The figure below shows a quadrilateral ABCD. Sides AB and DC are equal and parallel: A student wrote the following sentences to prove that quadrilateral ABCD is a parallelogram:

Side AB is equal to side DC, and DB is the side common to triangles ABD and CDB. Angle ABD is congruent to angle CDB by Alternate Interior Angles. Therefore, the triangles ABD and CDB are congruent by SAS postulate. By CPCTC, angles DBC and ADB are congruent and sides AD and BC are congruent. Angle DBC and angle ADB _. Therefore, AD is parallel and equal to BC. Quadrilateral ABCD is a parallelogram because its opposite sides are equal and parallel.

Which phrase best completes the student's proof?

are congruent by the AAS postulate
are congruent by the ASA postulate
form a pair of alternate interior angles that are congruent
form a pair of vertical angles that are congruent

Complex numbers multiply √-4 * √-25 and show all intermediate steps. alternative notation is sqrt(-4) * sqrt(-25).

Michelle and julie work at a catering company. they need to bake 264 cookies for a birthday party that starts in a little over an hour and a half. each tube of cookie dough claims to make 36 cookies, but michelle eats about 1/5 of every tube and julie makes cookies that are 1.5 times as large as the recommended cookie size. it takes about 8 minutes to bake a container of cookies, but since julie's cookies are larger, they take 12 minutes to bake. a. how many tubes should each girl plan to bake? how long does each girl use the oven? b. explain your solution process what did you make? c. what assumptions did you make?

Asystem of linear equations with more equations than unknowns is sometimes called an overdetermined system. can such a system be consistent? illustrate your answer with a specific system of three equations in two unknowns. choose the correct answer below. a. yes, overdetermined systems can be consistent. for example, the system of equations below is consistent because it has the solution nothing. (type an ordered pair.) x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 6 b. no, overdetermined systems cannot be consistent because there are fewer free variables than equations. for example, the system of equations below has no solution. x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 12 c. yes, overdetermined systems can be consistent. for example, the system of equations below is consistent because it has the solution nothing. (type an ordered pair.) x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 8 d. no, overdetermined systems cannot be consistent because there are no free variables. for example, the system of equations below has no solution. x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 24