The Diagonals of a Rhombus are Perpendicular

This post is dedicated to proving the diagonals of a rhombus are perpendicular. To complete this or any proof, it is good to make a plan. In this case, I am going to establish that the two triangles on the top of the rhombus are congruent. This is seen in steps 2 through 4 of the proof below. Next, I show that angles AEB and CEB right angles, which is modeled in steps 6 through 8. That is enough to state the diagonals of a rhombus are perpendicular.

I encourage you to click on the image to see it at full size. It looks much better at full size. I scanned the piece of notebook paper that I work this proof out on and messed around in photoshop.

Proof - Diagonals of a Rhombus are Perpendicular

Proof - Diagonals of a Rhombus are Perpendicular

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How to Write Indirect Proofs – Exterior Angle Inequality Theorem

A youtube viewer of mine requested a video on how to write an indirect proof. Before making the video, I thought it would be good to write a blog post about this topic before I make my video, because an indirect proof is best written as a paragraph proof. There are three steps to writing an indirect proof. 

  1. Assume that the conclusion is false, by negating the prove statement.
  2. Establish that the assumption in step #1 leads to a contradiction of some fact i.e. definition, postulate, corollary or theorem.
  3. State the assumption must be false, thus, the conclusion or prove statement is true.

Steps 1 and 2 involve all of the thought and memory skills and can be discussed separately. Do not make this harder than it is. Step 1 involves writing the negation of a statement. Step 2 requires you to pull on your knowledge of geometric definitions, postulates, theorems and corollaries to recognize the contradiction between a known geometric fact and the assumption in step 1. Step 3 involves stating the obvious: Since the assumption is false, the prove statement must be true. If you are confused, check out the examples. 

Example 1 – Prove the Exterior Angle Inequality Theorem with Indirect Proof

Given: \angle{1} is an exterior angle of \Delta{ABC} 

Prove: m\angle{1}>m\angle{4} 

Figure 1 - Indirect Proof Diagram

Figure 1 - Indirect Proof Diagram

Step 1 – Assume that m\angle{1} \not> m\angle{4},  this means that m\angle{1} \leq m\angle{4}.

Step 2 – We need to establish that m\angle{1} \leq m\angle{4} contradicts a mathematical fact.

m\angle{1} \leq m\angle{4} gives two different situations that need to be tested:

 m \angle{1}= \angle{4} or m \angle{1}<m \angle{4}.

m \angle{1}= \angle{4}

By the Exterior Angle Theorem, m \angle{3}+m \angle{4}=m \angle{1} and using substitution, m \angle{1}+m \angle{4}=m \angle{1}. Subtracting m \angle {1} from both sides gives m \angle{4}=0. This contradicts the fact an angle must have a measure greater than 0.

m \angle{1}<m \angle{4}

By the Exterior Angle Theorem, m \angle{3}+m \angle{4}=m \angle{1}. Angles must have a positive measure, the definition means m \angle{1}> m \angle{3} and m \angle{1}>m \angle {4}. 

Step 3 – In each instance, the assumption from step 1 is contradicted of a know mathematical fact. Thus, the assumption that m\angle{1} \leq m\angle{4} is false. So, the original prove statement,  m\angle{1}>m\angle{4} , must be true.

3.2 Proving The Converse of the Alternate Interior Angles Theorem

“What’s the point in proving theorems that have already been proved?” If your geometry class is anything like mine, then you have been asked to prove an existing theorem and you may have asked yourself the same question as above. Well there is the easy answer, because “I said so”. I agree, that is pretty lame and in reality being able to write a proof of an established theorem requires an ability to recall facts and to apply those facts and given information to arrive at a valid conclusion.

One such proof is of the Converse of the Alternate Interior Angles Theorem.

Converse of the Alternate Interior Angles Theorem

If two lines and a transversal form alternate interior angles that are congruent,

then the two lines are parallel.

One proof method of proof that I have stayed away from is the flow proof. The flow is too much work for the end product. It takes more time to complete the same problem. The image in this post took twice as much time to create than a simple image and writing a paragraph proof on my math blog. I came back to this section, because the a previous post on Parallel Lines is my top page receiving hits, so I wanted to do something more.

3.2 Proving Lines Parallel Paragraph Proof

3.2 Proving Lines Parallel Paragraph Proof

First off, the diagram is missing a label that is necessary for a paragraph proof of the converse of the alternate interior angles theorem. The line that is acting as the transversal of lines l and m will be called line t.

Given that line t is the transversal of lines l and m. By definition, Angle 3 and angle 1 are vertical angles and are congruent by the vertical angles theorem. It is given that angle 1 is congruent to angle 2. Applying the transitive property of congruence, it can be established that angle 2 is congruent to angle 3. It should be said that angle 2 and angle 3 are corresponding angles.Thus line l is parallel to line m because of the converse of the corresponding angles postulate. QED

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