Establish the AA criterion for similar triangles from the properties of similarity transformations, and use it

Learning goal: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar, and apply the criterion with its conditions stated.

The criterion. The AA (angle-angle) criterion says: if two angles of one triangle are equal in measure to two angles of another triangle, then the triangles are similar. Only two angles need to be checked, and no side at all. This lesson shows why the criterion follows from the definition of similarity in terms of similarity transformations, rather than asking you to accept it.

What is already established. Three facts carry the argument. (1) The angles of a triangle sum to 180°. (2) A dilation with scale factor k preserves every angle and multiplies every length by k. (3) Two triangles with two pairs of equal angles and the pair of included sides equal are congruent, meaning a sequence of rigid motions takes one onto the other; this is the ASA congruence criterion, itself established from rigid motions in the congruence unit.

The argument. Suppose triangle ABC and triangle DEF have angle A = angle D and angle B = angle E. Step 1: the third angles are equal. Angle C = 180° − angle A − angle B and angle F = 180° − angle D − angle E, and the subtracted quantities are the same, so angle C = angle F. Step 2: dilate triangle ABC from the vertex A with scale factor k = DE/AB. The image is a triangle AB′C′ with B′ on ray AB and C′ on ray AC. By the properties of dilations, its angles are unchanged: angle at A is still angle A = angle D, angle at B′ is angle B = angle E, and AB′ = k · AB = (DE/AB) · AB = DE. Step 3: triangles AB′C′ and DEF now have two pairs of equal angles, at A and D and at B′ and E, and the sides between those angles, AB′ and DE, are equal. By ASA they are congruent, so a sequence of rigid motions takes AB′C′ onto DEF. Step 4: the dilation followed by that sequence of rigid motions is a similarity transformation taking ABC onto DEF. By the definition of similarity, triangle ABC is similar to triangle DEF. That completes the argument, and every step used only the listed facts.

Why does ASA guarantee the rigid motions in step 3? Translate so that A lands on D. Rotate about D so that B′ lands on E, which is possible because AB′ = DE. If C′ and F are now on opposite sides of line DE, reflect across that line. The ray from D toward C′ makes the same angle with DE as the ray from D toward F, so it is the same ray; likewise the ray from E toward C′ is the ray from E toward F. Two distinct rays meet in at most one point, so C′ lands on F.

A worked model with numbers. Triangle ABC has angles 40° at A and 60° at B, and a protractor-and-ruler drawing gives sides AB = 6.9 cm, BC = 4.5 cm, CA = 6.1 cm, each to the nearest millimeter. Triangle DEF has angles 40° at D and 60° at E and side DE = 13.8 cm. By AA the triangles are similar, with C and F both 80°. The scale factor from ABC to DEF is DE/AB = 13.8/6.9 = 2, so EF = 2 · 4.5 = 9.0 cm and FD = 2 · 6.1 ≈ 12.2 cm, to the precision of the drawing: doubling also doubles the reading error, and a careful drawing of DEF itself measures FD ≈ 12.1 cm (the Law of Sines gives 12.14 cm). These are hypothetical ruler readings to the nearest millimeter; they are consistent with the angles (the Law of Sines, from a later lesson, gives BC ≈ 4.50 cm and CA ≈ 6.07 cm), as you can confirm by drawing the triangle. The point of the example is that two angles fixed the shape, and one side then fixed the size.

Exact families. Right triangles with a 30° angle all have angles 30°, 60°, 90°, so by AA every one is similar to every other. Their sides are always in the ratio 1 : √3 : 2, for example 5, 5√3, 10 or 8, 8√3, 16. Right triangles with a 45° angle are all similar, with sides in the ratio 1 : 1 : √2. Any two right triangles that share one acute angle are similar by AA, since the right angles supply the second pair. This last fact is the foundation of trigonometry in a later lesson.

Conditions and non-examples. The criterion needs two pairs of angles that correspond. One pair is not enough: triangles with angles 40°, 60°, 80° and 40°, 20°, 120° share a 40° angle but no similarity transformation can turn 60° into 20° or 120°. The criterion is for triangles only: a 2 by 3 rectangle and a 4 by 5 rectangle have four pairs of equal angles and are not similar. The reason AA works for triangles is step 3: a triangle is determined by two angles and the side between them, and the dilation supplies a side of the right length; step 1 adds that the third angles then agree as well. A quadrilateral is not determined by its angles.

