Question

1. match each figure with the statement that proves ℓ || m.

(diagrams with angles 1,4; 3,5; 1,6 and theorem boxes: ∠1 ≅ ∠4, m∠3 + m∠5 = 180°, ∠1 ≅ ∠6; theorem options: converse of the alternate exterior angles theorem, converse of the corresponding angles theorem, converse of the same - side interior angles postulate)

1. a student said that when x = 6, a || b. what error did

the student likely make? for what value of x is a || b?
(diagram with angles (5x + 12)° and (7x)°)

1. complete the flow proof of the converse of the

corresponding angles theorem. fill in the blanks.
given: ∠2 ≅ ∠3
prove: c || d
(flow proof diagram: ∠2 ≅ ∠3 (given), ∠1 ≅ ∠3 (transitive prop. of ≅ from ∠1 ≅ ∠2), then blank for converse of the corr. ∠s thm.; diagram with lines c, d and angles 1,2,3)

Explanation:

Problem 1: Matching Figures with Parallel Line Proofs

Step 1: Analyze the first figure (∠1 ≅ ∠4)

∠1 and ∠4 are corresponding angles. The Converse of the Corresponding Angles Theorem states that if corresponding angles are congruent, then the lines are parallel. So, ∠1 ≅ ∠4 matches with "Converse of the Corresponding Angles Theorem".

Step 2: Analyze the second figure (m∠3 + m∠5 = 180°)

∠3 and ∠5 are same - side interior angles. The Converse of the Same - Side Interior Angles Postulate states that if same - side interior angles are supplementary, then the lines are parallel. So, \(m\angle3 + m\angle5=180^{\circ}\) matches with "Converse of the Same - Side Interior Angles Postulate".

Step 3: Analyze the third figure (∠1 ≅ ∠6)

∠1 and ∠6 are alternate exterior angles. The Converse of the Alternate Exterior Angles Theorem states that if alternate exterior angles are congruent, then the lines are parallel. So, ∠1 ≅ ∠6 matches with "Converse of the Alternate Exterior Angles Theorem".

Problem 2: Error Analysis and Finding x for \(a\parallel b\)

Step 1: Identify the relationship between the angles

The two angles \((5x + 12)^{\circ}\) and \((7x)^{\circ}\) are same - side interior angles. For \(a\parallel b\), same - side interior angles should be supplementary, i.e., \((5x + 12)+7x = 180\).

Step 2: Analyze the student's error

The student likely thought the angles were equal (used the Converse of Corresponding Angles or Alternate Angles instead of Supplementary for same - side interior angles). If we set \(5x+12 = 7x\) (wrong assumption), solving \(5x + 12=7x\) gives \(12 = 2x\), \(x = 6\). But this is incorrect because same - side interior angles are supplementary, not equal.

Step 3: Solve for x correctly

Solve \((5x + 12)+7x=180\)
Combine like terms: \(12x+12 = 180\)
Subtract 12 from both sides: \(12x=180 - 12=168\)
Divide both sides by 12: \(x=\frac{168}{12}=14\)

Problem 3: Completing the Flow Proof

Step 1: Identify the missing angle relationship

We know that \(\angle1\cong\angle2\) (Vertical Angles Theorem, since \(\angle1\) and \(\angle2\) are vertical angles). Then, by the Transitive Property of Congruence (since \(\angle2\cong\angle3\) and \(\angle1\cong\angle2\)), we get \(\angle1\cong\angle3\). Then, by the Converse of the Corresponding Angles Theorem, we can conclude \(c\parallel d\).

The blank for \(\angle1\cong\angle2\) is filled with "Vertical Angles Theorem", and the blank after \(\angle1\cong\angle3\) is filled with "\(c\parallel d\)".

