Corners alone, sides alone: not enough

The idea

For polygons, equal corresponding angles alone, or corresponding sides in the same ratio alone, is not sufficient for similarity; both conditions are needed.

A square and a rectangle

A square has four corners of 90°. A rectangle that is 3 units long and 2 units wide also has four corners of 90°.

So all their corresponding angles are equal.

Is the rectangle similar to the square?

• Yes. All the corners are equal.
• No. The sides are not in the same ratio.
• We cannot tell.

Bend the frame

The blue square has sides 2. Shape the orange four-sided frame with the dials: PQ and PS are two of its sides and ∠P is its corner. The toy checks two things: are the corners equal to the square's, and are the sides in the same ratio?

Both conditions are needed

For polygons, both conditions are needed for similarity: corresponding angles equal and corresponding sides in the same ratio. One condition alone is not enough.

A rectangle has the square's corners, but its sides are not in the same ratio. A rhombus has sides in the same ratio, but its corners are not the square's. Only a bigger or smaller square has both.

Triangles will turn out to be special. You will see why later in this chapter.

Notes

For polygons, both conditions are needed: corresponding angles equal and corresponding sides in the same ratio. Either one alone is not enough.

Check yourself

Rectangle R₁ is 6 cm by 3 cm and rectangle R₂ is 8 cm by 6 cm. All their corners are 90°. Are they similar?

A rhombus has four sides of 4 cm and corners of 60° and 120°. A square has four sides of 2 cm. Each side of the rhombus is 2 times a side of the square. Are they similar?

Rectangle A is 6 cm by 4 cm. Rectangle B is similar to A and its longer side is 9 cm. What is its shorter side, in cm?

Answer: 6

9 ÷ 6 = 1.5, so the shorter side is 4 × 1.5 = 6 cm.

Which statement about two polygons is correct?

• Yes. All the corners are equal.. Equal corners are only one of the two conditions. Check the sides too.
• Yes. Both are rectangles.. Two rectangles can have different shapes: one nearly a square, one long and thin.
• No. The sides are not in the same ratio. — correct. Yes! The long sides give 8 ÷ 6 = 4/3 but the short sides give 6 ÷ 3 = 2. These are not equal.
• Yes. The sides are in the same ratio.. That is only one condition. The rhombus has corners of 60° and 120°, not 90°.
• No. The corners of the rhombus are not 90°. — correct. Yes! The sides are in the same ratio but the corners are not equal, so the two are not similar.
• Yes. Both have four equal sides.. Four equal sides is not enough. The corners must match as well.
• If the corresponding angles are equal, the polygons are similar.. A rectangle and a square have equal angles but are not similar.
• If the corresponding sides are in the same ratio, the polygons are similar.. A rhombus and a square can have sides in the same ratio but are not similar.
• They are similar if the angles are equal and the sides are in the same ratio. — correct. Yes! Both conditions are needed.