A parallelogram from a diagonal’s angle
Construct a parallelogram ABCD with AB = 7 cm, AD = 4 cm and the angle between AB and the diagonal AC = 30°.
Construct a parallelogram ABCD with AB = 6 cm, AD = 5 cm and the angle between AB and the diagonal AC = 45°.
Construct a parallelogram ABCD with AB = 4 cm, the diagonal AC = 8 cm and ∠CAB = 60°.
Construct a parallelogram ABCD with AB = 5 cm, the diagonal AC = 7 cm and ∠CAB = 30°.
1. Draw a base line and mark AB = 7 cm.
2. At A, construct ∠BAX = 30° with a compass.
3. With B as the center and radius AD = 4 cm, cut AX at C (the farther of the two points where the arc can cut it). Join BC.
4. With A as the center and radius 4 cm, and C as the center and radius 7 cm, draw arcs that meet at D.
5. Join AD and CD: ABCD is the parallelogram.
1. Draw a base line and mark AB = 6 cm.
2. At A, construct ∠BAX = 45° with a compass.
3. With B as the center and radius AD = 5 cm, cut AX at C (the farther of the two points where the arc can cut it). Join BC.
4. With A as the center and radius 5 cm, and C as the center and radius 6 cm, draw arcs that meet at D.
5. Join AD and CD: ABCD is the parallelogram.
1. Draw a base line and mark AB = 4 cm.
2. At A, construct ∠BAX = 60° with a compass.
3. With A as the center and radius 8 cm, cut AX at C. Join BC.
4. With A as the center and radius BC, and C as the center and radius AB = 4 cm, draw arcs that meet at D.
5. Join AD and CD: ABCD is the parallelogram.
1. Draw a base line and mark AB = 5 cm.
2. At A, construct ∠BAX = 30° with a compass.
3. With A as the center and radius 7 cm, cut AX at C. Join BC.
4. With A as the center and radius BC, and C as the center and radius AB = 5 cm, draw arcs that meet at D.
5. Join AD and CD: ABCD is the parallelogram.
Why it works
The angle at A fixes the line the diagonal AC lies on, and either BC (equal to AD) or AC itself fixes where C is on it. Then △CDA is △ABC turned round, so the opposite sides are equal and ABCD is a parallelogram. With two sides given, an arc from B can cut the line twice: the books take the farther point, which gives the longer diagonal.