1. Chapter at a Glance
Take any four-sided shape — even a lopsided one — and join the midpoints of its sides. Surprise: you always get a parallelogram! Take any quadrilateral tile, and copies of it can cover a floor with no gaps. In this chapter we study quadrilaterals precisely: the properties of parallelograms and how to prove a shape is one, the powerful midpoint theorem for triangles, medians and the centroid, and these surprising results about every quadrilateral.
- Angle sum of a quadrilateral; family of special quadrilaterals.
- Properties of a parallelogram and their proofs.
- Conditions that guarantee a parallelogram (converses).
- Rectangles, rhombuses and squares through their diagonals.
- The midpoint theorem and its converse.
- Medians and the centroid of a triangle.
- Varignon's theorem and tiling the plane with any quadrilateral.
2. Quadrilaterals — Basics
A quadrilateral ABCD has 4 sides, 4 angles and 2 diagonals (AC and BD). A diagonal splits it into two triangles, so:
∠A + ∠B + ∠C + ∠D = 360°
| Type | Defining property |
|---|---|
| Trapezium | At least one pair of parallel sides |
| Parallelogram | Both pairs of opposite sides parallel |
| Rectangle | Parallelogram with one right angle |
| Rhombus | Parallelogram with all sides equal |
| Square | Rectangle with all sides equal (also a rhombus) |
| Kite | Two pairs of equal adjacent sides |
3. Properties of a Parallelogram
- A diagonal divides a parallelogram into two congruent triangles.
- Opposite sides are equal.
- Opposite angles are equal.
- Adjacent angles are supplementary (sum 180°).
- The diagonals bisect each other.
Proof of 1: In parallelogram ABCD, draw diagonal AC. AB ∥ DC with transversal AC gives ∠BAC = ∠DCA (alternate angles). AD ∥ BC gives ∠DAC = ∠BCA. AC is common. By ASA, △ABC ≅ △CDA. Properties 2 and 3 follow by CPCT.
Proof of 5: Let the diagonals meet at O. In △AOB and △COD: AB = CD, ∠OAB = ∠OCD and ∠OBA = ∠ODC (alternate angles). By ASA, they are congruent, so OA = OC and OB = OD.
4. When Is a Quadrilateral a Parallelogram?
Each property above has a true converse. A quadrilateral is a parallelogram if:
- both pairs of opposite sides are equal; or
- both pairs of opposite angles are equal; or
- its diagonals bisect each other; or
- one pair of opposite sides is both equal and parallel.
In ABCD, ∠A = 3x + 10°, ∠C = 5x − 30° and ABCD is a parallelogram. Find all angles.
Solution: Opposite angles equal: 3x + 10 = 5x − 30 → x = 20. ∠A = ∠C = 70°, ∠B = ∠D = 110°.
5. Special Parallelograms and Their Diagonals
| Shape | Diagonals bisect each other | Diagonals equal | Diagonals perpendicular | Diagonals bisect angles |
|---|---|---|---|---|
| Parallelogram | ✓ | — | — | — |
| Rectangle | ✓ | ✓ | — | — |
| Rhombus | ✓ | — | ✓ | ✓ |
| Square | ✓ | ✓ | ✓ | ✓ |
Converses: a parallelogram with equal diagonals is a rectangle; with perpendicular diagonals, a rhombus; with both, a square.
6. The Midpoint Theorem
The segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
Converse: A line through the midpoint of one side, parallel to another side, bisects the third side.
Proof idea: Extend EF to D so that FD = EF. Then △AEF ≅ △CDF (SAS), giving CD = AE = EB and CD ∥ AB. So EBCD is a parallelogram (one pair of sides equal and parallel), hence ED ∥ BC and ED = BC, i.e. EF = ½BC.
The sides of a triangle are 8 cm, 10 cm and 12 cm. Find the perimeter of the triangle formed by joining the midpoints of its sides.
