Quadrilaterals Class 8: Properties of Rectangles, Parallelograms, Rhombuses, Kites and Trapeziums

What Quadrilaterals is about

Key concepts

Rectangles and the Carpenter's Problem

NCERT section 4.1 Rectangles and Squares

A rectangle is first defined as a quadrilateral whose angles are all 90° and whose opposite sides are equal.

The Carpenter's Problem: two strips of wood are joined so that a thread through their four ends makes a rectangle. The strips are the diagonals. Using congruent triangles:

• Diagonals are equal: in rectangle ABCD, $△ADC\cong △DAB$ (SAS: $AB=DC$, right angles at A and D, AD common), so $AC=BD$.
• Diagonals bisect each other: $△AOB\cong △COD$ (AAS), so $OA=OC$ and $OB=OD$.
• Converse: if two equal diagonals bisect each other, at any angle $x$, the triangles at O are isosceles; the base angles are $90°-\frac{x}{2}$ and $\frac{x}{2}$, and each corner is their sum, $90°$. So the shape is a rectangle.

So a rectangle can also be defined as a quadrilateral whose diagonals are equal and bisect each other. Carpenters use exactly this to make rectangular frames.

Deduction versus measuring: measuring a few rectangles only gives a conjecture; a proof shows the property holds for every rectangle.

Worked example

The diagonals of rectangle ABCD meet at O and $\angle AOB=50°$. Find $\angle OAB$ and $\angle OAD$, and check that $\angle A=90°$.

Try it first, then show the solutionSolution

1. Diagonals are equal and bisect each other, so $OA=OB$ and $△AOB$ is isosceles.
2. $\angle OAB=\frac{180°-50°}{2}=65°$.
3. $\angle AOD=180°-50°=130°$ (linear pair), and $OA=OD$, so $\angle OAD=\frac{180°-130°}{2}=25°$.
4. $\angle A=65°+25°=90°$, as it must be.

Answer: $\angle OAB=65°$, $\angle OAD=25°$; they add to 90°.

All right angles is enough: properties of a rectangle

NCERT section 4.1 Rectangles and Squares

Can a quadrilateral have four right angles but unequal opposite sides? No. Join diagonal BD in ABCD with all angles 90°. In $△BAD$ and $△DCB$, BD is common, $\angle A=\angle C=90°$, and the angles that BD makes with the sides match (each is $90°$ minus the same angle), so $△BAD\cong △DCB$ by AAS. Hence $AD=CB$ and $AB=DC$.

So the simplest definition is: a rectangle is a quadrilateral in which all angles are 90°.

Properties of a rectangle:
1. All angles are 90°.
2. Opposite sides are equal.
3. Opposite sides are parallel (co-interior angles $90°+90°=180°$).
4. Diagonals are equal and bisect each other.

Write congruences with matching vertex order: $△BAD\cong △DCB$ pairs B with D, A with C, D with B. Writing $△BAD\cong △CDB$ would wrongly pair the right angle at A with $\angle CDB$, which is not a right angle.

Worked example

In rectangle PQRS, diagonal PR = 13 cm and the diagonals meet at O. Find QS and OQ. If $\angle POQ=110°$, find $\angle OPQ$ and $\angle OPS$.

Try it first, then show the solutionSolution

1. Diagonals of a rectangle are equal: $QS=13$ cm.
2. They bisect each other: $OQ=6.5$ cm.
3. $OP=OQ$, so $\angle OPQ=\frac{180°-110°}{2}=35°$.
4. $\angle OPS=90°-35°=55°$.

Answer: $QS=13$ cm, $OQ=6.5$ cm, $\angle OPQ=35°$, $\angle OPS=55°$.

Squares

NCERT section 4.1 Rectangles and Squares

A square is a quadrilateral with all angles 90° and all sides equal. So every square is a rectangle, but not every rectangle is a square; in a Venn diagram the set of squares sits inside the set of rectangles.

