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## Proving Lines Parallel with Triangle Congruence SSS SAS AAS and ASA

I got a great question via personal message on youtube,

“hello, i need help on this one proof i am doing. Do you have any video on how to prove lines are parallel by sss, sas, asa, or aas? please tell me if you do. thankyou”

I have not done any problems like this yet, but I put this image together to help the student of geometry. I hope it helps. Below, I write a paragraph proof.

To really understand this problem you have to remember the ways to prove lines parallel: the converse of the corresponding angles postulate, the converse of the alternate interior angles theorem and the converse of the same-side interior angles theorem. So, to prove that segment AB is congruent to Segment CD. The shortest way to complete this proof is to show that either angels MAB and MDC or angles MBA and MCD are congruent. To prove those angles congruent, triangles ABM and DMC must be proved congruent.

With the given information, M is the midpoint of AD and BC, segments AM and DM are congruent as well as segments BM and CM because of the definition of midpoint. Since segments AD and BC intersect at M and form vertical angles AMB and DMC, those angles are congruent by the vertical angles theorem. That gives a SAS congruence in triangles ABM and DMC, thus leaving them congruent. Because of corresponding parts of congruent triangles are congruent(CPCTC), angle A is congruent to angle D. Angles D and A are congruent alternating interior angles, so segments AB and CD are parallel by the converse of the alternate interior angles theorem. QED

I color coded the markings on the diagram with the proof. I like to start with a blank diagram and mark my corresponding congruent parts as I go. It helps me develop my proof. I hope this technique helps you. I encourage you to use different colored pencils to mark your diagram is my coloring coding helped you follow this proof on how to prove lines congruent with triangle congruence.

Regards,

Mr. Pi

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## 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

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

## Geometry – Properties of Parallel Lines 3.1 Part 2

This lesson investigates and use the alternate interior angles theorem, the alternate exterior angles theorem, the corresponding angles postulate, the same side interior angles theorem and the same side exterior angles theorems. The other post titled, Geometry – Properties of Parallel Lines, would not allow me to put up my other video, so here is the 1st video lesson on properties of parallel lines.

## Proving Lines Parallel – Geometry – 3.2

Proving Lines Parallel – Geometry

This lesson investigates and use the converse of alternate interior angles theorem, the converse of alternate exterior angles theorem, the converse of corresponding angles postulate, the converse of same side interior angles theorem and the converse of same side exterior angles theorem.

The first problem in the video covers determining which pair of lines would be parallel with the given information. You are given that two same-side exterior angles are supplementary. There two pairs of lines that appear to parallel. You must determine which pair is parallel with the given information. One could argue that both pairs are parallel, because it could be used, but the problem is ONLY asking for what can be proved with the given information.

Much like the lesson on Properties of Parallel Lines the second problem models how to find the value of x that allow two lines to be parallel. You much write an equation
based on how the angles are related. The problem in the video show how to solve a problem that involves converse of alternate interior angles theorem, converse of alternate exterior angles theorem, converse of corresponding angles postulate. Which means an equal relationship. Remember, the supplementary relationship, where the sum of the given angles is 180 degrees.

Similar to the first problem, the third problem has you determining which lines are parallel, but the diagram is of a wooden frame with a diagonal brace. Two alternate interior angles are marked congruent. Remember, you are only asked for which sides are parallel by the given information.

If you have a specific question, please ask. Cite your book, I might have it and I can show the specific problem. Also, give your best description of the problem that you can. You must quote the question from your book, which means you have to give the name and author with copyright date. I have used digital images of problems I have worked out by hand for the Algebra 2 portion of my blog.