One check is enough for triangles

The idea

For triangles, unlike other polygons, equal corresponding angles and proportional corresponding sides each imply the other, so only one condition needs checking for similarity.

Frames made of sticks

For polygons you must check two things: the angles and the side ratios. A square and a rectangle have equal angles but are not similar. A square and a rhombus have sides in the same ratio but are not similar.

Pin four sticks at their ends and you get a frame that can wobble. Now pin three sticks.

Can a triangle of three pinned sticks wobble into a different shape?

• Yes, like the four-stick frame.
• No. Three sticks make only one shape.
• Only if the sticks are very long.

Wobble test

Left: a frame of 4 sticks. Right: a triangle of 3 sticks. Drag the corner S to bend the frame. Then try to bend the triangle by dragging the corner C. At the end, swap the stick CA for another one.

Triangles are rigid

For triangles, equal corresponding angles and proportional corresponding sides imply each other, so you need to check only one of the two.

The sides of a triangle fix its angles (SSS), and the angles fix the ratios of its sides (AAA). That is why AAA, AA and SSS each work on their own.

Other polygons can wobble. A rectangle and a square have equal angles but different shapes. A rhombus and a square have sides in the same ratio but different shapes. For them you must check both conditions.

Notes

For triangles, equal corresponding angles and proportional corresponding sides imply each other, so only one condition needs checking. Other polygons need both.

Check yourself

To show that two triangles are similar, what do you need to check?

A student checks that two quadrilaterals have all corresponding angles equal, and says they are similar. What is wrong?

In Δ ABC and Δ DEF, ∠A = ∠D, ∠B = ∠E and ∠C = ∠F. AB = 4 cm, BC = 6 cm and DE = 10 cm. Find EF, in cm.

Answer: 15

All three angles are equal, so DE/AB = EF/BC. DE/AB = 10/4 = 5/2, so EF = 6 × 5/2 = 15 cm.

Which frame keeps its shape even if its corners are only pinned, not fixed to the ground?

• Both the angles and the sides, every time.. For polygons in general, yes. For triangles, one condition makes the other true.
• Only the angles, or only the sides. One is enough. — correct. Yes! For triangles, equal angles give proportional sides (AAA) and proportional sides give equal angles (SSS).
• Nothing. All triangles are similar.. A thin tall triangle and a flat wide triangle are not similar.
• Nothing is wrong. Equal angles always make polygons similar.. Not for quadrilaterals: a square and a rectangle have equal angles but are not similar.
• For polygons other than triangles, equal angles alone are not enough. — correct. Yes! A square and a rectangle have equal angles, but their sides are not in the same ratio.
• The student should have checked the areas of the two shapes.. Areas are not used in the definition. The sides must be in the same ratio.
• A frame of 3 sticks (a triangle) — correct. Yes! Three sticks can make only one shape, so the angles cannot change.
• A frame of 4 sticks (a quadrilateral). It can wobble: the sides stay the same, but the angles change.
• A frame of 5 sticks (a pentagon). It can wobble too. Only the triangle is rigid.