Angle Side Angle (ASA) Theorem

In geometry, the Angle Side Angle Theorem states that if two angles and the non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then the two triangles are congruent. 

This theorem is helpful in a few different ways. First, it can be used as a shortcut for proving that two triangles are congruent without having to go through the entire Side Side Side (SSS) or Side Angle Side (SAS) process. Second, it can be used in reverse to help you solve problems. For example, if you're given ASA and two angles but only one side length, you can use the theorem to figure out the missing side length. 

How to Prove Triangle Congruence with ASA

There are a few steps you need to take in order to prove that two triangles are congruent using ASA. First, label each triangle with corresponding angles and sides. Next, set up a statement of congruence such as "Angle A equals Angle J," "Angle B equals Angle K," and "Side a equals Side j." Once you have your statement of congruence set up, you can begin making your individual proofs. 

To do this, you'll need to use one or more of the following postulates or theorems: 

Reflexive Property of Congruence: If point A is congruent to point A, then segment AB is congruent to segment AB. 

Substitution Property: If segment AB is congruent to segment CD and segment BC is congruent to segment AD, then triangle ABC is congruent to triangle CDA. 

CPCTC: Corresponding parts of congruent triangles are congruent. 

ASA: If angle A equals angle J, angle B equals angle K, and side a equals side j, then triangle ABC is congruent to triangle JKL. 

Once you've proved that all three corresponding parts are indeed congruent, then you can state that because ASA holds true, triangles ABC and JKLare therefore Congruent by ASA! Make sure to write this in your final proof so there's no confusion later on down the road. 

Conclusion

The Angle Side Angle Theorem is a powerful tool in geometry that can be used as a shortcut for proving SSS or SAS as well as helping you solve problems when given specific information. Just remember that when using ASA, all three corresponding parts must be proven congruent in order for the overall proof to hold water!

FAQ

What is side side angle in geometry?

Side Side Angle (SSA) is a theorem in geometry that states that if two sides and the angle between them of one triangle are congruent to two sides and the angle between them of another triangle, then the two triangles are congruent.

What is SSS SAS ASA AAS in geometry?

SSS, SAS, ASA, and AAS are all different theorems that can be used to prove triangle congruence. SSS states that if all three sides of one triangle are congruent to all three sides of another triangle, then the two triangles are congruent. SAS states that if two sides and the angle between them of one triangle are congruent to two sides and the angle between them of another triangle, then the two triangles are congruent. ASA states that if two angles and the side between them of one triangle are congruent to two angles and the side between them of another triangle, then the two triangles are congruent. AAS states that if two angles and the side not between them of one triangle are congruent to two angles and the side not between them of another triangle, then the two triangles are congruent.

What is the angle side angle formula?

The angle side angle (ASA) formula states that if two angles and the side between them of one triangle are congruent to two angles and the side between them of another triangle, then the two triangles are congruent.

What is the difference between angle side angle and side angle side?

The main difference between angle side angle (ASA) and side angle side (SAS) is that ASA requires two angles and the side between them to be congruent, while SAS only requires two sides and the angle between them to be congruent.