Diagonals of a square: they are equal and bisect each other (as in any rectangle), and also meet at 90°. Proof: $△BOA\cong △BOC$ by SSS ($BA=BC$, $OA=OC$, BO common), so $\angle BOA=\angle BOC$; together they make 180°, so each is 90°.

Diagonals bisect the angles: in $△ADC$, $AD=DC$ and $\angle D=90°$, so the other two angles are each 45°.

To build a square from two strips (Carpenter's Problem), make them equal, join them at their midpoints, and set them at right angles.

Worked example

Explain how to construct a square with a diagonal of 8 cm using only a ruler and a set-square, and give the angle between a diagonal and a side.

Try it first, then show the solutionSolution

1. Draw AC = 8 cm and mark its midpoint O (4 cm from A).
2. Using the set-square, draw a line through O perpendicular to AC, and mark B and D on it 4 cm from O on either side.
3. Join A, B, C, D. The diagonals are equal, bisect each other and are perpendicular, so ABCD is a square.
4. Each diagonal bisects a 90° corner, so it makes 45° with each side.

Answer: Equal diagonals of 8 cm bisecting each other at 90°; diagonal-to-side angle 45°.

Angle sum of a quadrilateral

NCERT section 4.2 Angles in a Quadrilateral

Draw one diagonal of quadrilateral SOME, say SM. It splits the quadrilateral into two triangles, each with angle sum 180°. The six triangle angles together make up exactly the four angles of the quadrilateral, so

the angles of any quadrilateral add up to 360°.

Consequences:

• If three angles are 90°, the fourth must be 90°; so 'three right angles' already makes a rectangle.
• A quadrilateral cannot have four obtuse angles (their sum would exceed 360°), and cannot have four acute angles (their sum would be less than 360°).
• The result also holds for a non-convex (arrowhead-shaped) quadrilateral, using the diagonal that lies inside it.

Worked example

The angles of a quadrilateral are in the ratio 2 : 3 : 4 : 6. Find them. Is any angle reflex?

Try it first, then show the solutionSolution

1. Let the angles be $2x,3x,4x,6x$. Their sum is $15x=360°$, so $x=24°$.
2. Angles: $48°,72°,96°,144°$.
3. Check: $48+72+96+144=360$. The largest is 144°, less than 180°, so none is reflex.

Answer: $48°,72°,96°,144°$; none is reflex.

Parallelograms

NCERT section 4.3 More Quadrilaterals with Parallel Opposite Sides

A parallelogram is a quadrilateral whose opposite sides are parallel. Every rectangle is a parallelogram, but a parallelogram need not have right angles.

Deduced properties:
1. Adjacent angles add to 180° (they are co-interior angles between parallel sides).
2. Opposite angles are equal: if $\angle P=x$, the next angle is $180°-x$ and the one after that is $180°-(180°-x)=x$.
3. Opposite sides are equal: diagonal BD makes $△ABD\cong △CDB$ (AAS: opposite angles equal, alternate angles equal, BD common).
4. Diagonals bisect each other: in parallelogram EASY, $△AOE\cong △YOS$ (ASA with alternate angles and $AE=YS$), so O is the midpoint of both.

The diagonals of a parallelogram are not equal in general (they are equal only in a rectangle), and need not be perpendicular.

Worked example

Two adjacent angles of a parallelogram are $(3x+15)°$ and $(2x-5)°$. Find $x$ and all four angles.

Try it first, then show the solutionSolution

1. Adjacent angles of a parallelogram add to 180°: $3x+15+2x-5=180$.
2. $5x+10=180$, so $x=34$.
3. The angles are $3(34)+15=117°$ and $2(34)-5=63°$.
4. Opposite angles are equal: $117°,63°,117°,63°$.

Answer: $x=34$; angles $117°,63°,117°,63°$.

Rhombuses

NCERT section 4.4 Quadrilaterals with Equal Sidelengths

A rhombus is a quadrilateral with all four sides equal. You can construct one with any angle less than 180° (not only 90°).

