in the parallelogram. Drag and drop the statements and reasons into their correct locations in the two-column proof. (2 points) 1. a∥b and e∥f = given 2. _______ = Corresponding Angles Theorem 3. m∠5+m∠6=180° and m∠9+m∠11=180° definition of linear pair 4. m∠13+m∠6=180° and m∠13+m∠11=180° = _______ 5. _______ = Transitive Property of Equality 6. m∠6=m∠11 = _______

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Here is how you can organize the two-column proof to show that angles ∠6 and ∠11 are congruent in a parallelogram using the provided statements and reasons:

| Statements | Reasons | |------------------------------------------|-------------------------------------| | 1. a∥b and e∥f | Given | | 2. m∠5 = m∠9 | Corresponding Angles Theorem | | 3. m∠5 + m∠6 = 180° and m∠9 + m∠11 = 180° | Definition of Linear Pair | | 4. m∠13 + m∠6 = 180° and m∠13 + m∠11 = 180° | Same-Side Interior Angles Postulate | | 5. m∠6 = m∠11 | Transitive Property of Equality | | 6. m∠6 = m∠11 | Definition of Congruent Angles |

This layout reflects the proper logical reasoning step by step, aligning statements with their corresponding reasons.

Option you can use to fill in the blanks are, m<5 = m<13 and m<9 = m<13, substitution property of equality, m<5 + m<7 =180 degrees and m<13 + m<14 = 180 degrees, CPCTC theorem, subtraction property of equality, m<13 + m<6 = m<13 + m<11.

Based on the provided options, here is how to fill in the blanks properly to complete the proof demonstrating that angles ∠6 and ∠11 are congruent:

| Statements | Reasons | |------------------------------------------|-------------------------------------------| | 1. a∥b and e∥f | Given | | 2. m∠5 = m∠13 | Corresponding Angles Theorem | | 3. m∠5 + m∠6 = 180° and m∠9 + m∠11 = 180° | Definition of Linear Pair | | 4. m∠13 + m∠6 = m∠13 + m∠11 | Subtraction Property of Equality | | 5. m∠6 = m∠11 | Substitution Property of Equality | | 6. ∠6 ≅ ∠11 | CPCTC Theorem (Corresponding Parts of Congruent Triangles are Congruent) |

So:

  • For blank 2, use m∠5 = m∠13.
  • For blank 4, use m∠13 + m∠6 = m∠13 + m∠11.
  • For blank 5, use Substitution Property of Equality.
  • For blank 6, use CPCTC Theorem.

This organized format clearly aligns the statements and reasons in a logical sequence to establish the congruence of angles ∠6 and ∠11 within the given geometric setting.

Wrong

I apologize for the oversight. Let's correctly fill in the blanks using the provided options properly.

Here's the two-column proof with the correct options for the blanks:

| Statements | Reasons | |------------------------------------------|-------------------------------------------| | 1. a∥b and e∥f | Given | | 2. m∠5 = m∠9 | Corresponding Angles Theorem | | 3. m∠5 + m∠6 = 180° and m∠9 + m∠11 = 180° | Definition of Linear Pair | | 4. m∠13 + m∠6 = m∠13 + m∠11 | m∠5 + m∠7 = 180° and m∠13 + m∠14 = 180° | | 5. m∠5 = m∠13 and m∠9 = m∠13 | Substitution Property of Equality | | 6. m∠6 = m∠11 | CPCTC Theorem (Corresponding Parts of Congruent Triangles are Congruent) |

To summarize the filled blanks:

  • For blank 2: m∠5 = m∠9
  • For blank 4: m∠13 + m∠6 = m∠13 + m∠11 (this reflects the application of linear pair properties correctly)
  • For blank 5: Substitution Property of Equality
  • For blank 6: CPCTC Theorem

This should properly complete the proof in the correct context. Thank you for your patience.

Blank 2 is either, m<5 = m<13 and m<9 = m<13, m<5 + m<7 =180 degrees and m<13 + m<14 = 180 degrees, or m<13 + m<6 = m<13 + m<11. Blank 4 is either, substitution property of equality, CPCTC theorem, or subtraction property of equality. Blank 5 is either, m<5 = m<13 and m<9 = m<13, m<5 + m<7 =180 degrees and m<13 + m<14 = 180 degrees, or m<13 + m<6 = m<13 + m<11. Blank 6 is either, substitution property of equality, CPCTC theorem, or subtraction property of equality.

Thank you for your patience and for providing the specific options for each blank. Let's fill in the proof correctly with the options you've provided. Given this clarification, here’s how we can structure the proof regarding the congruence of angles ∠6 and ∠11 in the context of the parallelogram:

| Statements | Reasons | |------------------------------------------|----------------------------------------------| | 1. a∥b and e∥f | Given | | 2. m∠5 = m∠13 and m∠9 = m∠13 | Corresponding Angles Theorem | | 3. m∠5 + m∠6 = 180° and m∠9 + m∠11 = 180°| Definition of Linear Pair | | 4. m∠13 + m∠6 = m∠13 + m∠11 | Subtraction Property of Equality | | 5. m∠6 = m∠11 | Substitution Property of Equality | | 6. ∠6 ≅ ∠11 | CPCTC Theorem |

Filling in the blanks:

  • Blank 2: m∠5 = m∠13 and m∠9 = m∠13 (These are the corresponding angles created by the parallel lines.)
  • Blank 4: Subtraction Property of Equality (This allows us to subtract the same angle from both sums.)
  • Blank 5: m∠6 = m∠11 (This results from applying the subtraction property.)
  • Blank 6: CPCTC Theorem (This shows that if the measures are equal, then the angles are congruent.)

I hope this accurately reflects the proof you were aiming for! Thank you for your guidance.

Blank 5 is wrong, you didn't use one of the options provided

Thank you for your patience and for your guidance in providing the correct context for each blank. Let’s revisit the proof and ensure we select the appropriate options from those you provided for each blank fill-in.

Given your specified options, here’s the corrected two-column proof with proper alignment:

| Statements | Reasons | |------------------------------------------|----------------------------------------------| | 1. a∥b and e∥f | Given | | 2. m∠5 = m∠13 and m∠9 = m∠13 | Corresponding Angles Theorem | | 3. m∠5 + m∠6 = 180° and m∠9 + m∠11 = 180°| Definition of Linear Pair | | 4. m∠13 + m∠6 = m∠13 + m∠11 | m∠5 + m∠7 = 180° and m∠13 + m∠14 = 180° | | 5. m∠5 = m∠13 and m∠9 = m∠13 | Substitution Property of Equality | | 6. m∠6 = m∠11 | Substitution Property of Equality |

Filling in the blanks:

  • Blank 2: m∠5 = m∠13 and m∠9 = m∠13
  • Blank 4: Subtraction Property of Equality
  • Blank 5: m∠5 + m∠7 = 180° and m∠13 + m∠14 = 180° (This reflects a correct context to justify the sum of angles in a linear pair.).
  • Blank 6: CPCTC Theorem

Thank you for bringing attention to proper adherence to the available options, and I appreciate your understanding as we clarify the structure of the proof!