Question

proving when a parallelogram is a rectangle
given: wxyz is a parallelogram.
(overline{zx}congoverline{wy})
prove: wxyz is a rectangle
statements

1. wxyz is a parallelogram
2. (overline{yx}congoverline{yx})
3. (overline{zy}congoverline{wx})
4. (mangle zyx + mangle wxy = 180)
5. (overline{zx}congoverline{wy})
6. (\triangle zyxcong\triangle wxy)

reasons

1. given
2. reflexive property
3. opposite sides of a parallelogram are congruent
4. consecutive angles of a parallelogram are supplementary
5. given
6. sss

Explanation:

Step1: Use congruent triangles property

Since \(\triangle ZYX\cong\triangle WXY\) (by SSS as \(ZY = WX\), \(YX=YX\), \(ZX = WY\)), then \(\angle ZYX\cong\angle WXY\) (corresponding parts of congruent triangles are congruent). So \(m\angle ZYX=m\angle WXY\)

Step2: Use angle - sum property of parallelogram

In parallelogram \(WXYZ\), \(\angle ZYX\) and \(\angle WXY\) are consecutive angles. We know that \(m\angle ZYX + m\angle WXY=180^{\circ}\) (consecutive angles of a parallelogram are supplementary). Substitute \(m\angle WXY\) with \(m\angle ZYX\) (from step 1), we get \(m\angle ZYX+m\angle ZYX = 180^{\circ}\), i.e., \(2(m\angle ZYX)=180^{\circ}\)

Step3: Solve for the angle measure

Divide both sides of \(2(m\angle ZYX)=180^{\circ}\) by 2. We have \(m\angle ZYX=\frac{180^{\circ}}{2}=90^{\circ}\)

Step4: Use the definition of a rectangle

A parallelogram with one right angle (\(m\angle ZYX = 90^{\circ}\)) is a rectangle. So \(WXYZ\) is a rectangle

Answer:

The statements in order: \(m\angle ZYX=m\angle WXY\); \(m\angle ZYX + m\angle ZYX=180\); \(\angle ZYX\cong\angle WXY\); \(2(m\angle ZYX)=180\); \(m\angle ZYX = 90\); \(WXYZ\) is a rectangle