A Stick's Shadow Can Measure a Pyramid You Cannot Climb
Around 600 BC, Thales is said to have found the height of the Great Pyramid using nothing but his own shadow. He waited until his shadow equalled his height, then measured the pyramid's shadow. The trick works because the sun strikes a man and a pyramid at the same angle, so the two triangles they cast are the same shape. That "same shape from equal angles" idea is exactly the AA criterion.
What Is the AA Criterion in Triangles?
The AA (Angle-Angle) criterion states that if two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. Similar means the triangles have the same shape: their corresponding angles are equal and their corresponding sides are in the same ratio.
For $\triangle ABC$ and $\triangle DEF$, the AA condition is:
$$\angle A = \angle D \qquad \text{and} \qquad \angle B = \angle E ;\Longrightarrow; \triangle ABC \sim \triangle DEF.$$
The symbol $\sim$ means "is similar to." When the AA criterion holds, the sides obey:
$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}.$$
Notice what AA does not claim: it does not say the triangles are the same size. Two equal angles fix the shape but leave the scale free, which is the crucial difference from congruence in triangles.
Why Is It Called AA and Not AAA?
A triangle has three angles, so you might expect all three to be needed. They are not, and the reason is the triangle sum theorem: the three angles of every triangle add to $180^\circ$.
If two pairs of angles are equal, the third pair is forced to be equal too:
$$\angle C = 180^\circ - \angle A - \angle B, \qquad \angle F = 180^\circ - \angle D - \angle E.$$
Since $\angle A = \angle D$ and $\angle B = \angle E$, subtracting the same amounts from $180^\circ$ leaves $\angle C = \angle F$ automatically. Checking the third angle would be redundant, so the criterion is named for the two angles you actually verify. Triangles with all three angles equal are called equiangular triangles, and by AA they are always similar.
How Do You Prove the AA Criterion?
The proof turns two equal angles into equal side-ratios using a parallel line and the basic proportionality theorem.
Given: in $\triangle ABC$ and $\triangle DEF$, $\angle A = \angle D$ and $\angle B = \angle E$. To prove: $\triangle ABC \sim \triangle DEF$.
Assume $\triangle DEF$ is the smaller triangle (if not, swap their roles). Mark a point $P$ on $AB$ so that $AP = DE$, and a point $Q$ on $AC$ so that $AQ = DF$. Join $PQ$.
Now compare $\triangle APQ$ and $\triangle DEF$:
$$AP = DE, \qquad \angle A = \angle D, \qquad AQ = DF.$$
Two sides and the included angle match, so by SAS, $\triangle APQ \cong \triangle DEF$. Congruent triangles have equal corresponding angles, so $\angle APQ = \angle E$. But $\angle E = \angle B$ was given, so:
$$\angle APQ = \angle B.$$
These are equal corresponding angles on line $AB$ cut by $PQ$ and $BC$, which means $PQ \parallel BC$. By the basic proportionality theorem, a line parallel to one side cuts the other two sides proportionally:
$$\frac{AP}{AB} = \frac{AQ}{AC}.$$
Replacing $AP = DE$ and $AQ = DF$ gives $\frac{DE}{AB} = \frac{DF}{AC}$, and the same argument on the third side completes the equal ratios. Therefore $\triangle ABC \sim \triangle DEF$. The AA criterion is proved.
How Is Similarity Different From Congruence?
This is the distinction the AA criterion lives or dies on, and it is where most errors start.
Similarity (AA) | Congruence (ASA) | |
|---|---|---|
What matches | Two angles | Two angles + included side |
Shape | Same | Same |
Size | Can differ (proportional) | Identical |
Symbol | $\sim$ | $\cong$ |
Sides | In equal ratio | Equal |
Two angles alone give similarity, because angles fix shape but not scale. Add an equal side and you upgrade to congruence, because the side locks the size. So the same two-angle information that makes triangles similar is one measurement short of making them congruent. For the congruence side of this story, see the batch companion on the ASA criterion, and for the broader idea across shapes, similar figures.
Where Is the AA Criterion Used?
The AA criterion is the engine behind indirect measurement, finding a length you cannot reach by comparing it with one you can:
Heights of tall objects from shadow lengths, as in the Thales pyramid story.
Distances across rivers or canyons using sightline triangles.
Scale drawings and maps, where every figure is similar to the real thing.
Optics, where an object and its image on a lens form similar triangles.
In each case, two equal angles guarantee the shapes match, and the proportional-sides relationship then converts the known length into the unknown one. The formal statement and further consequences appear at Wolfram MathWorld's Similar Triangles entry.
Examples of the AA Criterion in Triangles
The examples begin with spotting similarity and end with a measurement application.
Example 1
In $\triangle ABC$ and $\triangle PQR$, $\angle A = \angle P = 40^\circ$ and $\angle B = \angle Q = 75^\circ$. Are the triangles similar?
