Join four points and close the loop — you get a quadrilateral. Its four angles ALWAYS add up to 360°, and once you spot whether it is a parallelogram, the diagonals and angles start telling secrets!
Four sides
A quadrilateral has 4 sides, 4 vertices, 4 angles and 2 diagonals.
Angle sum
The four interior angles always total 360°.
Parallelogram
Both pairs of opposite sides parallel — the star of this chapter.
Midpoint magic
The line joining midpoints of two sides is parallel to the third and half of it.
1. What is a quadrilateral?
Take four points A, B, C and D in a plane, where no three of them lie on the same straight line. Join them in order — AB, BC, CD and DA. The closed figure you get is a quadrilateral ABCD. It has four sides (AB, BC, CD, DA), four vertices (A, B, C, D), and four angles (∠A, ∠B, ∠C, ∠D). The line segments AC and BD that join opposite vertices are called diagonals. Sides that share a vertex (like AB and BC) are adjacent sides; sides with no common vertex (like AB and CD) are opposite sides. Always name a quadrilateral going around the boundary in order — ABCD or BCDA is correct, but ACBD is wrong because it crosses the figure.
2. Angle sum property
The sum of the four interior angles of any quadrilateral is 360°. Why? Draw one diagonal, say AC. It splits the quadrilateral into two triangles, △ABC and △ACD. The angles of each triangle add to 180°, so the two triangles together give 180° + 180° = 360°. These angles are exactly the four angles of the quadrilateral, so ∠A + ∠B + ∠C + ∠D = 360°. This single fact lets you find a missing fourth angle whenever the other three are known.
3. Types of quadrilaterals
A trapezium has exactly one pair of parallel sides. A parallelogram has both pairs of opposite sides parallel. A rhombus is a parallelogram with all four sides equal. A rectangle is a parallelogram with all four angles equal to 90°. A square is both a rhombus and a rectangle — all sides equal AND all angles 90°. A kite has two pairs of adjacent sides equal. Notice the family ladder: every square is a rectangle and a rhombus, every rectangle and every rhombus is a parallelogram, and every parallelogram is a trapezium (under the inclusive definition). The special ones are just parallelograms with extra conditions.
4. Properties of a parallelogram
A parallelogram packs four key properties, each provable using congruent triangles. (a) Opposite sides are equal — in parallelogram ABCD, AB = CD and BC = AD. (b) Opposite angles are equal — ∠A = ∠C and ∠B = ∠D. (c) Adjacent (consecutive) angles are supplementary — they add to 180°, because the parallel sides are cut by a transversal making co-interior angles. (d) Diagonals bisect each other — the diagonals cut each other into two equal halves at their point of intersection. To prove opposite sides equal, draw diagonal AC: alternate angles give ∠BAC = ∠DCA and ∠BCA = ∠DAC, AC is common, so △ABC ≡ △CDA by ASA, hence AB = CD and BC = AD.
5. Tests — when is a quadrilateral a parallelogram?
The properties also work in reverse, giving handy tests. A quadrilateral is a parallelogram if ANY one of these is true: (i) both pairs of opposite sides are equal; (ii) both pairs of opposite angles are equal; (iii) the diagonals bisect each other; (iv) one pair of opposite sides is BOTH equal AND parallel. That last test is very powerful — you only need to check one pair of sides for two conditions to be sure.
6. Special parallelograms
In a rectangle every property of a parallelogram holds and, in addition, the diagonals are equal in length. In a rhombus all sides are equal and the diagonals bisect each other at right angles (90°); they also bisect the vertex angles. A square combines both — equal diagonals that bisect each other at right angles. These extra diagonal facts are favourite exam questions, so learn them well.
7. The Mid-point Theorem
This is the chapter’s most useful tool. The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it. If D and E are mid-points of sides AB and AC of △ABC, then DE ∥ BC and DE = ½ BC. The converse is equally important: the line drawn through the mid-point of one side, parallel to another side, bisects the third side. These two results let you prove many properties of quadrilaterals — for example, that the figure formed by joining the mid-points of the sides of any quadrilateral is always a parallelogram.