Final Answers

Problem 1:

• \(\angle1\cong\angle4\) - Converse of the Corresponding Angles Theorem
• \(m\angle3 + m\angle5 = 180^{\circ}\) - Converse of the Same - Side Interior Angles Postulate
• \(\angle1\cong\angle6\) - Converse of the Alternate Exterior Angles Theorem

Problem 2:

• Error: The student assumed the angles were equal (used the wrong angle relationship, thought they were corresponding/alternate instead of same - side interior)
• Value of \(x\) for \(a\parallel b\): \(x = 14\)

Problem 3:

• \(\angle1\cong\angle2\) is by "Vertical Angles Theorem"
• The conclusion is \(c\parallel d\)

Answer:

Problem 1: Matching Figures with Parallel Line Proofs

Step 1: Analyze the first figure (∠1 ≅ ∠4)

∠1 and ∠4 are corresponding angles. The Converse of the Corresponding Angles Theorem states that if corresponding angles are congruent, then the lines are parallel. So, ∠1 ≅ ∠4 matches with "Converse of the Corresponding Angles Theorem".

Step 2: Analyze the second figure (m∠3 + m∠5 = 180°)

∠3 and ∠5 are same - side interior angles. The Converse of the Same - Side Interior Angles Postulate states that if same - side interior angles are supplementary, then the lines are parallel. So, \(m\angle3 + m\angle5=180^{\circ}\) matches with "Converse of the Same - Side Interior Angles Postulate".

Step 3: Analyze the third figure (∠1 ≅ ∠6)

∠1 and ∠6 are alternate exterior angles. The Converse of the Alternate Exterior Angles Theorem states that if alternate exterior angles are congruent, then the lines are parallel. So, ∠1 ≅ ∠6 matches with "Converse of the Alternate Exterior Angles Theorem".

Problem 2: Error Analysis and Finding x for \(a\parallel b\)

Step 1: Identify the relationship between the angles

The two angles \((5x + 12)^{\circ}\) and \((7x)^{\circ}\) are same - side interior angles. For \(a\parallel b\), same - side interior angles should be supplementary, i.e., \((5x + 12)+7x = 180\).

Step 2: Analyze the student's error

The student likely thought the angles were equal (used the Converse of Corresponding Angles or Alternate Angles instead of Supplementary for same - side interior angles). If we set \(5x+12 = 7x\) (wrong assumption), solving \(5x + 12=7x\) gives \(12 = 2x\), \(x = 6\). But this is incorrect because same - side interior angles are supplementary, not equal.

Step 3: Solve for x correctly

Solve \((5x + 12)+7x=180\)
Combine like terms: \(12x+12 = 180\)
Subtract 12 from both sides: \(12x=180 - 12=168\)
Divide both sides by 12: \(x=\frac{168}{12}=14\)

Problem 3: Completing the Flow Proof

Step 1: Identify the missing angle relationship

We know that \(\angle1\cong\angle2\) (Vertical Angles Theorem, since \(\angle1\) and \(\angle2\) are vertical angles). Then, by the Transitive Property of Congruence (since \(\angle2\cong\angle3\) and \(\angle1\cong\angle2\)), we get \(\angle1\cong\angle3\). Then, by the Converse of the Corresponding Angles Theorem, we can conclude \(c\parallel d\).

The blank for \(\angle1\cong\angle2\) is filled with "Vertical Angles Theorem", and the blank after \(\angle1\cong\angle3\) is filled with "\(c\parallel d\)".

Final Answers

Problem 1:

• \(\angle1\cong\angle4\) - Converse of the Corresponding Angles Theorem
• \(m\angle3 + m\angle5 = 180^{\circ}\) - Converse of the Same - Side Interior Angles Postulate
• \(\angle1\cong\angle6\) - Converse of the Alternate Exterior Angles Theorem

Problem 2:

• Error: The student assumed the angles were equal (used the wrong angle relationship, thought they were corresponding/alternate instead of same - side interior)
• Value of \(x\) for \(a\parallel b\): \(x = 14\)

Problem 3:

• \(\angle1\cong\angle2\) is by "Vertical Angles Theorem"
• The conclusion is \(c\parallel d\)