Diagonals cut the corners

The idea

The diagonals of a square divide each of its angles into two equal halves of 45°.

How does a diagonal cut a corner?

Take a square and a diagonal that starts at one of its corners. The corner is a right angle, 90°. The diagonal cuts it into two parts.

How do you think the diagonal cuts the 90° corner?

• Into two equal parts
• Into a small part and a big part
• It cuts some corners equally and others not

Find the halving line

Choose a corner. Drag the round handle to swing a line out of it. The line cuts the corner into an orange part and a blue part. Make them equal, and see where the line goes.

Each half is 45°

The diagonals of a square divide each of its angles into two equal halves of 45°.

Every time, the halving line ran straight into the opposite corner: it was a diagonal.

Why 45°? In ∆ADC, ∠D = 90° and AD = DC, so the angles at A and C are equal. They add up to 180° − 90° = 90°, so each is 45°.

Notes

The diagonals of a square divide each of its angles into two equal halves of 45°.

Check yourself

The diagonal AC of a square ABCD cuts the angle at A into two equal parts. How large is each part?

Answer: 45 °

The corner of a square is 90°. The diagonal cuts it into two equal halves: 90° ÷ 2 = 45° each.

In the square ABCD, why is ∠DAC = ∠DCA?

ABCD is a square and its diagonals meet at O. What is ∠OAB + ∠OBA, in triangle AOB?

Answer: 90 °

The diagonals cut the corners in half, so ∠OAB = 45° and ∠OBA = 45°. Together: 45° + 45° = 90°.

In a rectangle ABCD the diagonal AC makes an angle of 45° with the side AB. What kind of rectangle is it?

• Because AD = DC, so the angles opposite these sides are equal — correct. Yes! In ∆ADC the sides AD and DC are equal, so the angles opposite them, ∠DCA and ∠DAC, are equal.
• Because AC is a diagonal. A diagonal alone does not make two angles equal. The equal sides AD and DC do.
• Because ∠D = 90°. ∠D = 90° tells us what the two other angles add up to. It is the equal sides that make them equal.
• Only a square can have that — correct. Yes! In ∆ABC, ∠B = 90° and ∠BAC = 45°, so ∠ACB = 45° too. Then BC = AB, and a rectangle with equal neighbouring sides is a square.
• A long thin rectangle. A long thin rectangle has a diagonal that lies close to the long side, much less than 45°.
• Any rectangle, the diagonal always makes 45°. Only in a square does the diagonal cut the corner into two equal 45° parts.