- Angle sum of a quadrilateral: ∠A + ∠B + ∠C + ∠D = 360°.
- Parallelogram: opposite sides equal, opposite angles equal, diagonals bisect each other.
- Consecutive angles of a parallelogram are supplementary (sum = 180°).
- Rectangle: diagonals are equal. Rhombus: diagonals bisect at 90°.
- Square: diagonals equal AND perpendicular bisectors of each other.
- Mid-point theorem: DE ∥ BC and DE = ½ BC when D, E are mid-points.
The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all four angles.
- Let the common multiple be x. The angles are 3x, 5x, 9x and 13x.
- By the angle sum property: 3x + 5x + 9x + 13x = 360°.
- Add the coefficients: 30x = 360°.
- Solve: x = 360° ÷ 30 = 12°.
- Now multiply back: 3x = 36°, 5x = 60°, 9x = 108°, 13x = 156°.
- Check: 36 + 60 + 108 + 156 = 360°. Correct.
In parallelogram ABCD, ∠A = 70°. Find ∠B, ∠C and ∠D.
- In a parallelogram opposite angles are equal, so ∠C = ∠A = 70°.
- Consecutive angles are supplementary, so ∠A + ∠B = 180°.
- Therefore ∠B = 180° − 70° = 110°.
- Opposite angles equal again: ∠D = ∠B = 110°.
- Check: 70 + 110 + 70 + 110 = 360°. Correct.
Remember the parallelogram with “SAD-B”: Sides opposite equal, Angles opposite equal, Diagonals Bisect each other. For the special ones: Rectangle — diagonals are Really equal; Rhombus — diagonals cross at a right angle.
Don’t confuse “diagonals bisect each other” (true for ALL parallelograms) with “diagonals are equal” (only rectangles and squares) or “diagonals are perpendicular” (only rhombuses and squares). Also, in proofs always state the congruence rule (SAS, ASA, SSS) and the reason for each step — missing reasons lose marks even with the right answer.
Q1. Three angles of a quadrilateral are 110°, 50° and 40°. Find the fourth angle.
Answer: Let the fourth angle be x. By the angle sum property, 110° + 50° + 40° + x = 360°. So 200° + x = 360°, which gives x = 360° − 200° = 160°.
Q2. The diagonals of a parallelogram ABCD intersect at O. If AO = 4 cm and BO = 5 cm, find the lengths of the diagonals AC and BD.
Answer: The diagonals of a parallelogram bisect each other, so O is the mid-point of both. Therefore AC = 2 × AO = 2 × 4 = 8 cm, and BD = 2 × BO = 2 × 5 = 10 cm.
Q3. Show that the diagonals of a rectangle are equal.
Answer: Let ABCD be a rectangle with diagonals AC and BD. In △ABC and △BAD: AB = BA (common), ∠ABC = ∠BAD = 90° (angles of a rectangle), and BC = AD (opposite sides of a rectangle / parallelogram are equal). So △ABC ≡ △BAD by the SAS rule. Hence AC = BD by CPCT (corresponding parts of congruent triangles). Therefore the diagonals are equal.
Q4. In △ABC, D and E are the mid-points of AB and AC. If BC = 9 cm, find DE. State the theorem used.
Answer: By the Mid-point Theorem, the segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it. So DE ∥ BC and DE = ½ × BC = ½ × 9 = 4.5 cm.
- ✅ A quadrilateral has 4 sides, 4 angles, 2 diagonals; angles add to 360°.
- ✅ A parallelogram: opposite sides and angles equal, diagonals bisect each other.
- ✅ Rectangle has equal diagonals; rhombus has perpendicular diagonals; square has both.
- ✅ Mid-point theorem: segment joining mid-points is parallel to and half the third side.
- ✅ To prove a parallelogram, use any one valid test (sides, angles, or diagonals).
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