Question

1. complete the flow proof shown. fill in the blanks. given: ∠1 and ∠4 are supplementary; c || d. prove: a || b given c || d same - side interior ∠s are suppl. ∠1 and ∠4 are suppl. m∠1 + m∠4 = 180 definition of suppl. ∠s m∠3 + m∠4 = m∠1 + m∠4 substitution m∠1 = m∠3 subtraction property of equality ∠1 ≅ ∠3 definition of congruent ∠s ∠2 ≅ ∠3 transitive property of = converse of corresponding angles theorem a || b

Explanation:

Step1: Recall parallel - line properties

Since \(c\parallel d\), the same - side interior angles are supplementary. So the first blank is "Same - side interior \(\angle s\) are suppl."

Step2: Use substitution

We know \(m\angle1 + m\angle4=180\) and \(m\angle3 + m\angle4=m\angle1 + m\angle4\) (by substitution).

Step3: Apply subtraction property

From \(m\angle3 + m\angle4=m\angle1 + m\angle4\), using the subtraction property of equality, we get \(m\angle1 = m\angle3\).

Step4: Use definition of congruent angles

Since \(m\angle1 = m\angle3\), by the definition of congruent angles, \(\angle1\cong\angle3\).

Step5: Apply transitive property

We have \(\angle2\cong\angle3\) and \(\angle1\cong\angle3\), so by the transitive property of congruence, \(\angle1\cong\angle2\).

Step6: Use corresponding - angles theorem

Since \(\angle1\cong\angle2\), by the converse of the corresponding - angles theorem, \(a\parallel b\).

Answer:

Same - side interior \(\angle s\) are suppl.; \(\angle1\cong\angle2\); \(\angle1\cong\angle2\) (from transitive property of congruence)