from an e-published set of sample STAR geometry questions
"Theorem: A triangle has at most one obtuse angle.
Eduardo is proving the theorem above by contradiction. He began by assuming that in
triangle ABC, ∠A and ∠B are both obtuse. Which theorem will Eduardo use to reach a
contradiction?
A. If two angles of a triangle are equal, the sides opposite the angles are equal.
B. If two supplementary angles are equal, the angles each measure 90°.
C. The largest angle in a triangle is opposite the longest side.
D. The sum of the measures of the angles of a triangle is 180°."
There's a few curiosities here. The student being tested must know what an obtuse angle is (greater
than 90 degrees for those of you who are a little rusty). Fair enough. He or she must also know not
only what a proof by contradiction is, but also what the word contradiction means. When being
taught months ago should a student have been able to hear the lesson in a language of his or her
choice? Sure. Why else have wireless networks and very small computers? Should the student be
able to take the STAR Geometry test in a language other than English? How about having the test
available in a larger font for visually impaired students? We're not done yet. Currently, Individual
Education Plans formally document what are called accommodations. These can be fairly
wide-ranging from having access to a folded towel to being able to pace across the back of the
classroom. There was a lawsuit filed against the State of California asking that whatever
accommodations were allowed during daily routines being allowed during testing. Seems fair to us.
After all, what are we testing here? Geometry. Not reading speed. So if a dyslexic child needs more
time, who cares? Is it safe or even sound strategy to wait a year to discover Eduardo did not
understand proof by contradiction 11 months ago? There's always a lot of chatter about making
education work like a business. Fine. Let's have quarterly STAR exams. Everyone takes them. No
exceptions. Even three months is far too long, so a second requirement is, with suitable security, to
post homework, quiz and test scores on the internet. This tells parents in near real-time how their
child is doing. It also tells principals, department chairs and mentors how well a teacher is doing.
By the way, only in Euclidean or parabolic geometry is the angle sum 180 degrees. In
Lobachevsky-Bolyai-Gauss or hyperbolic geometry the sum is less and in Riemannian or elliptic
geometry it is more.