Same ratios, same angles

The idea

If the sides of one triangle are proportional to the sides of another triangle, their corresponding angles are equal and the triangles are similar (SSS similarity criterion).

Sticks twice as long

Triangle ABC is made from three sticks: AB = 4, BC = 5 and CA = 6. Triangle DEF is made from sticks twice as long: DE = 8, EF = 10 and FD = 12.

The sides are in the same ratio: 8/4 = 10/5 = 12/6 = 2. Now think about the corners.

Are the angles of DEF equal to the angles of ABC?

• Yes. ∠D = ∠A, ∠E = ∠B and ∠F = ∠C.
• No. Longer sticks make bigger angles.
• Only the biggest angle is equal.

Build a triangle from three sticks

Triangle ABC is fixed. Build DEF by choosing its three sides DE, EF and FD with the dials. Watch the ratios and the angles.

Equal ratios give equal angles

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

This is the SSS criterion (Side-Side-Side) for similar triangles. If the sides of one triangle are proportional to the sides of another triangle, the corresponding angles are equal.

Zoom triangle ABC by the common ratio k. The zoomed triangle has sides k × AB, k × BC and k × CA, and these are exactly DE, EF and FD. Two triangles with three equal sides are congruent (Class IX), so DEF is the zoomed ABC. A zoom does not change angles, so ∠D = ∠A, ∠E = ∠B and ∠F = ∠C.

Notes

If DE/AB = EF/BC = FD/CA, then the corresponding angles are equal and Δ ABC ~ Δ DEF (SSS criterion).

Check yourself

The sides of Δ ABC are 3, 5 and 7. The sides of Δ DEF are 6, 10 and 14. Are the triangles similar?

The sides of Δ ABC are 4, 6 and 8. The sides of Δ DEF are 6, 9 and 11. Are the triangles similar?

In Δ ABC, ∠A = 80° and ∠B = 60°. In Δ PQR, PQ = 2 × AB, QR = 2 × BC and RP = 2 × CA. Find ∠R, in degrees.

Answer: 40

All ratios are 2, so Δ ABC ~ Δ PQR (SSS) and ∠R = ∠C = 180° − 80° − 60° = 40°.

Δ ABC ~ Δ DEF by SSS. AB = 6 cm, CA = 10 cm and DE = 12 cm. Find FD, in cm.

Answer: 20

DE/AB = 12/6 = 2, so FD = 2 × CA = 2 × 10 = 20 cm.

• Yes, they are similar. — correct. Yes! 6/3 = 10/5 = 14/7 = 2. All three ratios are equal, so the triangles are similar by SSS.
• No, the sides are different lengths.. Similar triangles can have different sizes. The ratios 6/3, 10/5 and 14/7 are all 2.
• We cannot tell without the angles.. We do not need the angles. By SSS, equal ratios are enough.
• Yes, two of the three ratios are equal.. All three ratios must be equal. 6/4 = 1.5 and 9/6 = 1.5, but 11/8 is only about 1.4.
• No, they are not similar. — correct. Yes! 6/4 = 9/6 = 1.5, but 11/8 is not 1.5. The sides are not proportional.
• Yes, the sides grow by about 1.5.. 11/8 is about 1.4, not 1.5. Every ratio must be exactly the same.