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### Course: Geometry (FL B.E.S.T.)>Unit 6

Lesson 2: Introduction to triangle similarity

# Triangle similarity postulates/criteria

Sal reviews all the different ways we can determine that two triangles are similar. This is similar to the congruence criteria, only for similarity! Created by Sal Khan.

## Want to join the conversation?

• Is K always used as the symbol for "constant" or does Sal really like the letter K?
• Since K is the mostly used constant alphabet that is why it is used as the symbol of constant...
I think this is the answer...
• so, for similarity, you need AA, SSS or SAS, right? so what about the RHS rule?
• If you have two right triangles and the ratio of their hypotenuses is the same as the ratio of one of the sides, then the triangles are similar. (You know the missing side using the Pythagorean Theorem, and the missing side must also have the same ratio.) So I suppose that Sal left off the RHS similarity postulate.
• At , why would we not worry about or need the AAS postulate for similarity? Same question with the ASA postulate. Also, what happened to the AAA postulate? Wouldn't that prove similarity too but not congruence?
• Howdy,

All we need to know about two triangles for them to be similar is that they share 2 of the same angles (AA postulate). You may ask about the 3rd angle, but the key realization here is that all the interior angles of a triangle must always add up to 180 degrees, so if two triangles share 2 angles, they will always share the 3rd.

That is why we only have one simplified postulate for similarity: we could include AAS or AAA but that includes redundant (useless) information. We don't need to know that two triangles share a side length to be similar.

Something to note is that if two triangles are congruent, they will always be similar.

So good questions! The key realization is that all we need to know for 2 triangles to be similar is that their angles are all the same, making the ratio of side lengths the same.

Hope this helps,
- Convenient Colleague
• Is SSA a similarity condition?
(1 vote)
• No. SSA alone cannot establish either congruency or similarity because, in some cases, there can be two triangles that have the same SSA conditions.
However, in conjunction with other information, you can sometimes use SSA. Specifically:
SSA establishes congruency if the given angle is 90° or obtuse.
SSA establishes congruency if the given sides are congruent (that is, the same length).

If the side opposite the given angle is longer than the side adjacent to the given angle, then SSA plus that information establishes congruency.

However, you shouldn't just say "SSA" as part of a proof, you should say something like "SSA, when the given sides are congruent, establishes congruency" or "SSA when the given angle is not acute establishes congruency". If the given angle is right, then you should call this "HL" or "Hypotenuse-Leg", which does establish congruency.

There are some other ways to use SSA plus other information to establish congruency, but these are not used too often. Some of these involve ratios and the sine of the given angle.
• Is RHS a similarity postulate?
• Actually, "Right-angle-Hypotenuse-Side" tells you, that if you have two rightsided triangles, with hypotenuses of the same length and another (shorter) side of equal length, these two triangles will be congruent (i.e. they have the same shape and size). Since congruency can be seen as a special case of similarity (i.e. just the same shape), these two triangles would also be similar.
• what is the difference between ASA and AAS
• The sequence of the letters tells you the order the items occur within the triangle.

ASA means you have 1 angle, a side to the right or left of that angle, and then the next angle attached to that side.

AAS means you have 1 angle, you skip the side and move to the next angle, then you include the next side.

Hope this helps.
• How I remember it is that it's like the congruency criterion except it allows dilation.
• What happened to the SSA postulate? Does that at least prove similarity but not congruence?
• No. If you fix two sides of a triangle and an angle not between them, there are two nonsimilar triangles with those measurements (unless the two sides are congruent or the angle is right.)

To see this, consider a triangle ABC, with A at the origin and AB on the positive x-axis. Say the known sides are AB, BC and the known angle is A. This angle determines a line y=mx on which point C must lie. C will be on the intersection of this line with the circle of radius BC centered at B. Because a circle and a line generally intersect in two places, there will be two triangles with the given measurements.