Four sides, 360°
A quadrilateral is any flat shape with four straight sides. Draw a diagonal from one corner to the opposite corner, and it cuts the quadrilateral into two triangles. The three angles of each triangle add to 180°, so the four angles of any quadrilateral add to 2 × 180° = 360°.
Some quadrilaterals have names of their own. Each name says which sides are parallel, which sides are equal and which angles are equal, and those facts are what you use to work out a missing angle or side.
The rectangle
A rectangle has four right angles. Its opposite sides are equal: the top and the bottom are the same length, and so are the two upright sides.
Its opposite sides are also parallel. The top and the bottom both meet the left side at a right angle, so they run in the same direction and never meet.
A square is a rectangle whose four sides are all equal.
A rectangle with width w and height h. Each side has an equal, parallel side opposite it, and every corner is a right angle.
The parallelogram
A parallelogram keeps both pairs of opposite sides parallel, but its corners need not be right angles. It looks like a rectangle that has been pushed over, so that its corners lean.
Its opposite sides are equal too. Push a rectangle over and its top side slides along the line it is on without changing length, while the two upright sides tilt by the same amount and stay equal to each other.
A parallelogram: the top and the bottom are parallel, and so are the two slanted sides.
The angles of a parallelogram
The parallel sides fix the angles. Each side of a parallelogram runs across from one side to the side parallel to it. So the two angles at its ends are co-interior angles: the pair between two parallel lines, on the same side of the line that crosses them. Co-interior angles add to 180°.
That makes any two neighboring angles of a parallelogram add to 180°. If one angle is 70°, the angle next to it is 180° − 70° = 110°. The angle after that is next to the 110°, so it is 180° − 110° = 70°, the same as the angle opposite it. Opposite angles are equal, and the four angles are 70°, 110°, 70° and 110°. Check them against the total: 70 + 110 + 70 + 110 = 360.
A line crossing two parallel lines. The two marked angles are between the parallels and on the same side of the crossing line, so they are co-interior: 70° + 110° = 180°.
The names nest
A rectangle is a parallelogram whose angles are all 90°, and a square is a rectangle whose sides are all equal. So anything true of every parallelogram is true of every rectangle and every square: in a rectangle, neighboring angles are 90° + 90° = 180°, and opposite sides are equal.
The names do not work the other way round. A parallelogram need not have a right angle, so it need not be a rectangle. And a shape that looks like a square is not known to be one until its sides and angles are shown to be equal. Use only what a name guarantees.
The trapezium
A trapezium keeps just one pair of sides parallel. In the United States it is called a trapezoid. Its two parallel sides are usually different lengths, and its other two sides, called the legs, are not parallel.
Each leg runs across from one parallel side to the other, so the two angles at the ends of a leg are co-interior and add to 180°. That is all the one parallel pair gives. The two angles at the bottom need not be equal, and the two legs need not be the same length.
A trapezium: the top and the bottom are parallel, and the two legs are not.
The isosceles trapezium
When the two legs of a trapezium are equal, it is called an isosceles trapezium. It has a line of symmetry through the middle of both parallel sides. Fold it along that line and one leg lands exactly on the other, so the two angles at the bottom are equal, and so are the two angles at the top.
If an isosceles trapezium has an angle of 72° at the bottom, the other bottom angle is 72° as well. Each top angle is co-interior with a bottom angle along a leg, so it is 180° − 72° = 108°. Check: 72 + 72 + 108 + 108 = 360.
The dashed line runs through the middle of both parallel sides. Folded along it, the isosceles trapezium lands on itself, leg on leg and angle on angle.
Worked example: Parallelogram: Adjacent and Opposite Angles
Question ABCD is a parallelogram. ∠ DAB is 40° more than ∠ ABC. Find ∠ ABC and ∠ BCD.
1.AD ∥ BC, so ∠ DAB + ∠ ABC = 180°.
AD and BC are parallel, so the angles at A and B add to 180°. 2.∠ DAB is 40° more than ∠ ABC: take the 40° off, 180° − 40° = 140° is two equal shares.
Take off the 40° difference: 140° shared equally. 3.∠ ABC = 140° ÷ 2 = 70° and ∠ DAB = 70° + 40° = 110°.
∠ ABC = 70°, ∠ DAB = 110°. 4.Opposite angles of a parallelogram are equal: ∠ BCD = ∠ DAB = 110°.
The angle at C is opposite A: 110°. 5.Check: 110 + 70 + 110 + 70 = 360.
110 + 70 + 110 + 70 = 360.
Answer: ∠ ABC = 70°; ∠ BCD = 110°
Common mistakes
- Halving 180° and then adding 40° to one angle: 90° and 130° add to 220°, not 180°.
- Giving ∠ BCD = ∠ ABC: C is opposite A, not B.
Worked example: Isosceles Trapezium and a Diagonal
Question PQRS is a trapezium with PQ parallel to SR and PS = QR. ∠ PSR = 72° and the diagonal PR makes ∠ PRS = 41°. Find ∠ SPR and ∠ QRP.
1.PS = QR, so the trapezium is isosceles and its base angles are equal: ∠ QRS = ∠ PSR = 72°.
Equal legs: the base angles at S and R are both 72°. 2.In triangle PSR: ∠ SPR = 180° − 72° − 41° = 67°.
Triangle PSR: ∠ SPR = 180° − 72° − 41° = 67°. 3.The diagonal PR splits ∠ QRS: ∠ QRP = 72° − 41° = 31°.
The diagonal splits the 72° at R: ∠ QRP = 72° − 41° = 31°. 4.Check with the Z shape: PQ ∥ SR, so ∠ QPR = ∠ PRS = 41°.
Z shape: ∠ QPR = ∠ PRS = 41°. 5.Then ∠ SPQ = 67° + 41° = 108°, and 108° + 72° = 180° on the leg PS.
At P: 67° + 41° = 108°, and 108° + 72° = 180° on the leg PS.
Answer: ∠ SPR = 67°; ∠ QRP = 31°
Common mistakes
- Assuming the diagonal bisects the angle at R and answering 36°: a trapezium's diagonal does not bisect its angles.
- Using ∠ QRS = 108° by confusing the base angle with the top angle of the trapezium.