The misconception. Students sometimes read AA as "if the triangles have two equal angles each, they are similar" and match a 50° angle in one triangle with a 70° angle in the other. The equal angles must be a pair, one from each triangle, and the second pair likewise. Write the correspondence explicitly, A ↔ D and B ↔ E, before naming the triangles similar, and list the sides in the same order: triangle ABC ~ triangle DEF means AB/DE = BC/EF = CA/FD.

Using AA to find lengths. Once similarity is established, corresponding sides are in one ratio. A triangle with angles 30°, 60°, 90° has its shortest side, opposite the 30° angle, equal to 5. A second triangle with the same angles has hypotenuse 16. By AA they are similar; the ratio of hypotenuses to shortest sides is 2 in the first triangle (10 to 5), so the shortest side of the second is 8 and the remaining side is 8√3 ≈ 13.86. The AA criterion did the geometric work; the ratio did the arithmetic.

Worked, applying the criterion: Triangle GHI has angles 55° at G and 35° at H. Triangle JKL has angles 35° at K and 90° at L. Is GHI similar to JKL? The third angle of GHI is 180° − 55° − 35° = 90°, at I. The third angle of JKL is 180° − 35° − 90° = 55°, at J. So G ↔ J (55°), H ↔ K (35°), I ↔ L (90°): two pairs of equal angles, and the triangles are similar by AA, with the correspondence GHI ~ JKL.

Worked, a 30-60-90 pair: Triangle 1 has angles 30°, 60°, 90° with shortest side 5, longer leg 5√3 and hypotenuse 10. Triangle 2 has angles 30°, 60°, 90° with hypotenuse 16. By AA the triangles are similar. Hypotenuse ratio 16/10 = 1.6, so the shortest side of triangle 2 is 5 · 1.6 = 8 and the longer leg is 5√3 · 1.6 = 8√3 ≈ 13.86. Check: 8² + (8√3)² = 64 + 192 = 256 = 16².

Worked, one angle is not enough: Triangle P has angles 40°, 60°, 80°; triangle Q has angles 40°, 20°, 120°. They share one angle, but the AA criterion needs two pairs. Since a similarity transformation cannot change an angle, and no angle of Q equals 60°, the triangles are not similar. A single shared angle says nothing.

Worked, the argument with a scale factor: Triangle ABC has angle A = 40°, angle B = 60°, AB = 6.9 cm; triangle DEF has angle D = 40°, angle E = 60°, DE = 13.8 cm. Dilate ABC from A by k = 13.8/6.9 = 2. The image AB′C′ has AB′ = 13.8 cm = DE and the same angles 40° at A and 60° at B′. By ASA, AB′C′ is congruent to DEF, so rigid motions take it onto DEF. The dilation followed by those rigid motions is a similarity transformation from ABC to DEF, which is what "similar" means.

Recording task: Draw triangle ABC with angle A = 40° and angle B = 60° on a base AB = 6.9 cm, and triangle DEF with angle D = 40° and angle E = 60° on a base DE = 13.8 cm. Measure the remaining sides of each and record them in a two-column table beside their ratios (expected: BC ≈ 4.5 cm and EF ≈ 9.0 cm; CA ≈ 6.1 cm and FD ≈ 12.1 cm; every ratio ≈ 2 within measuring error). Then trace ABC, dilate the tracing from A by 2 using a ruler along the rays AB and AC, and lay the result on DEF to confirm they match. Write the given and the four steps of the argument beneath the drawing, naming the fact each step uses.

Independent performance: Write out the argument that establishes the AA criterion for two triangles of your own choosing, given only two pairs of equal angles and one side of each triangle. State the scale factor of the dilation, the vertex used as its center, why the dilated triangle is congruent to the second triangle, and why a dilation followed by rigid motions is a similarity transformation. Then give one application: two right triangles that share an acute angle of 35°, with the hypotenuse of one equal to 20 and the hypotenuse of the other equal to 12, and find the ratio of any pair of corresponding sides. Finally give one non-example in which only one pair of angles agrees. Success criteria: the correspondence of vertices is written explicitly; every step names the property of dilations or rigid motions it uses; the third-angle step is justified by the 180° sum; the application states the ratio 12/20 = 0.6 and says why it applies to all three pairs of sides; and the non-example explains why one pair of angles is insufficient.