Clara writes the following proof for the theorem: If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram: Clara's proof

For triangles AOB and COD, angle 1 is equal to angle 2, as they are vertical angles.
AO = OC and BO = OD because it is given that diagonals bisect each other.
The triangles AOB and COD are congruent by SAS postulate.
Similarly, triangles AOD and COB are congruent.
By , angle ABD is equal to angle CDB and angle ADB is equal to angle CBD.
As the alternate interior angles are congruent, the opposite sides of quadrilateral ABCD are parallel.
Therefore, ABCD is a parallelogram.

Which is the missing phrase in Clara's proof?

A. CPCTC
B. property of parallelograms
C. transitive property
D. vertical angles theorem

b). 4320 have a nice day

a) 40 more; b) old factory:   10%, new factory:   16%; c) week 4

explanation:

a) at week 0, we see in the graph that the new factory produces 190 specialty items.  

using the function, substituting 0 in for w,

this is a difference of 230-190 = 40.

b) the function for the old factory is written in the form f(x) = a(1+r)ˣ, where a is the original amount, x is the amount of time and r is the growth rate.   the equation for the old factory is ; 230 is in place of a and is the original amount; w is the amount of time; which means 1+r = 1.1.   this means that r must be 0.1 or 10%.

for the new factory, we find the percent of increase to find the growth rate; the formula for this is

between the first two points, the amount of change is 220-190 = 30; this gives us

30/190 = 0.158 ≈ 0.16 = 16%

c) at week 0, the old factory has 230 while the new factory has 190.

at week 1, the old factory has 230(1.1) = 253 while the new factory has 220.

at week 2, the old factory has 230(1.1²) = 278.3 while the new factory has 252.

at week 3, the old factory has 230(1.1³) = 306.13 while the new factory has 290.

at week 4, the old factory has 230(1.1⁴) = 336.743 while the new factory has 337.   this is the first week that the new factory surpasses the old.