Geometric Proof for Quadrilateral

Question

Proof with input.Copy of proof.

There is an issue with this proof. Something isn't right as it is presented. This is from a student's test. It was sent to me like this (except the student's incorrect answers have been removed).

I believe the problem is that #4's reason - which was prewritten in the test - is slightly off. I think it should state that alternate interior angles are congruent instead of trying to use substitution. We haven't said anything about Angle B at this point.

I can't add any more lines to the proof or to the diagram. I can help the student argue a point to the teacher if I can make a case for a revised reason. Thank you in advance for some perspective.

Answer 1

I understand now. The reasoning for #4 simply needs to be that B and C are supplementary for the same reason that A and D are. I don't like pre-written proofs because it is so hard to see past what someone else has filled in. Anyway - thanks!

Answer 2

I'm good with this one. I can back this explanation up, so I am good.

Answer 3

Not "slightly off". It's dead wrong. We know nothing about $\angle B$ so we can't substitute it in, and nothing about angle $\angle B$ will tell us that $\angle C$ and $\angle D$ are supplementary.

This is simply completely wrong.

The key point of the proof is to substititute the $m\angle A$ in "$m\angle A + m\angle D=180$" with $m\angle C$ to get "$m\angle C + m\angle D = 180$". Without that lynchpin the proof goes nowhere.

We must give some to filling out $1,3,6$ but no more than $\frac 14$ credit but $\frac 18$ might be more appropriate.

But I'd criticize a fill-in-the-blank proof excercise where only one blank in four is worth 7/8s of the work.

....

Okay, to be less cruel to the poor student and to put the fault on these irritating fill-in-the-blank proofs, the poor student wanted to do a proof where s/he proved $m\angle B = m\angle D$ which the student can prove by noting $B$ is supplementary to $C$ and $A$ is supplementary to $B$ and $A$ and $C$ are congruent.

But the poor student never had the chance because it didn't fit the blanks.

So... I don't know what to do.

Failing marks to the exercise, I guess.

...

Okay.... half credit to the student but with full explanation as to why this is incorrect and completely wrong and how to make it right AND a full explanation as to how an alternative proof could have been done centered around the students correct (but non-conforming) claim of $m\angle B = m\angle D$.

The student deserves at least that much for the error of the exercise.