4-gons and their angle sum
A quadrilateral or 4-gon has 4 sides, 4 corners (vertices) and 4 angles. A line joining two opposite corners is a diagonal.
One diagonal cuts a 4-gon into two triangles. Each triangle has angles adding to 180°. So the 4 angles add to 360°.
Family: a trapezium has one pair of parallel sides. A parallelogram has two. A rectangle is a parallelogram with a right angle. A rhombus is a parallelogram with all sides equal. A square is both a rectangle and a rhombus.
Properties of a parallelogram
Take parallelogram ABCD and draw diagonal AC. AB ∥ DC, so ∠BAC = ∠DCA (alternate angles). AD ∥ BC, so ∠BCA = ∠DAC. AC is shared. By ASA, ΔABC ≅ ΔCDA. From CPCT we get:
- Opposite sides are equal: AB = CD and AD = BC.
- Opposite angles are equal: ∠B = ∠D (and in the same way ∠A = ∠C).
- Next angles add to 180°: ∠A + ∠B = 180° (co-interior angles).
- The diagonals bisect each other: if they meet at O, then OA = OC and OB = OD (prove with ΔAOB ≅ ΔCOD by ASA).
How to test if a 4-gon is a parallelogram
These converses are also true. A 4-gon is a parallelogram if any one of these holds:
- both pairs of opposite sides are equal;
- both pairs of opposite angles are equal;
- the diagonals bisect each other;
- one pair of opposite sides is both equal and parallel.
Special parallelograms: diagonals equal → rectangle. Diagonals perpendicular → rhombus. Diagonals equal and perpendicular → square.
Midpoint theorem and its converse
Midpoint theorem: in ΔABC, if D is the midpoint of AB and E is the midpoint of AC, then DE ∥ BC and DE = ½ BC.
Why: extend DE to F so that EF = DE, and join CF. ΔADE ≅ ΔCFE (SAS: AE = CE, DE = FE, vertically opposite angles at E). So CF = AD = DB and CF ∥ AB. Then DBCF has one pair of sides equal and parallel, so it is a parallelogram. Hence DF ∥ BC and DF = BC, so DE = ½ BC.
Converse: a line through the midpoint of one side, drawn parallel to a second side, cuts the third side at its midpoint.
Medians meet in the ratio 2 : 1
A median joins a corner of a triangle to the midpoint of the opposite side. Every triangle has three medians, and they always meet at one point called the centroid, G.
G is two-thirds of the way down each median: AG : GD = 2 : 1. A short reason: let medians BE and CF meet at G, and let P, Q be midpoints of BG and CG. By the midpoint theorem, FE and PQ are both parallel to BC and equal to ½ BC. So FEQP is a parallelogram, and its diagonals bisect each other: GE = GP = PB and GF = GQ = QC. So G cuts each median 2 : 1, and the third median must pass through the same point.
The centroid is the balancing point of a triangle cut from card.
Symmetry and tiling
A line of symmetry folds a shape into two matching halves.
- General parallelogram: 0 lines (but it looks the same after a half-turn).
- Rectangle: 2 lines (through midpoints of opposite sides).
- Rhombus: 2 lines (the diagonals).
- Square: 4 lines.
Tiling (tessellation) means covering a floor with copies of one shape, with no gaps and no overlaps. Every parallelogram tiles, because angles ∠A, ∠B, ∠C, ∠D meet at each corner of the floor and add to 360°. In fact any 4-gon, even an odd-looking one, can tile a floor by turning copies half a circle.
Try it at home
Cut any paper triangle. Fold to find the midpoints of two sides and mark them. Join them with a ruler and measure: the line is exactly half the third side. Then fold to find all three midpoints, draw the three medians and check that they meet at one point. Balance the triangle on a pencil tip at that point.
Cut 9 copies of one 4-gon from card and try to cover a page with no gaps. In the 3D free play, set the angle to 90° or make the sides equal and watch the name and the symmetry lines change.
Key formulas and definitions
- Angle sum of a 4-gon = 360°
- Parallelogram: AB = CD, AD = BC, ∠A = ∠C, ∠B = ∠D, ∠A + ∠B = 180°
- Diagonals of a parallelogram: OA = OC, OB = OD
- Midpoint theorem: DE ∥ BC, DE = ½ BC
- Centroid: AG : GD = 2 : 1, so AG = ⅔ AD
Worked examples
1. Three angles of a 4-gon are 80°, 95° and 110°. Find the fourth.
Step 1: all four add to 360°. Step 2: 80 + 95 + 110 = 285. Step 3: fourth angle = 360 − 285 = 75°.
2. In parallelogram ABCD, ∠A = 70°. Find ∠B, ∠C and ∠D.
Step 1: next angles add to 180°, so ∠B = 180 − 70 = 110°. Step 2: opposite angles are equal: ∠C = ∠A = 70°, ∠D = ∠B = 110°.
3. The diagonals of parallelogram PQRS meet at O. PO = 4.5 cm and QO = 3 cm. Find PR and QS.
Step 1: diagonals bisect each other, so OR = PO = 4.5 cm and OS = QO = 3 cm. Step 2: PR = 4.5 + 4.5 = 9 cm. Step 3: QS = 3 + 3 = 6 cm.
4. In ΔABC, D and E are midpoints of AB and AC. BC = 11 cm. Find DE.
Step 1: by the midpoint theorem DE = ½ BC. Step 2: DE = 11 ÷ 2 = 5.5 cm. Step 3: also DE ∥ BC.
5. The sides of a triangle are 6 cm, 8 cm and 10 cm. Find the perimeter of the triangle formed by joining the midpoints of its sides.
Step 1: each side of the small triangle is half of one side of the big one. Step 2: sides are 3, 4 and 5 cm. Step 3: perimeter = 3 + 4 + 5 = 12 cm (half of 24).
6. AD is a median of ΔABC with length 12 cm, and G is the centroid. Find AG and GD.
Step 1: AG : GD = 2 : 1, so AD has 3 equal parts. Step 2: one part = 12 ÷ 3 = 4 cm. Step 3: AG = 2 × 4 = 8 cm, GD = 4 cm.
7. In 4-gon ABCD, the diagonals are equal and bisect each other. What kind of 4-gon is it?
Step 1: diagonals bisect each other → parallelogram. Step 2: diagonals are also equal → rectangle. (If they were also perpendicular it would be a square.)
Common mistakes
- Thinking every parallelogram has equal diagonals. Only rectangles (and squares) do.
- Saying DE = ½ BC when D and E are not both midpoints. Check both points are exact middles.
- Writing AG : GD = 1 : 2. The longer part is next to the corner: AG : GD = 2 : 1.
- Thinking a rhombus has 4 lines of symmetry. It has only 2 (its diagonals); a square has 4.