Solution: Each side is half of a side of the big triangle: 4 + 5 + 6 = 15 cm.
7. Medians and the Centroid
A median joins a vertex to the midpoint of the opposite side. The three medians of a triangle meet at one point, the centroid G, which divides each median in the ratio 2 : 1 (from the vertex). The centroid is the triangle's balance point; a median divides a triangle into two triangles of equal area.
8. Varignon's Theorem and Tiling
The midpoints of the sides of any quadrilateral form a parallelogram. Its area is half the area of the quadrilateral.
Proof: Let P, Q, R, S be the midpoints of AB, BC, CD, DA. In △ABC, PQ ∥ AC and PQ = ½AC (midpoint theorem). In △ADC, SR ∥ AC and SR = ½AC. So PQ and SR are equal and parallel ⇒ PQRS is a parallelogram. ∎
Tiling: any quadrilateral (even a non-convex one) tiles the plane. Rotate a copy 180° about the midpoint of a side and fit it against the original; repeating this, the four angles at every meeting point are exactly the quadrilateral's four angles, which add to 360°.
9. Solved Problems (Board Pattern)
Show that if the diagonals of a parallelogram are equal, it is a rectangle.
Solution: In parallelogram ABCD with AC = BD: △ABC and △BAD have AB common, BC = AD, AC = BD → congruent (SSS), so ∠ABC = ∠BAD. These are adjacent angles of a parallelogram, summing to 180°, so each is 90°. ∎
ABCD is a trapezium with AB ∥ DC, AB = 14 cm and DC = 8 cm. Find the length of the segment joining the midpoints of AD and BC.
Solution: The midline of a trapezium is parallel to the bases and equals half their sum: (14 + 8)/2 = 11 cm.
Show that the bisectors of two adjacent angles of a parallelogram meet at a right angle.
Solution: ∠A + ∠B = 180°, so ½∠A + ½∠B = 90°. In the triangle formed, the third angle = 180° − 90° = 90°. ∎
10. Master Formula Table
| Result | Statement |
|---|---|
| Angle sum | ∠A + ∠B + ∠C + ∠D = 360° |
| Parallelogram angles | Opposite equal; adjacent sum 180° |
| Midpoint theorem | EF ∥ BC, EF = ½BC |
| Trapezium midline | ½(sum of parallel sides) |
| Centroid | Divides each median 2 : 1 |
| Varignon area | Area(PQRS) = ½ Area(ABCD) |
| Medial triangle | Perimeter = ½, area = ¼ of the original |
11. Master Summary Table
| Result | Proof tool | Exam focus |
|---|---|---|
| Parallelogram properties | ASA via diagonal | Find angles/sides |
| Parallelogram tests | Converses | Prove a quadrilateral is a parallelogram |
| Rectangle/rhombus/square | Diagonal properties | Identify shapes |
| Midpoint theorem | Construction + SAS | Lengths, perimeters |
| Varignon | Midpoint theorem twice | Midpoint quadrilateral |
| Tiling | Angle sum 360° | Explain why any quadrilateral tiles |
12. Mind Map
- Opposite sides, angles equal
- Diagonals bisect
- Rectangle, rhombus
- Square
- Midpoint theorem
- Centroid 2 : 1
- Varignon
- Tiling
13. Tips and Tricks
- Whenever midpoints appear, think "midpoint theorem".
- To prove a parallelogram, the quickest test is often "one pair of opposite sides equal and parallel".
- Draw a diagonal to split a quadrilateral into triangles.
- Assuming the diagonals of every parallelogram are equal (only rectangles).
- Assuming diagonals bisect the angles in every parallelogram (only rhombus/square).
- One pair of equal sides plus a different pair of parallel sides does not prove a parallelogram (isosceles trapezium).
14. Important One-liners
- The angles of a quadrilateral add up to 360°.
- The diagonals of a parallelogram bisect each other.
- A parallelogram with equal diagonals is a rectangle.
