Question

Consider the diagram and proof below.

Answer 1

Given the following statements and Reasons

1. WXYZ is a ▱; ZX ≅ WY 1. given
2. ZY ≅ WX 2. opp. sides of ▱ are ≅
3. YX ≅ YX 3. reflexive
4. △ZYX ≅ △WXY 4. SSS ≅ thm.
5. ∠ZYX ≅ ∠WXY 5. CPCTC
6. m∠ZYX ≅ m∠WXY 6. def. of ≅
7. m∠ZYX + m∠WXY = 180° 7. ?
8. m∠ZYX + m∠ZYX = 180° 8. substitution
9. 2(m∠ZYX) = 180° 9. simplification
10. m∠ZYX = 90° 10. div. prop. of equality
11. WXYZ is a rectangle 11. rectangle ∠ thm.

The missing reason in step 7 is "consecutive ∠s in a ▱ are supplementary"

Answer 2

Observe the given figure.

Given: WXYZ is a parallelogram and $ZX \cong WY$

To Prove: WXYZ is a rectangle

  Statements                                                                                      

1. WXYZ is a parallelogram and $ZX \cong WY$                

Reason: Given

2.  $ZY \cong WX$                                          

Reason: Opposite sides of parallelogram are equal.

3.  $YX \cong YX$

Reason: Reflexive

4.  $\Delta ZYX \cong \Delta WXY$

Reason: By SSS congruence theorem

5.  $\angle ZYX \cong \angle WXY$

Reason: CPCT

6.  $m \angle ZYX \cong m \angle WXY$

Reason: Def of congruency

7.  $m \angle ZYX + m \angle WXY= 180^\circ$

Reason: Consecutive angles in a parallelogram are supplementary.

"If a quadrilateral is a parallelogram, it has consecutive angles then they are supplementary".

8.  $m \angle ZYX + m \angle ZYX= 180^\circ$

Reason: Substitution.

9. $2(m \angle ZYX)= 180^\circ$

Reason: Simplification

10. $m \angle ZYX = 90^\circ$

Reason: Division property of Equality

11. WXYZ is a rectangle

Reason: Rectangle angle theorem