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SSC CGL Maths · Quadrilaterals

Properties of Quadrilaterals

The four angles of any quadrilateral add to 360°; each special quadrilateral (trapezium, parallelogram, rhombus, kite) is known by its sides, and its diagonals settle most questions.

All Quadrilaterals notes

Why this matters

Eleven questions in the bank, six EASY and five MODERATE: six from the 2024 shifts and five from the 2025 shifts. The EASY ones test a definition or the 360° sum. The MODERATE ones hide a pair of congruent or similar triangles inside the figure; once you see which sides are equal, the answer is one line.

Concept 1 of 4: Angle sum and the diagonals of any quadrilateral

One diagonal cuts any quadrilateral into two triangles. Their angles together make up the four corners, so the corners add to 2 × 180° = 360°. The diagonals also cut the figure into triangles, and equal parts in those triangles tell you where the diagonals meet.

Definition

  • The four interior angles of a quadrilateral add to 360∘360^\circ.
  • Angles given in a ratio: if the ratio covers all four angles, one part =360∘total parts= \frac{360^\circ}{\text{total parts}}. If one angle is given in degrees, take it away from 360∘360^\circ first and share the rest among the parts.
  • A cyclic quadrilateral has all four corners on a circle. Its opposite angles add to 180∘180^\circ. This is true only of a cyclic one.
  • Two triangles are congruent (equal in every part) by SSS, SAS, ASA or AAS. If the diagonals of ABCD meet at O and the perpendiculars from B and D to AC are equal, the two right triangles at O are congruent (AAS: right angle, vertically opposite angles, equal perpendiculars), so OB=ODOB = OD.

Quadrilateral angles

∠A+∠B+∠C+∠D=360∘cyclic: ∠A+∠C=∠B+∠D=180∘\angle A + \angle B + \angle C + \angle D = 360^\circ \qquad \text{cyclic: } \angle A + \angle C = \angle B + \angle D = 180^\circ

Worked example

The angles of a quadrilateral are xx, x+20x + 20, 2x2x and 2x+402x + 40 degrees. Find the largest angle.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 24 Sep 2025, Shift 1 · Q65Easy

Example 1 · Quadrilaterals · Properties of Quadrilaterals

In a quadrilateral ABCD, angles A,B,C are in the ratio 1:2:31:2:3. If angle D=120∘D=120^\circ, what is angle A?

Sharing 360° when one angle is already known

If three angles are in a ratio and the fourth is given in degrees, take the fourth away from 360∘360^\circ before sharing. Dividing 360∘360^\circ by the three parts gives a wrong answer that is often printed.

Using the cyclic rule on any quadrilateral

Opposite angles add to 180∘180^\circ only when all four corners lie on one circle. A general quadrilateral, or a parallelogram that is not a rectangle, does not have this property.
Drill 2 more on angle sum and the diagonals of any quadrilateral

Concept 2 of 4: Trapezium: area and the triangles made by its diagonals

A trapezium has one pair of parallel sides. Its area is the average of the two parallel sides times the gap between them. Its diagonals cut out two triangles, one on each parallel side, and because the sides are parallel those two triangles have the same angles: one is a scaled copy of the other.

Definition

  • Area =12×(a+b)×h= \frac{1}{2} \times (a + b) \times h, where a and b are the parallel sides and h is the perpendicular distance between them.
  • In a right trapezium, one non-parallel side is perpendicular to the parallel sides, so that side is the height.
  • If AB∥CDAB \parallel CD and the diagonals meet at E, triangles ABE and CDE are similar with scale AB:CDAB : CD.
  • So the diagonals cut each other in the same ratio: AE:EC=BE:ED=AB:CDAE : EC = BE : ED = AB : CD.
  • Areas of similar triangles go as the square of the scale: area ABE:area CDE=AB2:CD2\text{area } ABE : \text{area } CDE = AB^2 : CD^2.

Trapezium

area=12(a+b)hAE:EC=AB:CDarea ABEarea CDE=AB2CD2\text{area} = \frac{1}{2}(a + b)h \qquad AE : EC = AB : CD \qquad \frac{\text{area } ABE}{\text{area } CDE} = \frac{AB^2}{CD^2}
  • a, bthe two parallel sides
  • hthe perpendicular distance between them

Worked example

A trapezium has parallel sides 12 cm and 18 cm, and they are 8 cm apart. Find its area.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 2 · Q69Easy

Example 2 · Quadrilaterals · Properties of Quadrilaterals

In a trapezoid ABCD with AB parallel to CD, the diagonals AC and BD intersect at E. What is the ratio of the area of △ABE\triangle ABE to the area of △CDE\triangle CDE?

Giving the side ratio for an area ratio

The triangles on the parallel sides have their sides in the ratio AB:CDAB : CD, but their areas are in the ratio AB2:CD2AB^2 : CD^2. The plain side ratio is printed as a wrong option.

Using a slanted side as the height

The height of a trapezium is the perpendicular distance between the parallel sides. A slanted side is longer than that. Only in a right trapezium is a side the height.
Drill 1 more on trapezium: area and the triangles made by its diagonals

Concept 3 of 4: Parallelogram: opposite sides, angles and diagonals

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Slide one side along the other and it lands on its opposite side, so opposite sides are equal and opposite angles are equal. Each diagonal splits it into two congruent triangles.