In rhombus GAME, diagonal AE gives two isosceles triangles, and $△GAE\cong △MAE$ (SSS). So the four angles that AE makes with the sides are all equal. These equal alternate angles show $EM\parallel GA$ and $GE\parallel AM$, so every rhombus is a parallelogram and has all parallelogram properties.

Properties of a rhombus:
1. All sides equal.
2. Opposite sides parallel.
3. Adjacent angles add to 180°; opposite angles equal.
4. Diagonals bisect each other.
5. Diagonals bisect the angles of the rhombus.
6. Diagonals meet at 90° ($△GEO\cong △MEO$, so the two angles at O are equal and add to 180°).

A square is both a rectangle and a rhombus; in the Venn diagram it lies where the rectangle and rhombus sets overlap, inside the parallelogram set.

Worked example

In rhombus ABCD, $\angle A=70°$ and the diagonals meet at O. Find the angles of $△AOB$ and all the angles of the rhombus.

Try it first, then show the solutionSolution

1. Diagonal AC bisects $\angle A$: $\angle OAB=35°$.
2. Diagonals meet at right angles: $\angle AOB=90°$, so $\angle OBA=180°-90°-35°=55°$.
3. So $\angle B=2\times 55°=110°$ (diagonal BD bisects it); check $70°+110°=180°$.
4. Rhombus angles: $70°,110°,70°,110°$.

Answer: $△AOB$: $35°,55°,90°$; rhombus: $70°,110°,70°,110°$.

Kites and trapeziums

NCERT section 4.6 Kite and Trapezium

Kite: a quadrilateral ABCD with $AB=BC$ and $CD=DA$ (two pairs of equal adjacent sides). Joining two copies of a scalene triangle along a side, mirror-fashion, gives one. Since $△ABD\cong △CBD$ (SSS), the diagonal BD (joining the vertices where equal sides meet):

• bisects $\angle ABC$ and $\angle ADC$;
• bisects the other diagonal AC and is perpendicular to it;
• and the angles at A and C are equal.

Trapezium: a quadrilateral with at least one pair of parallel opposite sides. If $PQ\parallel SR$, the co-interior angles give $\angle P+\angle S=180°$ and $\angle Q+\angle R=180°$. Every parallelogram is a trapezium under this definition.

Isosceles trapezium: the non-parallel sides are equal. Dropping perpendiculars from the shorter parallel side makes a rectangle in the middle and two congruent right triangles, so the base angles are equal: $\angle U=\angle V$.

Family tree (Venn diagram): squares lie inside both rectangles and rhombuses; rectangles and rhombuses lie inside parallelograms; parallelograms lie inside trapeziums; rhombuses (and squares) are also kites.

Worked example

ABCD is a trapezium with $AB\parallel DC$, $\angle A=2x$ and $\angle D=x+30°$. Find $\angle A$ and $\angle D$. If $\angle B=75°$, find $\angle C$.

Try it first, then show the solutionSolution

1. AB and DC are parallel with transversal AD, so $\angle A+\angle D=180°$.
2. $2x+x+30=180$ gives $x=50$: $\angle A=100°$, $\angle D=80°$.
3. Similarly $\angle B+\angle C=180°$, so $\angle C=105°$.
4. Check: $100+75+105+80=360$.

Answer: $\angle A=100°$, $\angle D=80°$, $\angle C=105°$.

Playing with quadrilaterals: geoboards and joined triangles

NCERT section 4.5 Playing with Quadrilaterals

The activities in 4.5 let you test the properties with your hands and then justify them.

• Geoboard: two rubber bands of equal length crossing at their midpoints at right angles give a square (equal diagonals, bisecting, perpendicular). Lengthen one band equally at both ends and the diagonals are now unequal but still perpendicular bisectors of each other: a rhombus.
• Two equilateral triangles (side 8 cm) joined along a side make a quadrilateral with all sides 8 cm and angles $60°,120°,60°,120°$: a rhombus.
• Two isosceles triangles (8, 8, 6 cm) can be joined to give a rhombus (all sides 8), a kite (sides 8, 8, 6, 6 adjacent in pairs) or a parallelogram (sides 8, 6, 8, 6).
• Two scalene triangles (6, 9, 12 cm) give parallelograms or kites, depending on how they are flipped.

