One angle and the sides around it

The idea

If one angle of a triangle equals one angle of another triangle and the sides including these angles are proportional, the triangles are similar (SAS similarity criterion).

A hinge with two sticks

Triangle ABC has ∠A = 60°, with AB = 5 and AC = 8. Triangle DEF has ∠D = 60° too, with DE = 10 and DF = 16.

The two sides around the 60° angle are in the same ratio: 10/5 = 16/8 = 2. You know nothing about EF or the other angles.

Is Δ DEF similar to Δ ABC?

• Yes. The third side and the other angles must match too.
• No. We would also need to know EF.
• Only if EF is measured and found in the same ratio.

Two sticks round a 60° hinge

Triangle ABC is fixed, with ∠A = 60°. Build DEF with the same 60° angle at D. Choose the two sticks DE and DF around it with the dials. The third side EF follows.

Angle and the sides around it

If ∠A = ∠D and AB/DE = AC/DF, then Δ ABC ~ Δ DEF.

This is the SAS criterion (Side-Angle-Side). One angle is equal, and the two sides including that angle are in the same ratio. The angle must be between the two sides.

Mark P on DE with DP = AB, and Q on DF with DQ = AC. Then Δ DPQ is a copy of Δ ABC: two sides and the angle between them are equal. Since AB/DE = AC/DF, we get DP/DE = DQ/DF, so PQ is parallel to EF (converse of the Basic Proportionality Theorem). Then ∠P = ∠E and ∠Q = ∠F, and these are equal to ∠B and ∠C. So all the angles of the two triangles are equal.

Notes

If ∠A = ∠D and AB/DE = AC/DF (the sides including the equal angles), then Δ ABC ~ Δ DEF (SAS criterion).

Check yourself

In Δ ABC, ∠A = 50°, AB = 3 cm and AC = 4 cm. In Δ DEF, ∠D = 50°, DE = 9 cm and DF = 12 cm. Are the triangles similar?

In Δ ABC, ∠A = 50°, AB = 3 cm and AC = 4 cm. In Δ DEF, ∠D = 50°, DE = 6 cm and DF = 9 cm. Are the triangles similar?

In Δ ABC, AB = 4, BC = 6 and ∠A = 40°. In Δ DEF, DE = 8, EF = 12 and ∠D = 40°. The sides are in the same ratio 2 and the angles are equal. Can we say Δ ABC ~ Δ DEF by SAS?

In Δ ABC and Δ DEF, ∠A = ∠D and AB/DE = AC/DF. AB = 4 cm, AC = 6 cm and DE = 10 cm. Find DF, in cm.

Answer: 15

AB/DE = 4/10 = 2/5, so AC/DF = 2/5 and DF = 6 × 5/2 = 15 cm.

• Yes, they are similar. — correct. Yes! ∠A = ∠D, and 9/3 = 12/4 = 3. An equal angle with the sides around it in the same ratio: SAS.
• No, we do not know the third side.. SAS does not need the third side. It follows from the other two sides and the angle.
• No, the triangles have different sizes.. Similar triangles can have different sizes.
• Yes, because the angles at A and D are equal.. One equal angle is not enough. The sides around it must be in the same ratio too.
• No, they are not similar. — correct. Yes! 6/3 = 2 but 9/4 = 2.25. The sides around the 50° angle are not in the same ratio, so SAS does not hold.
• Yes, because DE and DF are bigger than AB and AC.. Being bigger is not enough. The two ratios must be equal.
• Yes. There is an equal angle and the sides are in the same ratio.. The angle ∠A is between AB and AC, not between AB and BC. The sides given do not include the equal angle.
• No. The equal angle ∠A is not between the sides AB and BC. — correct. Yes! SAS needs the equal angle to be between the two sides. Here AB and BC include ∠B, not ∠A.
• No. The triangles are different sizes.. Size does not matter for similarity. The problem is where the angle sits.