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Monday, October 18, 2010

Counterexamples in Geometry

Yesterday on Twitter I asked for false Geometry statements for which it's easy to draw a counterexample. Twitter is brilliant for this - everybody can come up with a-couple-a-three no problem, but it would be a pain to sit and think of a dozen. And even when you did, they might not be the best dozen for your purposes. After waiting a day, I got to pick from lots.

I'm consciously trying to do a better job of motivating proof this year. Many times this will take the form of asking the students to provide a conjecture before we try and prove anything. For example, I have this planned for next week: Start with an isosceles triangle. Make an exterior angle at the vertex. Bisect the exterior angle. What appears to be true? Now prove it. I'm looking forward to using the Nspire for this purpose - they'll quickly be able to look dynamically at tons of examples and bring the inductive reasoning to bear towards a conjecture.

Anyway, I'm getting ahead of myself. On Wednesday we'll be learning about the Triangle Sum Theorem, which they already "know." But a proof is very accessible and uses what we just learned about angles made by parallel lines and a transversal. So I was challenged with how to motivate proving something they already "know."

Hence, my need for false statements.

True or false? If false, draw a counterexample.

1.              All right triangles are isosceles.

2.              All rectangles are similar figures. 

3.              All pentagons are regular polygons.

4.              Altitudes are always inside a triangle.

5.              All quadrilaterals with four congruent sides are squares.
6.              For any two lines cut by a transversal, corresponding angles are congruent.

7.              All quadrilaterials have congruent diagonals.

8.              Diagonals of a quadrilateral always intersect.

9.       The three angles in the interior of a triangle sum to 180 degrees.

10.     The acute angles in a right triangle are always complementary.

Kids draw counterexamples. Miss Nowak sticks them up on the document camera. Everyone gets in counterexample drawing mode, remembering that vocabulary, thinking, drawing, etc. Then they get to #9.
"But wait, the angles in a triangle do add up to 180."
"How do you know? Did you draw every possible triangle? And measure their angles? And add them up?"
"No. We learned it in eighth grade."
"But how do you KNOW?"
"My teacher told me."
"Okay, I believe you, but that's not math. That's like maybe some weird sort of religion."

I LOVE THIS. I can't wait for Wednesday. I'm going to leap out of bed and run to school to teach it.

Here are other false statements offered by Twitterzens. There are some great ones here but I had to pick an appropriate mix for my crowd. (Also, if you're wondering who actually answers the questions you throw out into the Twitter-void, follow these rock stars.) Enjoy!

angle 1 is supplementary to angle 2. Angle 2 is supplementary to angle 3. Therefore, angle 1 is supp to angle 3.

all equilateral triangles are congruent to each other.

if it has two pairs of congruent sides, it's a rectangle.

3 coplanar points/lines always form a triangle.

all isosceles right triangles are congruent.

all coplanar points are collinear.


The perimeter of a rectangle is larger than its area.

An altitude of a triangle is also a median.

The center of a triangle's circumcircle is inside of the triangle.

Pairs of the following kinds are similar: isosceles triangles, scalene triangles, rhombi, isotraps, equiangular hexagons.

If two circles are the same size, then lines tangent to both of them are parallel.

If two circles intersect, then their two common tangents are parallel.


All trapezoids have 2 congruent sides.

All triangles have a line of symmetry.

Quadrilaterals can't have two obtuse angles.


A circle's circumference is equal to its area.

Every rectangle is a square.

Every scalene triangle contains a right angle.


2 complementary angles are adjacent.

Adjacent angles are supplementary.

if 3 angles of one triangle are congruent to the corr angles of another triangle, then those two triangles are congruent.

if diagonals of a quadrilateral are perpendicular, then the quad is a kite.

All similar shapes are congruent.

All corresponding angles formed by a line transversing 2 other lines are congruent

the sum of the interior angles of a triangle is 180 degrees (see earlier post about non-Euclidean space)

All squares are kites.

Two lines that are perpendicular to a third line are parallel (not true in 3D.)

The diagonals of a quadrilateral always intersect.

Diagonals of parallelograms are also angle bisectors.

corresponding angles for two lines cut by a transversal are congruent

if a triangle does not have all equal angles, it is not an isosceles triangle.

No squares have greater area than circles.

All rectangles are squares.

All equilateral triangles are right (or obtuse.)

Any figure with four equal sides is a square.

All pentagons are concave.

All parallelograms are rectangles.


  1. I don't twitter.

    If a pair of sides and a pair of angles are congruent, then the quadrilateral is a parallelogram.

    You might like this post: Proving a Quadrilateral is a Parallelogram both for the content, and because I seem to have anticipated your need for a mix of true and false statements.


  2. Actually jd I think the whole different approach to proofs was more inspired by this post of yours. Thank you.

  3. Actually, the really scary thing is how many of these statements I have heard... from... yes, you guessed it...

  4. #9 and #10 motivating discussion of non-Euclidean geometry?


Hi! I will have to approve this before it shows up. Cuz yo those spammers are crafty like ice is cold.