How to justify: list the side lengths around the shape in order, then use the definitions (all sides equal: rhombus; two pairs of equal adjacent sides: kite; opposite sides equal: parallelogram).

Worked example

Two cardboard right-angled isosceles triangles, each with equal sides of 5 cm, are joined along their longest sides (the hypotenuses). Name the quadrilateral and find all its sides and angles.

Try it first, then show the solutionSolution

1. The joined hypotenuse is inside, so the outline is made of the four 5 cm sides: all sides equal.
2. At the two right-angle vertices, the angle is 90°.
3. At the two ends of the joint, two 45° angles meet: $45°+45°=90°$.
4. All sides equal and all angles 90°, so it is a square. Check: $4\times 90°=360°$.

Answer: A square with sides 5 cm and all angles 90°.

Maths in real life: Quadrilaterals

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Parallelograms, Rhombuses, Kites and Trapeziums
• Rectangles, Squares and the Angle Sum of a Quadrilateral

Quadrilaterals formula sheet

Angle sum

$\angle A+\angle B+\angle C+\angle D=360°$

Parallelogram angles

$\angle A=\angle C,\angle B=\angle D,\angle A+\angle B=180°$

Rectangle diagonals

$AC=BD,OA=OC=OB=OD$

Base angles at the diagonal crossing (rectangle)

$\text{if }\angle AOB=x:\angle OAB=90°-\frac{x}{2},\angle OAD=\frac{x}{2}$

Square diagonals

$AC=BD,AC\perp BD,\text{diagonal-side angle}=45°$

Rhombus diagonals

$AC\perp BD,\text{each bisects two angles}$

Trapezium ($AB \parallel DC$)

$\angle A+\angle D=180°,\angle B+\angle C=180°$

Isosceles trapezium

$\text{base angles equal}$

Kite ($AB = BC$, $CD = DA$)

$BD\perp AC,AO=OC,\angle A=\angle C$

How the school exam asks Quadrilaterals

The question patterns to expect, and the method that works for each.

Tricks to remember Quadrilaterals

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Rectangle diagonals: Equal and Halved

  Rectangle = 'E and H': Equal diagonals that Halve each other.

  Use it when: Carpenter's Problem and rectangle proofs.

• Rhombus diagonals: Perpendicular and Bisecting

  Rhombus = 'Plus sign': diagonals cross like a plus, at 90°, cutting each other in half.

  Use it when: Rhombus angle questions.

• Square = rectangle + rhombus

  A square has every diagonal property: equal, halved, perpendicular, angle-bisecting.

  Use it when: Square questions and Venn diagrams.

• Next door 180, opposite same

  In a parallelogram, neighbours share 180°, opposites are twins.

  Use it when: Finding parallelogram angles.

• Two triangles make 360

  One diagonal turns any quadrilateral into two triangles: $180+180$.

  Use it when: Recalling and proving the angle sum.

Quadrilaterals: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 2 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Quadrilaterals

The slips examiners see most often, and what to write instead.

• ✗Mistake: Assuming the diagonals of a parallelogram are equal

  Rectangle properties are carried over to every parallelogram.

  Wrong working

  ABCD is a parallelogram with AC = 10 cm, so BD = 10 cm.

  Right working

  Parallelogram diagonals only bisect each other; they are equal only if it is a rectangle. BD cannot be found from AC alone.

  ✓Do this: Check which family the shape belongs to before using a diagonal property.

  Usually costs: The answer and method marks.

• ✗Mistake: Adding opposite angles of a parallelogram to 180°

  Adjacent and opposite are confused.

  Wrong working

  $\angle A=65°$, so $\angle C=115°$.

  Right working

  Opposite angles are equal: $\angle C=65°$; the adjacent angle $\angle B=115°$.

  ✓Do this: Adjacent: add to 180°. Opposite: equal.