Definition

  • Both pairs of opposite sides are parallel and equal.
  • Opposite angles are equal; neighbouring angles add to 180∘180^\circ (they are co-interior angles between parallel sides).
  • The diagonals bisect each other: they cut each other in half. They need not be equal, and they need not be perpendicular.
  • Special parallelograms: a rectangle has all angles 90∘90^\circ and equal diagonals; a rhombus has all sides equal and perpendicular diagonals; a square has both.
  • Midpoint theorem: in a triangle, a line through the midpoint of one side parallel to a second side meets the third side at its midpoint, and the piece between is half the second side. It settles questions where a side of a parallelogram is extended by its own length.

Parallelogram ABCD

AB=CD, BC=DA∠A=∠C, ∠A+∠B=180∘AO=OC, BO=ODAB = CD,\ BC = DA \qquad \angle A = \angle C,\ \angle A + \angle B = 180^\circ \qquad AO = OC,\ BO = OD
  • Othe point where the diagonals meet

Worked example

In parallelogram ABCD, angle A is 3 times angle B. Find all four angles.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 19 Sep 2025, Shift 1 · Q68Easy

Example 3 · Quadrilaterals · Properties of Quadrilaterals

A parallelogram is a quadrilateral in which:

Calling a trapezium a parallelogram

A quadrilateral with only one pair of parallel sides is a trapezium. A parallelogram needs both pairs of opposite sides parallel, and then they are equal too.

Giving a parallelogram perpendicular or equal diagonals

The diagonals of a parallelogram only bisect each other. They are perpendicular only in a rhombus or a square, and equal only in a rectangle or a square.

Taking a diagonal as an angle bisector

A diagonal of a general parallelogram does not halve the corner angles. It does so only when all four sides are equal, in a rhombus or a square.
Drill 1 more on parallelogram: opposite sides, angles and diagonals

Concept 4 of 4: Rhombus and kite: equal sides give congruent halves

Equal sides make mirror images. In a kite, two pairs of neighbouring sides are equal, so the diagonal between the two places where the equal sides meet is a line of symmetry: the triangles on either side of it are congruent. A rhombus has all four sides equal, so both diagonals are lines of symmetry.

Definition

  • Rhombus: all four sides equal. Its diagonals bisect each other at right angles, so each diagonal is the perpendicular bisector of the other.
  • So a point inside a rhombus that is equally far from two opposite corners lies on the other diagonal (a point equally far from P and R lies on the perpendicular bisector of PR).
  • Side of a rhombus from its diagonals d1,d2d_1, d_2: the half-diagonals and a side form a right triangle, so side2=(d12)2+(d22)2\text{side}^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2.
  • Kite: two pairs of equal neighbouring sides, say AB=ADAB = AD and CB=CDCB = CD. The diagonal AC is a line of symmetry: triangles ABC and ADC are congruent (SSS), AC bisects the angles at A and C, the angles at B and D are equal, and AC is the perpendicular bisector of BD.
  • First decide which sides are equal. In a triangle, the equal sides face equal angles, so unequal base angles tell you those two sides are not the equal pair.

Rhombus side from its diagonals

side2=(d12)2+(d22)2\text{side}^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2
  • d1,d2d_1, d_2the two diagonals of the rhombus

Worked example

A rhombus has diagonals 16 cm and 12 cm. Find the length of its side.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 25 Sep 2024, Shift 2 · Q54Moderate

Example 4 · Quadrilaterals · Properties of Quadrilaterals

In a quadrilateral ABCD, AB = BC, AD = DC. ∠ABD=68∘\angle ABD = 68^\circ, ∠ADB=(2y−7)∘\angle ADB = (2y-7)^\circ, ∠BDC=33∘\angle BDC = 33^\circ, ∠DBC=(3x+2)∘\angle DBC = (3x+2)^\circ. Then the value of 2x+3y is:

Taking the kite's symmetry line from the wrong diagonal

Only the diagonal joining the two corners where equal sides meet is a line of symmetry. Find the equal sides first: if a triangle on one diagonal has unequal base angles, its two sides at that diagonal are not equal.

Giving a rhombus equal diagonals

The diagonals of a rhombus are perpendicular and bisect each other, but they are equal only when the rhombus is a square.

Putting whole diagonals into Pythagoras

The right triangle inside a rhombus has legs equal to half of each diagonal. With diagonals 6 cm and 8 cm the side is 9+16=5\sqrt{9 + 16} = 5 cm, not 36+64=10\sqrt{36 + 64} = 10 cm.
Drill 3 more on rhombus and kite: equal sides give congruent halves

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

  • Angle sum and the diagonals of any quadrilateral

    Quadrilateral angles

    ∠A+∠B+∠C+∠D=360∘cyclic: ∠A+∠C=∠B+∠D=180∘\angle A + \angle B + \angle C + \angle D = 360^\circ \qquad \text{cyclic: } \angle A + \angle C = \angle B + \angle D = 180^\circ
  • Trapezium: area and the triangles made by its diagonals

    Trapezium

    area=12(a+b)hAE:EC=AB:CDarea ABEarea CDE=AB2CD2\text{area} = \frac{1}{2}(a + b)h \qquad AE : EC = AB : CD \qquad \frac{\text{area } ABE}{\text{area } CDE} = \frac{AB^2}{CD^2}
  • Parallelogram: opposite sides, angles and diagonals

    Parallelogram ABCD

    AB=CD, BC=DA∠A=∠C, ∠A+∠B=180∘AO=OC, BO=ODAB = CD,\ BC = DA \qquad \angle A = \angle C,\ \angle A + \angle B = 180^\circ \qquad AO = OC,\ BO = OD
  • Rhombus and kite: equal sides give congruent halves

    Rhombus side from its diagonals

    side2=(d12)2+(d22)2\text{side}^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2

Watch out for (10)

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