Some Strategies for Achieving Correct Proofs

Strategy 1. Use diagrams and notes to clarify what is given and what is to be proved (and the difference between the two).

Example from Major Quiz 1.

Problem 1. Parallelogram

If ABCD is a quadrilateral with AB parallel to CD and with AB = CD, prove that ABCD is a parallelogram.

Strategy.

DRAW GIVENS. Draw one figure that shows the givens.

In this case, we are given a couple of parallel segments of the same length.

DRAW SOME IMMEDIATE CONSEQUENCES. Since a major feature of parallel lines is their relation to transversals, we try drawing in a natural transversal for this figure and mark the equal angles.

DRAW GOAL. Now draw a separate figure showing what one need to prove. To satisfy the definition of parallelogram, it suffices to prove that the sides BC and AD are parallel. Continuing to use the transversal AC, we draw the equal angles imply the parallelism.

PUT THEM TOGETHER. If these pictures are put together, it suggests an idea for making the link between givens and goal.

In the combined figure, one can see two apparently congruent triangles, ABC and CDA.

Let's analyze. If these triangles are congruent, then angle ACB = angle CAD, which implies BC parallel to DA.

Can we prove they are congruent. Looking at the "given" figure, we see a pair of congruent angles and two pairs of equal sides. Angle BAC = angle DCA; segments AB = CD and AC = CA. Thus triangles ABC and CDA are congruent by SAS.

Comment: Notice that the first figure forms a figure like the letter Z. The equality of angles can be viewed as point symmetry (180-degree rotational symmetry). Keeping this in mind helps keep the various angles straight.

 

 

Strategy 2. Look for consequences of all the hypotheses. Examine critically whether all the hypotheses are used in a potential proof.

 

Example 1.

Given points non-collinear points A, B, C, D with AB = BC = CD and angle ABC = angle BCD, prove that there is a circle passing through ABCD.

Consider alternate versions. Are they true or false

Alternate 1

Given points non-collinear points A, B, C, D with AB = BC = CD, prove that there is a circle passing through ABCD.

Alternate 2

Given points non-collinear points A, B, C, D, prove that there is a circle passing through ABCD.

Alternate 3

Given points non-collinear points A, B, C, D with AB = CD and angle ABC = angle BCD, prove that there is a circle passing through ABCD.

Now consider various proofs introduced in class.