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?
cpctc
property of parallelograms
transitive property
vertical angles theorem

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step-by-step explanation:

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irst, find the z score of a score of 80. this is given by the formula z = (x-u)/o, where x is the score, u is the mean, and o is the standard deviation. this gives

z = (80-78)/12 = 1/6 = .1667

now, find what percentile a z score of 1/6 correlates to in the normal distribution.

a z score of 1/6 = .1667 is equivalent to the 56.36th percentile of the distribution. so the probability a score is over this is 1-.5636 = .4364

the mean of a sample mean is equal to the original mean. however, the standard deviation of a sample mean is equal to the original standard deviation divided by the square root of n (sample size). so the new standard deviation is 12/sqrt(16) = 3.

the new z score is 80-78/3 = 2/3 = .667. this is equivalent to the 74.86th percentile. so the probability that the mean will be greater than this score is 1-.7486 = .2514.