Diagonals cut the angles

The idea

In a rhombus, a diagonal makes four equal angles with the sides, so the diagonals bisect the angles of the rhombus.

Do the diagonals halve the angles?

This rhombus leans: its angles are 50° and 130°. In a square a diagonal cuts each corner in half. What about a leaning rhombus?

Does a diagonal of this rhombus cut the corners it passes through into two equal parts?

• Yes, at all four corners
• Only at the 50° corners
• No, not at any corner

Find the halving lines

Choose a corner. Drag the round handle until the orange and blue parts are equal. Where does the line go? Do all four corners.

Equal angles with the sides

In a rhombus, a diagonal makes four equal angles with the sides, so the diagonals bisect the angles of the rhombus.

Every halving line was a diagonal. The two halves were 25° at the 50° corners and 65° at the 130° corners.

Why? In ∆ADB, AD = AB, so the angles at D and B are equal. Call each a: a + a + 50° = 180°, so a = 65°.

Notes

In a rhombus, a diagonal makes four equal angles with the sides, so the diagonals bisect the angles of the rhombus.

Check yourself

In rhombus ABCD, ∠A = 50°. The diagonal AC cuts ∠A into two equal parts. How large is each part?

Answer: 25 °

The diagonal AC bisects ∠A, so each part is 50° ÷ 2 = 25°.

In rhombus ABCD, ∠B = 100°. The diagonal BD passes through B. How large is ∠ABD?

Answer: 50 °

The diagonal BD bisects ∠B, so ∠ABD = 100° ÷ 2 = 50°.

In rhombus ABCD, the diagonal AC makes an angle of 35° with the side AB. How large is ∠B?

Answer: 110 °

AC bisects ∠A, so ∠A = 2 × 35° = 70°. The neighbouring angles add up to 180°, so ∠B = 180° − 70° = 110°.

Which diagonal bisects the angle at B in rhombus ABCD?

• AC. AC joins A and C. It bisects the angles at A and at C.
• BD — correct. Yes! BD passes through B, and the diagonals of a rhombus bisect the angles at their ends.
• Neither: a diagonal never bisects an angle of a leaning rhombus. In every rhombus the diagonals bisect the angles, leaning or not.