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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Proving Angles Congruent – Geometry Proof

If you have been reading my math blog at all, then you know I have been posting my youtube videos here and giving each one a brief description. I am ready to get back in action and prove some angles congruent! One of the easiest ways to prove angles congruent is with knowledge of the Vertical Angles Theorem. The vertical angle theorem states that vertical angles are congruent.

Vertical Theorem and a Proof of the Vertical Angle Theorm

Vertical Theorem and a Proof of the Vertical Angle Theorm

In the proof of the vertical angles theorem, you have to establish a relationship between angles 1 and 3 and angles 2 and 3. Both pairs of angles are are supplementary pairs, thus their sum is 180 degrees, which can be seen in statement 2 of the above proof. Now that statement 2 is established, you can state that the sum of the measures of angles 1 and 3 is equal to the sum of the measures of angles 2 and 3. The previous is shown in statement 3 in the above proof. Now this equation is really cool, because it can be changed into the measure of angle 1 is equal to the measure of angle 2, which is very close to what must be proved. Since the measures are equal, the angles are also congruent. See statements for and 5 in the above prove.

I hope to incorporate this into my class some how. I will get back to you and let you know.

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