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AA Criterion from
Similarity Transformations

Discover how two matching angles guarantee that triangles are similar, using the language of dilations, rotations, and reflections.

Geometry

What Does Similar Mean?

Two figures are similar when one can be mapped onto the other by a sequence of similarity transformations: translations, rotations, reflections, and dilations. In symbols we write \(\triangle ABC \sim \triangle DEF\).

Same Shape
All corresponding angles are equal
Proportional Sides
All corresponding side lengths share a common ratio \(k\)

The Similarity Transformations

Rigid Motions
Translations, rotations, reflections: preserve both shape and size.
Dilation
Scales every length by factor \(k\): preserves shape but changes size.
Combined
Rigid motion + dilation = similarity transformation.

Why Two Angles Are Enough

Every triangle has angles summing to \(180^\circ\). If two angles of one triangle equal two angles of another, the third angle is forced to be equal as well:

Because all three pairs of angles match, the triangles have the same shape. A dilation can then scale one onto the other, proving similarity.

The AA Similarity Criterion

A B C
\(50^\circ\)
\(70^\circ\)
\(\sim\)
 D E F
\(50^\circ\)
\(70^\circ\)

Since \(\angle A = \angle D = 50^\circ\) and \(\angle B = \angle E = 70^\circ\), the AA Criterion gives \(\triangle ABC \sim \triangle DEF\).

Proof via Transformations

1
Dilate \(\triangle DEF\) by factor \(k = \frac{AB}{DE}\) so that side \(D'E' = AB\).
2
Translate & rotate the dilated triangle so \(D'\) lands on \(A\) and \(D'E'\) aligns with \(AB\).
3
Because \(\angle A = \angle D\) and \(D'E' = AB\), ray \(D'F'\) coincides with ray \(AC\).
4
Because \(\angle B = \angle E\) and the rays from both \(B\) and \(E'\) coincide, \(F'\) must land on \(C\).
5
We mapped \(\triangle DEF\) onto \(\triangle ABC\) using a similarity transformation, so \(\triangle ABC \sim \triangle DEF\). ✓

What AA Guarantees

Once we establish similarity via AA, two powerful consequences follow automatically:

Equal Angles
All three pairs of corresponding angles are congruent: \(\angle A = \angle D\), \(\angle B = \angle E\), \(\angle C = \angle F\).
Proportional Sides
All corresponding sides share the same ratio: \(\displaystyle\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k\).

AA vs. Other Similarity Tests

Criterion What You Need Key Feature
AA 2 pairs of equal angles No side info required
SSS 3 pairs of proportional sides Needs all three side ratios
SAS 2 proportional sides + included angle Mix of sides and angles

How to Apply the AA Criterion

1
Identify the two triangles you want to compare. Label their vertices consistently.
2
Find two pairs of equal angles. Look for given values, vertical angles, parallel-line angle pairs, or shared angles.
3
State the similarity. Write \(\triangle ABC \sim \triangle DEF\) with vertices in matching order.
4
Set up proportions if you need to find missing side lengths: \(\displaystyle\frac{AB}{DE} = \frac{BC}{EF}\).

Example 1: Given Angles

P Q R
\(40^\circ\)
\(65^\circ\)
 6 8 X Y Z
\(40^\circ\)
\(65^\circ\)
 12
\(x\)

Example 2: Parallel Lines

D E A B C
\(DE \parallel BC\)
 4 6 5
\(x\)

Example 3: Shared Angle

A B E C D
\(\alpha\)
\(90^\circ\)
\(90^\circ\)
 3 7 5

Describing the Transformation Sequence

When a problem asks you to explain similarity using transformations, you must name the specific steps that map one triangle onto the other.

1
Dilation: Center at a convenient point, scale factor \(\displaystyle k = \frac{\text{new side}}{\text{old side}}\).
2
Translation: Slide a vertex of the image onto its corresponding vertex.
3
Rotation: Rotate so a pair of corresponding sides align.
4
Reflection (if needed): Flip if the orientation is reversed.

When AA Does Not Apply

AA works only for triangles. Two rectangles can share two \(90^\circ\) angles yet have completely different shapes. Why?

Triangles
3 angles summing to \(180^\circ\): two angles determine the third. Shape is locked.
Quadrilaterals+
4+ angles give extra degrees of freedom. Matching angles alone do not guarantee the same proportions.

Common Mistakes to Avoid

✕ Wrong Vertex Order
Writing \(\triangle ABC \sim \triangle FDE\) when the correct correspondence is \(\triangle ABC \sim \triangle DEF\). Vertex order must match angle correspondence.
✕ Assuming Equal Sides
AA proves similarity (proportional sides), not congruence (equal sides). Similar triangles can be different sizes.
✕ AA on Non-Triangles
Applying the AA criterion to rectangles, pentagons, or other polygons where it does not hold.

Summary at a Glance

Concept Details
AA Criterion 2 equal angle pairs → triangles are similar
Transformation Basis Dilation + rigid motions map one triangle onto the other
Consequences All 3 angle pairs equal, all 3 side pairs proportional
Scale Factor \(k\) \(\displaystyle\frac{\text{corresponding side of image}}{\text{corresponding side of pre-image}}\)
Limitations Applies only to triangles, not general polygons
Writing the Statement Vertices listed in corresponding order: \(\triangle ABC \sim \triangle DEF\)

Question 1 of 5

In \(\triangle JKL\), \(\angle J = 55^\circ\) and \(\angle K = 80^\circ\). In \(\triangle MNP\), \(\angle M = 55^\circ\) and \(\angle P = 45^\circ\). Are the triangles similar by AA?

Question 2 of 5

\(\triangle ABC \sim \triangle DEF\) by AA with \(AB = 9\), \(DE = 6\), and \(EF = 10\). What is \(BC\)?

Question 3 of 5

Line \(DE\) is drawn parallel to base \(BC\) of \(\triangle ABC\), intersecting \(AB\) at \(D\) and \(AC\) at \(E\). Which statement best explains why \(\triangle ADE \sim \triangle ABC\)?

Question 4 of 5

Which sequence of transformations maps \(\triangle PQR\) onto a similar triangle \(\triangle P'Q'R'\) that is twice as large?

Question 5 of 5

Two rectangles each have four \(90^\circ\) angles. A student claims they must be similar by "AA." What is wrong with this reasoning?