- A parallelogram with perpendicular diagonals is a rhombus.
- The segment joining midpoints of two sides of a triangle is half the third side.
- Midpoints of any quadrilateral form a parallelogram.
15. Exam-pattern Questions with Detailed Answers
Three angles of a quadrilateral are 75°, 90° and 105°. Find the fourth.
Show answer
360° − 270° = 90°.
Answer: 90°
In parallelogram ABCD, ∠A = 65°. Find ∠B, ∠C and ∠D.
Show answer
∠B = 115°, ∠C = 65°, ∠D = 115°.
Answer: 115°, 65°, 115°
Which quadrilateral has diagonals that are equal and bisect each other at right angles?
- Rectangle
- Rhombus
- Square
- Kite
Show answer
Equal + perpendicular + bisecting → square.
Answer: (C)
The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find them.
Show answer
30k = 360 → k = 12: 36°, 60°, 108°, 156°.
Answer: 36°, 60°, 108°, 156°
In △ABC, D and E are midpoints of AB and AC, and DE = 4.5 cm. Find BC.
Show answer
BC = 2 × DE = 9 cm.
Answer: 9 cm
The diagonals of a rhombus are 10 cm and 24 cm. Find its side.
Show answer
Halves 5 and 12 meet at 90°: side = 13 cm.
Answer: 13 cm
In a parallelogram, two adjacent angles differ by 30°. Find all angles.
Show answer
x + (x + 30) = 180 → x = 75. Angles: 75°, 105°, 75°, 105°.
Answer: 75°, 105°, 75°, 105°
The diagonals of a quadrilateral are 12 cm and 16 cm. Find the perimeter of the quadrilateral formed by joining midpoints of its sides.
Show answer
By Varignon/midpoint theorem, sides are 6, 8, 6, 8 → 28 cm.
Answer: 28 cm
Prove that the quadrilateral formed by joining the midpoints of the sides of a rectangle is a rhombus.
Show answer
Each side of PQRS equals half a diagonal of the rectangle (midpoint theorem). The rectangle's diagonals are equal, so all four sides of PQRS are equal; it is a parallelogram (Varignon) with equal sides → rhombus. ∎
Answer: Proof as shown
ABCD is a parallelogram. P and Q are points on diagonal BD with DP = BQ. Prove APCQ is a parallelogram.
Show answer
Let diagonals meet at O. OB = OD and BQ = DP, so OQ = OB − BQ = OD − DP = OP. Also OA = OC. The diagonals AC and PQ of APCQ bisect each other at O → parallelogram. ∎
Answer: Proof as shown
In △ABC, AD is a median and E is the midpoint of AD. BE extended meets AC at F. Show that AF = ⅓AC.
Show answer
Through D draw DG ∥ BF meeting AC at G. In △BCF, D is the midpoint of BC and DG ∥ BF, so G is the midpoint of CF: FG = GC. In △ADG, E is the midpoint of AD and EF ∥ DG, so F is the midpoint of AG: AF = FG. Thus AF = FG = GC, so AF = ⅓AC. ∎
Answer: Proof using the converse of the midpoint theorem twice
Explain why copies of any quadrilateral can tile a floor without gaps or overlaps.
Show answer
Rotate a copy by 180° about the midpoint of one side; the two copies share that side exactly. Repeating across all sides arranges copies so that at every corner one copy of each angle A, B, C, D meets. Since A + B + C + D = 360°, they fill the full turn with no gap or overlap.
Answer: Angle sum 360° at each vertex of the tiling
16. Quick Revision Summary
- Quadrilateral angles sum to 360°.
- Parallelogram: opposite sides and angles equal; diagonals bisect each other — and conversely.
- Rectangle: equal diagonals; rhombus: perpendicular diagonals; square: both.
- Midpoint theorem: joining midpoints gives a parallel segment half as long.
- Centroid divides medians 2 : 1; midpoints of any quadrilateral form a parallelogram; every quadrilateral tiles.