Triangle Congruence by AAS and ASA Part 2

If you have not read my previous post on triangle congruence, make sure you do. Anyway back to the angle-side-angle and angle-angle-side theorems. I will prove even more triangles congruent in this post! Like I wrote before, that is all I really remembering doing in high school geometry.

Example 1 Triangle Congruence Proof with AAS

The first example is example 3 in the video. If you are wondering why I am writing about something I already did in a video, let me explain. It has to do with different learning styles. It is pretty important to understand your learning style. I will have to blog about learning styles soon. Anyway, through my blog, I try to reach a variety of learning styles. Thus, I create graphics, write and create videos toTriangle Congruence with AAS get my math point across. Anyway, on with example 3 from the video.

Given: angle S is congruent to angle Q and segment RP bisects angle SRQ

Prove: triangle SRT congruent to triangle QRP

When I look at what is given and the diagram, I notice the two triangles share a side which will be a corresponding congruent side, segment PR. Since, segment RP bisects angle SRQ, that means angles SRP and QRS are congruent corresponding parts by the definition of angle bisector. Those two facts taken with the given, provides enough proof to establish AAS in both triangles SRT and QRP.

Example 2 Triangle congruence with AAS

Given: angles B and D are congruent and segment AB is parallel to segment CD

Prove: triangles ABC and CDA are congruent

As is true with any proof, you need to understand the given and how it will help you identify a pair of correspond angles or sides of a triangle congruent to use one of the methods to proving triangles congruent. From the given, we have a pair of corresponding angles congruent, D and B. From the diagram a pair of corresponding sides can be established. AC is a shared side and with the reflexive property of congruence it can stated AC is congruent to AC. Also in the given, it is stated that segments AB and DC are parallel. Whenever you hear the words parallel lines, you must remember the special angle pairs formed by two parallel lines and a transversal. In this case, angle BAC is congruent to angle DCA because of the alternate interior angles theorem. With all of that, it can be said that triangle ABC is congruent to triangle CDA by the AAS theorem.

Example 3 Proving Triangles Congruent with AAS

Given: Segments XQ and TR are parallel and segment XR bisects QT

Prove: triangle XMQ congruent to triangle RMT

This proof requires the most work of all the proofs I have done on this blog. There are no corresponding parts given as congruent, which means we have to establish three pairs of corresponding parts congruent. The two keywords in the given are parallel and bisects. Angle X and angle R are alternate interior angles and are congruent because the two angles are formed by two parallel lines and a transversal. Angles XMQ and RMT are congruent because all vertical angles are congruent. I have two angles and need to prove 1 pair of corresponding sides congruent. Those sides will be segment TM and segment QM by the definition of segment bisector. Since the sides are the non-included sides, triangle XMQ is congruent to triangle RMT by the AAS theorem.


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