AA Criterion: Understanding and Proof

Deck 1 of 2 · HSG.SRT.A.3

This deck covers:

• Why two angle pairs are sufficient to guarantee similarity
• Formal proof using similarity transformations

Deck 2 covers: applying AA and solving real-world problems

Learning Objectives: Complete Series

By the end of this two-deck series, you should be able to:

1. State the AA criterion for triangle similarity
2. Prove AA using properties of similarity transformations
3. Apply AA to determine if two triangles are similar
4. Explain why two angle pairs are sufficient
5. Distinguish between AA for similarity and ASA/AAS for congruence
6. Use AA to solve real-world problems

This deck focuses on Objectives 1, 2, and 4.

To Verify Similarity, Must We Check Everything?

We know similar triangles have:

• All corresponding angles equal
• All corresponding sides proportional

Question: To confirm similarity, must we check all three angle pairs AND all three side ratios?

Or is there a shortcut?

Similarity: A Quick Recap

Definition: △ABC ~ △DEF when a similarity transformation maps one onto the other.

This guarantees:

• ∠A = ∠D, ∠B = ∠E, ∠C = ∠F (equal corresponding angles)
• AB/DE = BC/EF = AC/DF (proportional corresponding sides)

Our shortcut question: can we verify less and still guarantee all of this?

Triangle Angle Sum: The Key Constraint

In any triangle:

Consequence: Knowing two angles determines the third completely:

If ∠A = 50° and ∠B = 60°, then ∠C must equal 70°. No other value is possible.

Third Angle Is Determined Automatically

• △ABC: ∠A = 50°, ∠B = 60° → ∠C = 70°
• △DEF: ∠D = 50°, ∠E = 60° → ∠F = 70°

All three angle pairs match — even though we only specified two.

Quick Check: Third Angle Pairs

△PQR: ∠P = 40°, ∠Q = 80° → ∠R = ?

△STU: ∠S = 40°, ∠T = 80° → ∠U = ?

Do all three angle pairs match between these two triangles?

Calculate ∠R and ∠U before the next slide...

Two Angles → Three Match. What About Sides?

✓ Two angle pairs match → all three angle pairs match (angle sum theorem)

But: Similar triangles also need proportional sides.

? Equal angles → proportional sides automatically?

The AA Criterion says: YES — but we need to prove it.

AA Criterion: Statement and Scope

AA Criterion: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Formally: If ∠A ≅ ∠D and ∠B ≅ ∠E, then △ABC ~ △DEF.

Triangles only: AA works because of the triangle angle sum theorem. Equal angles alone do not guarantee similarity for quadrilaterals or other polygons.

Quick Check: Identifying AA

Which pairs can we confirm similar by AA?

Pair 1: △ABC with ∠A = 35°, ∠B = 80°
   △DEF with ∠D = 35°, ∠E = 80°

Pair 2: △PQR with ∠P = 60°, ∠Q = 60°
   △XYZ with ∠X = 60°, ∠Z = 60°

For Pair 2: are you matching corresponding angles?

Proof Strategy: Construct the Transformation

Recall: △ABC ~ △DEF means a similarity transformation maps △ABC onto △DEF.

Proof approach: Given ∠A ≅ ∠D and ∠B ≅ ∠E, construct the transformation.

Transformation sequence: Translate → Rotate → Dilate

If we build this successfully, similarity is proven by definition.

Proof Setup: Given and Goal

Given: △ABC and △DEF with ∠A ≅ ∠D and ∠B ≅ ∠E

To prove: △ABC ~ △DEF

Method: Build a transformation sequence in three steps:

1. Translate △ABC so vertex A coincides with D
2. Rotate the image so side A'B' aligns with ray DE
3. Dilate from D to scale the triangle to match △DEF exactly

Step 1: Translate A to D

Translation: Move △ABC so A lands on D.

• Image: △A'B'C' with A' = D
• Rigid motion → all angles and distances preserved: ∠A' = ∠A, ∠B' = ∠B

Step 2: Rotate to Align Ray A'B' with Ray DE

Rotation: Rotate △A'B'C' around D so ray A'B' aligns with ray DE.

