We saw a series of shortcut methods for establishing that two triangle are congruent, without having to prove all 3 corresponding angles are congruent and all 3 corresponding sides are congruent. These shortcut methods included SSS, ASA, SAS, AAS and HL

So, how do we prove triangles similar?

reminder
 Two triangles are similar if and only if the corresponding sides are in proportion and the corresponding angles are congruent.

Fortunately, we also have shortcut methods for establishing that two triangles are similar, without having to prove all 3 corresponding angles are congruent and all 3 sets of corresponding sides are in proportion. These shortcut methods include AA, SSS, and SAS.

Wow! Some of the same "letterings", but not the same "theorems".
Let's take a look!

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There are three accepted methods for proving triangles similar:
While three methods exist, the AA Similarity Theorem is by far the most popular method.
Always look for the AA method first.

AA Similarity Theorem:

AA for
similar triangles
To prove two triangles are similar, it is sufficient to show that two angles of one triangle are congruent to the two corresponding angles of the other triangle.
theorem
 If two angles of one triangle are congruent to the corresponding angles of another triangle, the triangles are similar. (proof of this theorem is shown below)

AAtext1
AAtri
Yes, this could also be AAA. But if you have 2 sets of angles, you automatically have the 3rd set.
So, proving 2 sets congruent is sufficient to prove similar triangles.

PROOF:
AAgiven2

* This will be a transformational proof.
We will search for a sequence of transformations that will map ΔABC onto ΔDEF .
Atri222saa

1. Since the triangles are different sizes, we will start by dilating ΔABC to a smaller size. The smaller size needs to match the side lengths of ΔDEF, so we match AB with DE by setting the scale factor to be DEAB. The new dilated triangle will be ΔA'B'C'.

2. After the dilation, we know that ΔA'B'C' ΔABC because a dilation is a similarity transformation.

3. After the dilation, ∠A congruent∠A' and ∠B congruent∠B' since dilations preserve angle measure.

4. After the dilation, A'B' = k•AB = DE, which gives us DE.

5. We are given that ∠A congruent∠D and ∠B congruent∠E, and from step 3 that ∠A congruent∠A' and ∠B congruent∠B'. Using the transitive property, we get ∠A' congruent∠D and ∠B' congruent∠E.

6. ΔA'B'C' congruent ΔDEF by angle side angle (ASA) for congruent triangles.

7. Since ΔA'B'C' congruent ΔDEF and ΔA'B'C' ΔABC, we have ΔDEF ΔABC.

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beware      beware      beware
The next two methods for proving similar triangles are
NOT the same theorems used to prove congruent triangles.


SSS Similarity Theorem:

SSS
for similar triangles
To prove two triangles are similar, it is sufficient to show that the three sets of corresponding sides are in proportion.
theorem
 If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar. (proof of this theorem is shown below)
AAtext2
AAtri2

PROOF:

* This will be a traditional 2-column proof.
We will organize our statements and support with reasons.
  Statements Reasons
1. Given
2. Auxiliary line, Constructions of copy segment and parallel lines.
3. ∠1 congruent ∠A; ∠2 congruent ∠B If 2 parallel lines are cut by a transversal, the corresponding angles are congruent.
4. ΔMCN ΔACB If two corresponding angles of two triangles are congruent, the triangles are similar.
5. Corresponding sides of similar triangles are proportional.
6. Substitution (MC = DF)
7. Copy from step 1 (to add on the last 2 ratios).
8. AB • DE = MN • AB
BC • EF = NC • BC		 In a proportion, the product of means equals the product of the extremes.
(cross-multiply)
9. DE = MN; EF = NC Division (of AB, BC on both sides in step 8)
10. Congruent segments are segments of equal measure.
11. ΔDEF congruent ΔMNC  SSS - If 3 sides of 1 Δ are congruent to the corresponding sides of another Δ, the Δs are congruent.
12. ∠1congruent∠D; ∠2congruent∠E  CPCTC - Corresponding parts of congruentΔs are congruent.
13. ∠Acongruent∠D; ∠Bcongruent∠E  Substitution
14. ΔABC ΔDEF AA - If 2 ∠s of one Δ, are congruent to the corresponding ∠s of another Δ, the Δs are.

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SAS Similarity Theorem:

SAS for similar triangles
 To prove two triangles are similar, it is sufficient to show that two sets of corresponding sides are in proportion and the angles they include are congruent.
theorem
 If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar. (proof of this theorem is shown below)
AAtext3
Atri3

PROOF:

* This will be a traditional 2-column proof.
We will organize our statements and support with reasons.
  Statements Reasons
1. Given
2. Auxiliary line: Constructions of copy segment and parallel lines.
3. ∠1 congruent ∠A If 2 parallel lines are cut by a transversal, the corresponding angles are congruent.
4. ∠1 congruent ∠D  Substitution
5. ΔMCN ΔACB If two corresponding angles of two triangles are congruent, the triangles are similar.
6. Corresponding sides of similar triangles are proportional.
7. Substitution (MC = DF)
8. Copy from step 1 (to add on the last ratio).
9. AB • DE = MN • AB
 In a proportion, the product of means equals the product of the extremes.
(cross-multiply)
10. DE = MN Division (of AB on both sides in step 9)
11. Congruent segments are segments of equal measure.
12. ΔDEF congruent ΔMNC  SAS - If 2 sides and the included ∠of 1 Δ are congruent to the corresponding parts of another Δ, the Δs are congruent.
13. ∠C congruent ∠F  CPCTC - Corresponding parts of congruentΔs are congruent.
14. ΔABC ΔDEF AA - If 2 ∠s of one Δ, are congruent to the corresponding ∠s of another Δ, the Δs are.


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Remember, how, after proving triangles congruent, we could state that the "left-over parts"
that were not directly used would also be congruent?
CPCTC - Corresponding parts of congruent triangles are congruent.

We can also use this idea of "left-over parts" when working with similar triangles.
After proving the triangles similar by AA, we can state:
Corresponding sides of similar triangles are in proportion.
 The "angles" have already been addressed in AA, so they are not mentioned.
This statement is just a reiteration of part of the definition of similar triangles.
You might think the letters CSSTP may be used, but letters are not traditionally assigned to this statement.

Once the triangles are similar by AA ...
the remaining set of angles will be congruent and the remaining corresponding sides will be in proportion. Just be sure the triangles are similar before using the following statement.
Corresponding sides of similar triangles are in proportion.

sstritext4
SStri


When working with similar triangle proofs, we will even be going one step further and
"cross-multiplying" the proportions to form products.

• Corresponding sides of similar triangles are in proportion.
DE • AC = AB • DF • In a proportion, the product of the means equals the product of the extremes.



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