Question

select the correct answers from each drop down. quadrilateral rect has vertices r(0,4), e(1,6), c(5,4), and t(4,2). what are the missing steps to complete the proof to show quadrilateral rect is a rectangle? statements reasons 1. quadrilateral rect has vertices r(0,4), e(1,6), c(5,4), and t(4,2). 1. given 2. ( re=sqrt{5}, ec = 2sqrt{5}, ct=sqrt{5} ), and ( tr = 2sqrt{5} ) 2. distance formula 3. slope of ( re = 2 ), slope of ( ec=-\frac{1}{2} ), slope of ( ct = 2 ), and slope of ( tr=-\frac{1}{2} ) 3. slope formula 4. ( overline{re}paralleloverline{ct} ) and ( overline{ec}paralleloverline{tr} ) 4. 5. ( overline{re}perpoverline{ec} ) and ( overline{ec}perpoverline{ct} ) 5. 6. ( re = ct ) and ( ec = tr ) 6. definition of distance 7. rect is a rectangle. 7. definition of a rectangle

Explanation:

Step1: Analyze parallel sides

If two lines have the same slope, they are parallel. Since slope of \(RE = 2\) and slope of \(CT=2\), and slope of \(EC =-\frac{1}{2}\) and slope of \(TR =-\frac{1}{2}\). So, for statement 4, the reason is "If two non - vertical lines have the same slope, then they are parallel".

Step2: Analyze perpendicular sides

If the product of the slopes of two non - vertical lines is \(- 1\), then the lines are perpendicular. For \(RE\) and \(EC\), slope of \(RE\times\)slope of \(EC=2\times(-\frac{1}{2})=-1\). For \(EC\) and \(CT\), slope of \(EC\times\)slope of \(CT =(-\frac{1}{2})\times2=-1\). So, for statement 5, the reason is "If the product of the slopes of two non - vertical lines is \(-1\), then the lines are perpendicular".

Answer:

1. If two non - vertical lines have the same slope, then they are parallel.
2. If the product of the slopes of two non - vertical lines is \(-1\), then the lines are perpendicular.