  Usually costs: 1 mark per angle.

• ✗Mistake: Saying perpendicular diagonals make a rhombus

  The converse of a rhombus property is assumed.

  Wrong working

  AC ⊥ BD, so ABCD is a rhombus.

  Right working

  A kite also has perpendicular diagonals. You also need the diagonals to bisect each other.

  ✓Do this: Look for a counterexample before accepting a converse.

  Usually costs: 1-2 marks in a true/false item.

• ✗Mistake: Wrong vertex order in a congruence

  Triangles are named in any order.

  Wrong working

  $△ABD\cong △CBD$ in parallelogram ABCD.

  Right working

  $△ABD\cong △CDB$: A↔C, B↔D, D↔B, so equal parts match.

  ✓Do this: Write matching vertices in the same positions.

  Usually costs: Up to 1 mark in a proof.

• ✗Mistake: Proof steps without reasons

  Statements are written bare.

  Wrong working

  OA = OC, ∠AOB = ∠COD, so the triangles are congruent.

  Right working

  OA = OC (diagonals bisect each other), ∠AOB = ∠COD (vertically opposite), OB = OD (diagonals bisect each other); so $△AOB\cong △COD$ (SAS).

  ✓Do this: Put a reason in brackets after every statement and name the congruence rule.

  Usually costs: 1-2 marks.

NCERT exercise guide: Quadrilaterals

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Practice questions on Quadrilaterals

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   The diagonals of a rhombus meet at
   (a) 45°
   (b) 60°
   (c) 90°
   (d) 120°

   Hint

   Deduction 10.

   Answer

   (c) 90°

2. Q2 · 1 mark · MCQExam level

   A quadrilateral with at least one pair of parallel opposite sides is a
   (a) kite
   (b) trapezium
   (c) rhombus
   (d) rectangle

   Hint

   Book's definition.

   Answer

   (b) trapezium

3. Q3 · 2 marks · Very short answerExam level

   One base angle of an isosceles trapezium is 68°. Find the other three angles.

   Hint

   Base angles equal; co-interior angles add to 180°.

   Answer

   $68°$, $112°$, $112°$

4. Q4 · 1 mark · Very short answerExam level

   In rectangle ABCD, AC = 9 cm. Find BD and OB, where O is the point where the diagonals meet.

   Hint

   Equal and bisecting.

   Answer

   $BD=9$ cm, $OB=4.5$ cm

5. Q5 · 3 marks · Short answerHarder exam level

   In parallelogram ABCD, $\angle A=(3x+15)°$ and $\angle B=(2x-5)°$. Find all four angles.

   Hint

   Adjacent angles add to 180°.

   Answer

   $x=34$: $117°,63°,117°,63°$

6. Q6 · 2 marks · Very short answerHarder exam level

   In rhombus ABCD, $\angle ABD=27°$. Find all angles of the rhombus and $\angle AOB$.

   Hint

   Diagonals bisect the angles.

   Answer

   $\angle B=\angle D=54°$, $\angle A=\angle C=126°$, $\angle AOB=90°$

7. Q7 · 3 marks · Short answerHarder exam level

   PQRS is a rectangle whose diagonals meet at O. If $\angle POQ=110°$, find $\angle OPQ$, $\angle OPS$ and $\angle OSR$.

   Hint

   Isosceles triangles at O.

   Answer

   $\angle OPQ=35°$, $\angle OPS=55°$, $\angle OSR=35°$ (alternate to $\angle OQP$).

8. Q8 · 3 marks · Short answerHarder exam level

   ABCD is a kite with $AB=AD$ and $CB=CD$. If $\angle BAD=100°$ and $\angle BCD=40°$, find $\angle ABC$. Which diagonal is cut in half by the other?

   Hint

   Equal angles at B and D.

   Answer

   $\angle ABC=\frac{360°-140°}{2}=110°$. Diagonal AC (through the vertices where equal sides meet) bisects BD at right angles.