• Image: △A''B''C'' with ray A''B'' on ray DE
• Rigid motion preserves angles: ∠A'' = ∠A = ∠D and ∠B'' = ∠B = ∠E

After Rigid Motions: Current State

After translation and rotation:

• A'' = D (by construction)
• Ray A''B'' lies on ray DE (by construction)
• ∠A'' = ∠A = ∠D (rigid motions preserve angles + given ∠A = ∠D)
• ∠B'' = ∠B = ∠E (rigid motions preserve angles + given ∠B = ∠E)

What remains: the triangle is correctly positioned. Now adjust the size.

Quick Check: Why Rigid Motions Preserve Angles

Question: Why do translations and rotations preserve angle measures?

• A) They scale all distances proportionally
• B) They preserve all distances, so shapes — and angles — stay identical
• C) They only preserve distances along one axis
• D) Angles are independent of all distances

Think about what "rigid" means before selecting...

Angle Equality Forces Ray Alignment

After rigid motions, both rays start at D and make the same angle with ray DE:

• Ray A''C'' makes angle ∠A'' with ray DE
• Ray DF makes angle ∠D with ray DE
• Since ∠A'' = ∠D, these two rays are the same ray

Conclusion: Ray A''C'' lies along ray DF — the angle equality forces this.

Step 3: Dilate to Match Sizes

Dilation: Center D, scale factor

• B'' scales along ray DE → lands at E (distance DE from D)
• C'' scales along ray DF → lands at F

Dilation Result: Triangle Maps Exactly

After dilation from D with :

• A'' → D (center of dilation stays fixed)
• B'' → E (scaled along ray DE to distance DE from D)
• C'' → F (scaled along ray DF to correct distance)

Result: △A''B''C'' maps exactly onto △DEF.

Proof Conclusion

Transformation sequence applied to △ABC:

1. Translate (A → D)
2. Rotate (ray A'B' → ray DE)
3. Dilate from D with scale factor

This sequence is a similarity transformation (rigid motions + dilation).

Therefore: △ABC ~ △DEF ∎

AA is a proven theorem, not an observation.

Check-In: Explain the Proof Steps

Challenge: Explain what each step accomplishes and why it's possible.

1. Why can we always translate A to D?
2. Why can we always rotate to align ray A'B' with ray DE?
3. Why does angle equality force ray A''C'' to lie on ray DF?
4. Why does the dilation send B'' to E and C'' to F?

Try explaining steps 3 and 4 in particular — those carry the proof.

Worked Example: Scale Factor from the Proof

Given: △ABC with ∠A = 50°, ∠B = 60°; △DEF with ∠D = 50°, ∠E = 60°
Also: AB = 5, DE = 10

Step 1: Two angle pairs match → △ABC ~ △DEF by AA ✓

Step 2: Scale factor from the dilation step:

Result: Every side in △DEF is 2 × the corresponding side in △ABC.

AA as Theorem, Not Axiom

Why does the proof approach matter?

AA is derived from:

• The transformation definition of similarity (HSG.SRT.A.2)
• Properties of rigid motions (preserve angles and distances)
• Properties of dilations (scale distances from a center, preserve angles)

AA is not assumed — it is proved. This connects AA to the transformation framework you built in HSG.SRT.A.1 and A.2.

Key Takeaways: Understanding and Proof

✓ Two angles determine three — angle sum theorem makes the third automatic
✓ AA Criterion proven — translate + rotate + dilate maps any two triangles with equal angle pairs
✓ AA for triangles only — the angle sum constraint is what makes the proof work

Two angle pairs are enough — "AA" means exactly that, not "AAA"
AA proves similarity (same shape) — Deck 2 will show: not necessarily the same size

Next: Deck 2 — Applying the AA Criterion

Coming up in Deck 2:

• Apply AA efficiently: identify angle pairs → conclude similarity
• Three example contexts: direct angles, parallel lines, right triangles
• Distinguish AA (similarity) from ASA/AAS (congruence) — with a concrete counterexample
• Three real-world problems: shadows, map scaling, parallel segments

AA is your most efficient tool for identifying similar triangles. Let's use it.