Same angles, same ratios

The idea

If the corresponding angles of two triangles are equal (equiangular triangles), their corresponding sides are in the same ratio and the triangles are similar (AAA similarity criterion).

Two triangles, equal angles

Triangle ABC is small and triangle DEF is big. Their angles are equal: ∠A = ∠D, ∠B = ∠E and ∠C = ∠F. Triangles like this are called equiangular.

Corners with the same number of tick marks have equal angles.

Are the sides of ABC in the same ratio to the sides of DEF?

• Yes. AB/DE = BC/EF = CA/FD
• No. Equal angles say nothing about the sides.
• Only if the two triangles are the same size.

Slide the triangle into the corner

The small triangle ABC has the same angles as the big triangle DEF. Drag ABC up to the big triangle, or use the slide dial, until its corner A sits on the corner D. Do it for each of the three sizes.

Equal angles give equal ratios

If ∠A = ∠D, ∠B = ∠E and ∠C = ∠F, then AB/DE = BC/EF = CA/FD, and Δ ABC ~ Δ DEF.

This is the AAA criterion (Angle-Angle-Angle) for similar triangles. In two equiangular triangles the ratio of any two corresponding sides is always the same. This was the truth that the Greek mathematician Thales found.

Slide ABC into the corner D, as you did. A sits on D, B on DE and C on DF. Because ∠B = ∠E (corresponding angles), the line BC is parallel to EF. So BC cuts DE and DF in the same ratio (the Basic Proportionality Theorem): DB/BE = DC/CF. This gives AB/DE = AC/DF. In the same way, AB/DE = BC/EF.

Notes

If ∠A = ∠D, ∠B = ∠E and ∠C = ∠F, then AB/DE = BC/EF = CA/FD and the triangles are similar (AAA criterion).

Check yourself

Two triangles both have angles 30°, 60° and 90°. In the first, the shortest side is 2 cm. In the second, the shortest side is 5 cm. What can you say?

Δ ABC and Δ DEF have equal corresponding angles. AB = 3 cm, DE = 6 cm and BC = 4 cm. Find EF, in cm.

Answer: 8

AB/DE = 3/6 = 1/2, so BC/EF = 1/2 and EF = 2 × 4 = 8 cm.

Slide Δ ABC into the corner D of Δ DEF, with ∠A = ∠D and ∠B = ∠E. Which line is parallel to EF?

A friend says: "These two triangles have equal angles, so their sides must be equal too." What is wrong?

• They are similar, but not congruent. — correct. Yes! Equal angles make them similar (AAA). Their sizes are different, so they are not congruent.
• They are congruent.. Equal angles do not make equal sizes. One has shortest side 2 cm and the other 5 cm.
• They are not similar.. All three angles are equal, so they are similar by the AAA criterion.
• AB. AB lies along DE, so it meets EF at E.
• BC — correct. Yes! ∠B and ∠E are corresponding angles and they are equal, so BC ∥ EF.
• AC. AC lies along DF, so it meets EF at F.
• Nothing. Equal angles mean equal sides.. A small and a big triangle can have the same angles. Their sides are in the same ratio, but not equal.
• Equal angles give sides in the same ratio, not equal sides. — correct. Yes! The triangles are similar. They are congruent only if the ratio is 1.
• Equal angles tell us nothing about the sides.. They tell us a lot: by AAA the sides are in the same ratio.