MathsCBSE Class 8 · Quadrilaterals

Parallelograms, Rhombuses, Kites and Trapeziums for Class 8

Updated 17 October 2026 · Fact-checked

A parallelogram has both pairs of opposite sides parallel; its opposite sides and opposite angles are equal, adjacent angles add to 180 degrees and the diagonals bisect each other. A rhombus (all sides equal) is a parallelogram whose diagonals meet at 90 degrees and bisect its angles. A kite has two adjacent pairs of equal sides; a trapezium has at least one pair of parallel sides.

Understand Parallelograms, Rhombuses, Kites and Trapeziums

Parallelogram. A quadrilateral with both pairs of opposite sides parallel. A rectangle is a special parallelogram with all angles $9 0^{\circ}$ .

Properties (each is deduced, not assumed):
1. Opposite sides are parallel (definition) and equal (a diagonal gives two congruent triangles).
2. **Adjacent angles add up to $18 0^{\circ}$ (interior angles on the same side of a transversal to parallel lines), and opposite angles are equal**.
3. The diagonals bisect each other (triangles formed by the two diagonals are congruent by the alternate angles and equal opposite sides). The diagonals need not be equal.

If a quadrilateral has opposite sides equal, it is a parallelogram (join a diagonal, use SSS congruence, and alternate angles give parallel sides). A quadrilateral whose diagonals bisect each other is also a parallelogram.

Rhombus. A quadrilateral with all four sides equal. Then:

it is a parallelogram (opposite sides parallel), so adjacent angles add to $18 0^{\circ}$ , opposite angles are equal and the diagonals bisect each other;
the diagonals bisect the angles of the rhombus;
the diagonals meet at ** $9 0^{\circ}$ **.

If one angle is $5 0^{\circ}$ , the angles are $5 0^{\circ}, 13 0^{\circ}, 5 0^{\circ}, 13 0^{\circ}$ . A square is a rhombus with all angles $9 0^{\circ}$ , so it is a rectangle, a rhombus and a parallelogram.

Kite. A quadrilateral with two non-overlapping adjacent pairs of equal sides, for example $A B = A D$ and $C B = C D$ . Properties: the diagonal joining the vertices where the equal sides meet (AC) bisects the other diagonal (BD) at right angles, and bisects the angles at A and C. The angles at the two ends of the other diagonal, $∠ B$ and $∠ D$ , are equal (the triangles $A B C$ and $A D C$ are congruent by SSS). A kite need not be a rhombus.

Trapezium. A quadrilateral with at least one pair of parallel opposite sides. Angles on the same non-parallel side add up to $18 0^{\circ}$ (co-interior angles). In an isosceles trapezium the non-parallel sides are equal and the base angles (at the two ends of a parallel side) are equal; hence the other two angles are also equal.

Classification and Venn diagram. Squares lie inside rectangles and rhombuses; rectangles and rhombuses lie inside parallelograms; a kite that is also a parallelogram is a rhombus. A square is the only quadrilateral that is both a rectangle and a rhombus.

For rectangles, squares and the angle sum, see the previous topic page of this chapter.

Parallelograms, Rhombuses, Kites and Trapeziums: formulas to remember

Adjacent angles of a parallelogram: $∠ A + ∠ B = 18 0^{\circ}, ∠ A = ∠ C$; Adjacent supplementary; opposite equal.
Parallelogram diagonals: $O A = O C, O B = O D$; They bisect each other but need not be equal.
Rhombus diagonals: $A C ⊥ B D, each bisects the vertex angles$; Right-angled triangles AOB and so on.
Right triangle in a rhombus: $∠ O A B + ∠ O B A = 9 0^{\circ}$; Triangle AOB has a right angle at O.
Trapezium co-interior angles: $∠ A + ∠ D = 18 0^{\circ} (A B ∥ D C)$; Angles on the same non-parallel side.
Kite angles: $∠ B = ∠ D (A B = A D, C B = C D)$; Angles between unequal sides are equal.

How to solve Parallelograms, Rhombuses, Kites and Trapeziums questions

A routine for angle and side questions on special quadrilaterals.

1Name the quadrilateral from the given information and recall its properties.
2For parallelograms, set adjacent angles equal to 180 degrees; for opposite angles, set them equal.
3For rhombus diagonals, use the right angle at O and the angle bisecting property to find the angles in triangle AOB.
4For a kite, use the perpendicular diagonal and the equal angles at B and D; check with the 360 degree sum.
5For a trapezium, find co-interior pairs on each non-parallel side.
6Verify by adding the four angles to 360 degrees.