9. Q9 · 3 marks · Short answerHarder exam level

   Construct a rhombus whose diagonals are 6 cm and 8 cm. Write the steps and justify why your figure is a rhombus.

   Hint

   Perpendicular bisectors of each other.

   Answer

   Draw AC = 8 cm, mark midpoint O, draw a perpendicular through O, mark B and D 3 cm on each side, join ABCD. Diagonals bisect each other at 90°, so the four triangles at O are congruent (SAS) and all sides are equal: a rhombus.

10. Q10 · 5 marks · Long answerHarder exam level

    ABCD is a trapezium with $AB\parallel DC$, $\angle A=3x$ and $\angle D=x+20°$. (a) Find $\angle A$ and $\angle D$. (b) If $\angle B=70°$, find $\angle C$. (c) Is ABCD an isosceles trapezium? Give a reason.

    Hint

    Co-interior angles.

    Answer

    (a) $4x+20=180$, $x=40$: $\angle A=120°$, $\angle D=60°$. (b) $\angle C=180°-70°=110°$. (c) No: in an isosceles trapezium the base angles are equal, but $\angle A=120°\ne \angle B=70°$.

11. Q11 · 3 marks · Short answerHOTS / competency

    Prove that a parallelogram with one right angle is a rectangle.

    Hint

    Use the angle properties of a parallelogram.

    Answer

    Let $\angle A=90°$. Adjacent angles of a parallelogram add to 180°, so $\angle B=\angle D=180°-90°=90°$. Opposite angles are equal, so $\angle C=\angle A=90°$. All four angles are 90°, so by the definition it is a rectangle.

12. Q12 · 3 marks · Short answerHOTS / competency

    True or false, with reason: (i) If one pair of opposite sides of a quadrilateral is equal and the other pair is parallel, it is a parallelogram. (ii) A rectangle whose diagonals cross at 90° is a square.

    Hint

    Think of an isosceles trapezium; use congruent triangles at the crossing point.

    Answer

    (i) False: an isosceles trapezium has one pair of opposite sides parallel and the other pair equal, but is not a parallelogram. (ii) True: the diagonals are equal and bisect each other, so the four triangles at the crossing point are congruent (SAS with the 90° angles); hence all four sides are equal, and a rectangle with equal sides is a square.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.What is the simplest definition of a rectangle?Show answer

   A quadrilateral in which all angles are 90°.

2. 2.Give a definition of a rectangle using diagonals.Show answer

   A quadrilateral whose diagonals are equal and bisect each other.

3. 3.What is the angle sum of a quadrilateral?Show answer

   360°.

4. 4.In a parallelogram, how are adjacent and opposite angles related?Show answer

   Adjacent angles add to 180°; opposite angles are equal.

5. 5.Do the diagonals of a parallelogram bisect each other?Show answer

   Yes, always.

6. 6.At what angle do the diagonals of a rhombus meet?Show answer

   90°.

7. 7.What extra diagonal property does a square have over a rectangle?Show answer

   Its diagonals are perpendicular and bisect its angles (45° each).

8. 8.Define a kite.Show answer

   A quadrilateral ABCD with AB = BC and CD = DA.

9. 9.Define a trapezium as in the book.Show answer

   A quadrilateral with at least one pair of parallel opposite sides.

10. 10.Which quadrilateral is both a kite and a parallelogram?Show answer

    A rhombus.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you prove the diagonals of a rectangle are equal?
• ☐Can you show that equal, bisecting diagonals always give a rectangle?
• ☐Can you find angles at the crossing point of rectangle diagonals?
• ☐Can you prove the angle sum of a quadrilateral is 360°?
• ☐Can you find all angles of a parallelogram from one angle or an equation?
• ☐Can you prove the diagonals of a parallelogram bisect each other?
• ☐Can you find the angles formed by the diagonals of a rhombus?
• ☐Can you state the kite diagonal properties?
• ☐Can you find the angles of an isosceles trapezium from one angle?
• ☐Can you draw the Venn diagram of quadrilateral families?

Quadrilaterals: frequently asked questions