Two pairs of angles are equal, so by AA the triangles are similar. The third angles are equal automatically: $180^\circ - 40^\circ - 75^\circ = 65^\circ$ in each.
Final answer: yes, $\triangle ABC \sim \triangle PQR$ by AA.
Example 2
Two triangles each have angles $50^\circ$ and $60^\circ$. A student concludes they are congruent. Is that right?
The first instinct is that equal angles mean identical triangles. Test it with sizes: a triangle with angles $50^\circ, 60^\circ, 70^\circ$ and sides 3, 4, 5 has the same three angles as one with sides 6, 8, 10, yet the second is twice as big. Same angles, different sizes.
So they cannot be congruent. Equal angles give the same shape, not the same size, which is exactly similarity. The rule is AA similarity, not congruence.
Final answer: similar by AA, not congruent.
Example 3
$\triangle ABC \sim \triangle DEF$ with $AB = 6$, $DE = 3$, and $BC = 8$. Find $EF$.
Similar triangles have proportional sides, so $\frac{AB}{DE} = \frac{BC}{EF}$:
$$\frac{6}{3} = \frac{8}{EF} ;\Rightarrow; EF = \frac{8 \times 3}{6} = 4.$$
Final answer: $EF = 4$.
Example 4
In a figure, $PQ \parallel BC$ with $P$ on $AB$ and $Q$ on $AC$. Are $\triangle APQ$ and $\triangle ABC$ similar?
Since $PQ \parallel BC$, the corresponding angles are equal: $\angle APQ = \angle ABC$ and $\angle AQP = \angle ACB$. Two equal angles give similarity by AA.
Final answer: yes, $\triangle APQ \sim \triangle ABC$ by AA.
Example 5
A 2 m pole casts a 3 m shadow while a building casts a 45 m shadow at the same time. How tall is the building?
The sun's rays make equal angles, so the pole-triangle and building-triangle are similar by AA. Corresponding sides are proportional:
$$\frac{\text{height}}{\text{shadow}}: \quad \frac{2}{3} = \frac{h}{45} ;\Rightarrow; h = \frac{2 \times 45}{3} = 30 \text{ m.}$$
Final answer: the building is 30 m tall.
Example 6
Right triangles $\triangle ABC$ and $\triangle DEF$ each have a right angle, and $\angle A = \angle D$. Are they similar?
Both have a $90^\circ$ angle, and $\angle A = \angle D$ is the second equal pair. Two equal angles give similarity by AA.
Final answer: yes, similar by AA.
What Mistakes Do Students Make With the AA Criterion?
The errors nearly all come from confusing similarity with congruence.
Mistake 1: Calling AA-similar triangles congruent
Where it slips in: any problem where two triangles share two angles.
Don't do this: conclude the triangles are identical and set their sides equal.
The correct way: AA gives similarity, so set up a ratio of sides, not an equality. The students who slip here are usually the ones who treat the $\sim$ symbol as if it means $\cong$; the two say very different things about size.
Mistake 2: Checking all three angles unnecessarily
Where it slips in: verifying similarity when two angles are already known equal.
Don't do this: hunt for the third angle before concluding, wasting time.
The correct way: stop at two. The angle sum forces the third, so two equal pairs already settle it.
Mistake 3: Mismatching corresponding sides in the ratio
Where it slips in: writing the proportion for side lengths.
Don't do this: pair sides by their position on the page rather than by the equal angles they sit opposite.
The correct way: match sides that are opposite equal angles, then write the ratio. A scrambled correspondence gives a wrong length even when the similarity is correct.
Conclusion
The AA criterion in triangles proves two triangles similar when two pairs of angles are equal.
Only two angles are needed because the angle sum forces the third to match.
The proof cuts off equal segments to build a triangle congruent to the smaller one, then uses the basic proportionality theorem to get equal side-ratios.
AA gives similarity (same shape, proportional sides), not congruence (same shape and size).
To master similarity proofs and indirect measurement with a teacher, explore Bhanzu's geometry tutor or high school math tutor sessions, or browse math classes online.
Practise These Yourself
Work through the exercises below to solidify your understanding. Decide whether pairs of triangles are similar from their angles; use a similarity ratio to find a missing side; and solve one shadow-measurement problem end to end. If you find yourself setting sides equal instead of proportional, reread the similarity-versus-congruence table. To work through similar-triangle problems live with a Bhanzu trainer, book a free demo class.
Read More
Similar Triangles — all the similarity criteria (AA, SAS, SSS) with properties and proofs.
Areas of Similar Triangles — how areas scale with the square of the similarity ratio.
Properties of a Triangle — angle sum, side relationships, and classification recap.
Types of Triangles — a refresher on classifying triangles by angle and side.
Triangle Congruence Theorem — the congruence counterpart to these similarity rules.
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