Parallelograms, Rhombuses, Kites and Trapeziums in real life

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.

Rhombus as a folding gate

Suppose a folding gate is made of rods that form rhombuses of side 30 cm, and the gate is opened until one angle of every rhombus is 60 degrees.

Find the other angles of the rhombus and the length of the shorter diagonal.

Try it first, then show the solutionSolution

Adjacent angles add to $18 0^{\circ}$ : the angles are $6 0^{\circ}, 12 0^{\circ}, 6 0^{\circ}, 12 0^{\circ}$ .
The shorter diagonal joins the ends of the two sides (30 cm each) that meet at the $6 0^{\circ}$ angle. That triangle has two equal sides and a $6 0^{\circ}$ angle between them, so its other two angles are $6 0^{\circ}$ each and it is equilateral.
The shorter diagonal is therefore 30 cm.

Answer: Angles 60, 120, 60, 120 degrees; shorter diagonal 30 cm.

Memory hook: A rhombus with a 60 degree angle splits into two equilateral triangles.

Parallelogram angles

Imagine a window frame that has been pushed out of square so that it is a parallelogram.

One corner measures 72 degrees. Find the other three corners.

Try it first, then show the solutionSolution

Opposite corner is also $7 2^{\circ}$ .
Adjacent corners are $18 0^{\circ} - 7 2^{\circ} = 10 8^{\circ}$ , and so is the fourth.

Answer: 72, 108, 72, 108 degrees

Memory hook: Adjacent supplementary, opposite equal.

Tricks to remember Parallelograms, Rhombuses, Kites and Trapeziums

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

Parallelogram: opposite equal, adjacent 180
Opposite sides and opposite angles are equal; neighbours in the angles add to 180 degrees.
Use it when: Setting up equations for unknown angles.
Rhombus: diagonals are perpendicular bisectors
The rhombus diagonals cut each other at right angles and cut the corners in half.
Use it when: Finding angles in triangle AOB.
Kite: one diagonal is the axis
The line joining the corners where equal sides meet is the axis; it bisects the other diagonal at 90 degrees.
Use it when: Kite angle questions.
Trapezium: at least one pair
"At least one pair" means a parallelogram is also a trapezium in the chapter's definition.
Use it when: Venn diagram and true-or-false questions.

Parallelograms, Rhombuses, Kites and Trapeziums: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Opposite angle of a parallelogram · 1 mark · MCQBoard level

In a parallelogram ABCD, $∠ A = (2 x + 10)^{\circ}$ and $∠ B = (3 x - 5)^{\circ}$ . The measure of $∠ C$ is: (a) 70 degrees (b) 80 degrees (c) 100 degrees (d) 110 degrees

Try it first, then show the solutionSolution

Adjacent angles: $(2 x + 10) + (3 x - 5) = 180$ , so $5 x + 5 = 180$ and $x = 35$ .
$∠ A = 8 0^{\circ}$ , and $∠ C = ∠ A = 8 0^{\circ}$ .

Answer: (b) 80 degrees

Example 2: Angle in a rhombus triangle · 2 marks · Short answerBoard level

ABCD is a rhombus with angle A = 64 degrees, and the diagonals meet at O. Find angle ABO.

Try it first, then show the solutionSolution

Adjacent angles of a rhombus add to $18 0^{\circ}$ : $∠ B = 18 0^{\circ} - 6 4^{\circ} = 11 6^{\circ}$ . (1 mark)
The diagonal $B D$ bisects $∠ B$ , so $∠ A B O = 11 6^{\circ} \div 2 = 5 8^{\circ}$ . Check: in right triangle $A O B$ , $3 2^{\circ} + 5 8^{\circ} + 9 0^{\circ} = 18 0^{\circ}$ . (1 mark)

Answer: Angle ABO = 58 degrees

Example 3: Sides and angles of a parallelogram · 3 marks · Short answerHarder board level

A parallelogram has perimeter 56 cm. Two adjacent sides differ by 4 cm, and two adjacent angles are in the ratio 4 : 5. Find the lengths of the sides and all four angles.

Try it first, then show the solutionSolution

Let the sides be $x$ and $x + 4$ . Perimeter $= 2 (2 x + 4) = 56$ , so $2 x + 4 = 28$ and $x = 12$ . Sides: 12 cm and 16 cm. (1 mark)
Adjacent angles are supplementary: $4 k + 5 k = 18 0^{\circ}$ gives $k = 2 0^{\circ}$ , so the angles are $8 0^{\circ}$ and $10 0^{\circ}$ . (1 mark)
Opposite angles are equal: $8 0^{\circ}, 10 0^{\circ}, 8 0^{\circ}, 10 0^{\circ}$ . (1 mark)

Answer: Sides 12 cm and 16 cm; angles 80, 100, 80, 100 degrees.

