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.

Two column proof proving lines parallel with triangle congruenceTo 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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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.

Parallel and Perpendicular Lines – Geometry – 3.3

This lesson is about the parallel lines theorem, the perpendicular to same line theorem and the perpendicular to one line theorem. Now, these are shorter names that I have given these theorems.

The Parallel Lines Theorem

If two lines are parallel to the same line,
then they are parallel to each other.

The Perpendicular to Same Line Theorem

In a plane, if two lines are perpendicular to the same line,
then they are parallel to each other.

The Perpendicular to One Line Theorem

In a plane, if a line is perpendicular to one of two parallel
lines, the it is also perpendicular to the other.

The first problem covered in the video is working with frame making. It is not the best I have ever done, but I will be putting something together in the near future to add to this
post.

The second problem is a paragraph proof involves proving two lines parallel. The diagram involves three lines that appear to be parallel that are in the order from top to
bottom, lines a, b and c. There are two transversals, but only one is labeled line s. Line is is perpendicular to lines a and c and lines a and b are parallel. We must prove c is parallel to b. From given, it can be established that b is perpendicular to s because of the Perpendicular to One Line Thm. Now we can state that c is parallel to b because of the
Perpendicular to Same Line Thm. QED

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.

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.

Geometry – Properties of Parallel Lines – 3.1

Geometry – Properties of Parallel Lines

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 use of a two column proof to show that 2 angles are supplementary, but the angles are not able to be proved with a theorem or postulate. The proof takes three or four steps.

Another problem in the video lesson below is finding the measure of angles when give two parallel lines and a transversal. There are two types of these problems. The first type of finding the measure of an angle is when you are given the actual measure of the angle such as 50 degrees. You are able to find the measure of every angle formed by 2 parallel lines and a transversal, when you know the measure of one. If one angle measures 50 degrees, all of the every angle in the diagram will be 50 or 130 degrees. All angles are either congruent or supplementary in this situation.

The other type of find the measure of an angle problem is when you are given two or more algebraic expressions instead of real numbers as the measurements. For example you are given the

measure of angle 1 = 3x + 7  (1)

measure of angle 6 = 5x + 15 (2).

You need to find the value of x that makes the lines parallel. It will be

3x + 7 = 5x + 15, (3)

Equation 3 is the result of the angles having a congruent relationship as with alternate interior angles, alternate exterior angles and corresponding angles. If the angles are same side interior angles or same side exterior angles, then the equation would be:

(3x + 7) + (5x + 15) = 180 (4)

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.

Here is the second video.

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