Algebra and Geometry Help

Proof: Vertical angles are equal | Angles and intersecting lines | Geometry | Khan Academy
Proof: Vertical angles are equal | Angles and intersecting lines | Geometry | Khan Academy

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.

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.

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

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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