Example 4: Angles of a kite · 3 marks · Short answerHarder board level

In kite ABCD, AB = AD and CB = CD. Angle BAD = 64 degrees and angle BCD = 100 degrees. Find angle ABC and angle ADC, and check using the diagonals.

Try it first, then show the solutionSolution

Triangles $A B C$ and $A D C$ are congruent (SSS), so $∠ A B C = ∠ A D C$ . (1 mark)
Angle sum: $64 + 100 + 2∠ A B C = 360$ , so $∠ A B C = \frac{196}{2} = 9 8^{\circ}$ and $∠ A D C = 9 8^{\circ}$ . (1 mark)
Check: the diagonal AC is the axis, perpendicular to BD at O. In triangle $A O B$ : $∠ O A B = 3 2^{\circ}$ , so $∠ A B O = 5 8^{\circ}$ . In triangle $C O B$ : $∠ O C B = 5 0^{\circ}$ , so $∠ C B O = 4 0^{\circ}$ . Then $∠ A B C = 5 8^{\circ} + 4 0^{\circ} = 9 8^{\circ}$ . (1 mark)

Answer: Angle ABC = angle ADC = 98 degrees.

Example 5: Isosceles trapezium · 4 marks · Long answerHarder board level

ABCD is an isosceles trapezium with AB parallel to DC and AD = BC. If angle A = 68 degrees, find the other three angles and explain each step.

Try it first, then show the solutionSolution

In an isosceles trapezium the angles at the ends of a parallel side are equal: $∠ B = ∠ A = 6 8^{\circ}$ . (1 mark)
$A B ∥ D C$ and $A D$ is a transversal, so $∠ A + ∠ D = 18 0^{\circ}$ and $∠ D = 18 0^{\circ} - 6 8^{\circ} = 11 2^{\circ}$ . (1 mark)
Similarly $∠ C = 18 0^{\circ} - 6 8^{\circ} = 11 2^{\circ}$ . (1 mark)
Check: $68 + 68 + 112 + 112 = 360$ . (1 mark)

Answer: B = 68 degrees, C = 112 degrees, D = 112 degrees.

Where students lose marks in Parallelograms, Rhombuses, Kites and Trapeziums

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

Mistake: Treating adjacent angles of a parallelogram as equal
Adjacent angles are supplementary; opposite angles are equal.
Wrong working
" $∠ A = ∠ B = 8 0^{\circ}$ ."
Right working
$∠ A = 8 0^{\circ}$ gives $∠ B = 10 0^{\circ}$ .
Do this: Write adjacent = 180 and opposite = equal at the top.
Usually costs: The answer.
Mistake: Saying the diagonals of a parallelogram are equal
They bisect each other but are equal only if it is a rectangle.
Wrong working
"In parallelogram PQRS, PR = QS."
Right working
Only $O P = O R$ and $O Q = O S$ hold in general.
Do this: Say "bisect" for parallelograms, "equal and bisect" for rectangles.
Usually costs: 1 mark.
Mistake: Forgetting the right angle at O in a rhombus
The right angle makes triangle AOB a right triangle with acute angles adding to 90 degrees.
Wrong working
"In triangle AOB, angles are 32, 58 and 80."
Right working
Angles are $3 2^{\circ}, 5 8^{\circ}, 9 0^{\circ}$ .
Do this: Mark the right angle on the figure at once.
Usually costs: 1 to 2 marks.
Mistake: Calling a trapezium isosceles because two angles are equal
The equal angles must be at the two ends of a parallel side.
Wrong working
" $∠ A = ∠ C$ so it is isosceles."
Right working
Check $∠ A = ∠ B$ (ends of $A B ∥ D C$ ).
Do this: Say which pair of sides is parallel before comparing angles.
Usually costs: 1 mark.

Exam tips

In angle problems, state the property you use: "adjacent angles of a parallelogram", "diagonals of a rhombus bisect the angles".
Mark the right angle at the intersection of rhombus and kite diagonals before any angle chase.
Check every set of four angles against 360 degrees.
In classification answers, give a counter-example, such as the 3 cm and 5 cm kite.
Draw the Venn diagram of quadrilaterals once and use it for all true-or-false items.

