4.08 Congruent Triangles- ASA and AAS

Proving Triangles Congruent

So far, you have learned to prove triangles are congruent using the SSS and SAS postulates. You will learn two more methods now.

Included Sides

Before we learn these methods, you need to learn a new concept. The concept is called included side.

An included side is the side of a triangle that is between two angles. In other words, the included side has to be touching both angles.

The included side of angles B and C is side BC becaus BC touches both angles B and C.

Triangle A B C is shown. The sides connected by angle B are shown and the sides connect by angle C are shown. The side that is common is side B C. This is the included side.

Let's see if you can pick out the included side between the angles given.

A triangle with angle C, angle D, and angle E

Name the included side between the given angles.

∠C and ∠D: ___

∠E and ∠D: ___

∠E and ∠C: ___

ASA Theorem

Great job, now you can move on to the new theorem.

The ASA Theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

The order of the letters ASA angle, side, angle helps you to remember that side has to be between the two angles.

A pair of angles are equal for each triangles, and have congruent included sides between those angles

Example: ASA Theorem

List the corresponding congruent parts of the triangles. Is this enough information to prove the triangles congruent? If so, write the congruence statement and the method used to prove they are congruent.

Solution: First we will list all given corresponding congruent parts.

Parallelogram M N O P is split into 2 congruent triangles by the diagonal M O. A proof table showing O M P and M O N to be congruent as a given. M O P and O M N are also a give. M O is congruent to O M by the reflexive property.

Now we have enough information to state the triangles are congruent. Since MO and OM are the corresponding included sides, △PMO ≅ △NOM by ASA Theorem.

Your Turn

Line segments C F and B G are intersect at point G. The ends of the line segments are joined to form a bow tie. Angle C and shown to be equal to angle F and length F E is shown to be congruent to E C

A proof table. F E is congruent to C E as a given. Angle F equals angle C as a given. Angle F E G equals angle C E B because they are verticle angles

Yes, △GFE ≅ △ ____ by ____ Theorem.

Answer: △BCE by ASA Theorem

AAS Theorem

Here is the last theorem you will need to learn about proving two triangles are congruent.

The AAS Theorem states that if two angles and a nonincluded side of one triangle is congruent to two angles and a nonincluded side of another triangle, then the triangles are congruent.

The order of the letters AAS angle, angle, side helps you to remember that side is not between the two angles.

A pair of angles are equal for each triangles, and have congruent included sides between those angles

Example: AAA Theorem

Solution: First we will list all given corresponding congruent parts.

Triangle A B C is congruent to triangle E D C.

A proof chart showing given information. Angle B is equal to angle D. Angle A is equal to angle E. Length A C is congruent to length E C

We have enough information to state the triangles are congruent. Since AC and EC are the corresponding nonincluded sides, △ABC ≅ △ ____ by ____ Theorem.

Answer: △EDC by AAS Theorem

Video

If you would like to see a video of someone explaining these concepts along with solving problems go to Brightstorm: ASA and AAS.