4.3 Triangle Congruence by AAS and ASA

In the video, it is not labeled example 1, but the first bit of information is critical to this lesson.

Angle Side Angle Postulate

It two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Angle Angle Side Theorem

It two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of another triangle, then the two triangles are congruent.

It would be good to remember that a postulate is something that is assumed to be true without question. Theorems are facts that have been proved true using definitions, postulates and other already proved theorems. We cannot use a theorem until we learn about it.

Example 2

In a recent post about proving The Converse of the Alternate Interior Angles Theorem, I created my first flow proof. It was a short proof, much like in this example using ASA. I color coded my markings, but on the vertical angles APX and BPY I did not get the correct shade of blue.

Paragraph Proof Angle-Side-Angle

Example 3

As you can see, it is a fairly complicated diagram for a relatively easy proof.

Given: angle CAB is congruent to angle DAE, segment AS is congruent to segments AE and angle ABC and AED are right triangles

Prove: triangle ABC is congruent to AED

Proving Triangles Congruent with ASA

Planning this proof requires you to understand your given information and the prove statement. Since the diagram has an added dimension of difficulty because extra triangle in the middle, you need to focus on the two outer right triangles ABC and AED. These are the angles that need to be proved congruent.

It is given that angles CAB and DAE are congruent and are corresponding angles in each triangle. Also, the corresponding sides AB and AE are congruent. Finally, there is information relating to the corresponding angles ABC and AED. They are both right angles. Angles ABC and AED are the angle that gives Angle-Side-Angle in each triangle. Since all right angles are congruent, angles ABC and AED are congruent. Therefore, both triangles ABC and AED are congruent by ASA. QED

Example 4

Given: segments NM and NP are congruent and angles M and P are congruent

Prove: triangles NML and NPO congruent

Proving Triangles Congruent with ASA

There are many ways to prove the triangles congruent in this diagram, but I like to produce the most concise proof possible. Hopefully you can see the ASA relationship that will be able to be proved from this set up. Angles LNM and ONP can be proved congruent to establish the ASA congruence in these two triangles.

Since it is given that segments NM and NP are congruent and angles M and P are congruent, a ASA congruence can be established with triangles NML and NPO. Angle LNM and angle ONP are congruent because they are vertical are congruent. Finally, it can be stated that triangles NML and NPO congruent because of the Angle-Side-Angle Theorem. QED

As usual, I hope this has been helpful. Be sure to check back for more geometry.

Regards,

Mr. Pi

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