Practice questions on Parallelograms, Rhombuses, Kites and Trapeziums

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

Q1 · 1 mark · Assertion-reasonBoard level
Assertion (A): A rectangle is a parallelogram.
Reason (R): The opposite sides of a rectangle are parallel.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Recall the definition of a parallelogram.
Answer
(a). A parallelogram is defined as a quadrilateral with both pairs of opposite sides parallel. A rectangle has that property.
Q2 · 2 marks · Very short answerBoard level
The diagonals AC and BD of a parallelogram ABCD meet at O. If AC = 14 cm and BD = 9.6 cm, find OA, OB, OC and OD.
Hint
The diagonals bisect each other.
Answer
$O A = O C = 14 \div 2 = 7$ cm and $O B = O D = 9.6 \div 2 = 4.8$ cm.
Q3 · 3 marks · Short answerHarder board level
Is every kite a rhombus? Which quadrilateral is both a kite and a parallelogram? Justify your answers and give an example kite that is not a rhombus.
Hint
A parallelogram has opposite sides equal.
Answer
Not every kite is a rhombus: a kite with $A B = A D = 3$ cm and $C B = C D = 5$ cm has unequal adjacent sides. If a kite is also a parallelogram, its opposite sides are equal, so $A B = C D$ and $A D = C B$ ; together with $A B = A D$ this makes all four sides equal, so it is a rhombus. Every rhombus is a kite because its adjacent sides are equal in pairs.
Q4 · 4 marks · Long answerHOTS / competency
ABCD is a rectangle and P, Q, R, S are the midpoints of AB, BC, CD and DA. Prove that PQRS is a rhombus.
Hint
Show that four right triangles at the corners are congruent.
Answer
In rectangle $A B C D$ , $A B = C D$ and $B C = D A$ . So $A P = P B = C R = R D$ (halves of equal sides) and $B Q = QC = D S = S A$ . The corner triangles $P B Q$ , $QC R$ , $R D S$ and $S A P$ each have a right angle between sides equal to these halves, so they are congruent by SAS. Hence $P Q = QR = R S = S P$ (corresponding sides). A quadrilateral with four equal sides is a rhombus.
Q5 · 4 marks · Case-basedHOTS / competency
Case study: the cross-section of a shed roof is a trapezium ABCD with AB parallel to DC. The angles A and D are in the ratio 2 : 3 and the angles B and C are in the ratio 5 : 7.
(i) Find angle A and angle D. (1 mark)
(ii) Find angle B and angle C. (1 mark)
(iii) Verify that the four angles add to 360 degrees and decide whether the trapezium is isosceles, giving a reason. (2 marks)
Hint
Angles A and D are co-interior angles for the parallel lines AB and DC, as are B and C.
Answer
(i) $2 k + 3 k = 18 0^{\circ}$ gives $k = 3 6^{\circ}$ , so $∠ A = 7 2^{\circ}$ and $∠ D = 10 8^{\circ}$ . (ii) $5 m + 7 m = 18 0^{\circ}$ gives $m = 1 5^{\circ}$ , so $∠ B = 7 5^{\circ}$ and $∠ C = 10 5^{\circ}$ . (iii) $72 + 108 + 75 + 105 = 360$ . In an isosceles trapezium the angles at the ends of a parallel side are equal, but $∠ A = 7 2^{\circ} \neq = 7 5^{\circ} = ∠ B$ , so it is not isosceles.

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.What is a parallelogram?Show answer
A quadrilateral in which both pairs of opposite sides are parallel.
2.What is a rhombus?Show answer
A quadrilateral with all four sides equal.
3.What is a kite?Show answer
A quadrilateral with two non-overlapping adjacent pairs of sides of equal length.
4.What is a trapezium?Show answer
A quadrilateral with at least one pair of parallel opposite sides.
5.What do the diagonals of a rhombus do?Show answer
They bisect each other at right angles and bisect the angles of the rhombus.
6.What is special about the angles in an isosceles trapezium?Show answer
The angles at the two ends of a parallel side are equal.

Parallelograms, Rhombuses, Kites and Trapeziums: frequently asked questions

Is a square a rhombus?

Yes. A square has all sides equal, so it is a rhombus; it is also a rectangle and a parallelogram.

Is a parallelogram a trapezium?

By the chapter's definition (at least one pair of parallel sides), yes.

Do the diagonals of a kite bisect each other?

Only one of them is bisected: the axis diagonal bisects the other at right angles, but the other does not bisect the axis in general.

How do I prove a quadrilateral is a parallelogram?

Show opposite sides are equal, or that the diagonals bisect each other, or that opposite sides are parallel. The chapter proves the first two statements.