MathsICSE Class 8 · Chapter 15

Understanding Quadrilaterals ICSE Class 8: Angle Sum and Parallelogram Properties

Updated 15 October 2026 · Fact-checked

A quadrilateral is a closed figure with four sides, and its four angles always add up to 360°. ICSE Class 8 verifies the properties of a parallelogram: opposite sides and opposite angles are equal and the diagonals bisect each other, with equal diagonals in a rectangle and perpendicular ones in a rhombus.

What Understanding Quadrilaterals is about

A quadrilateral is a closed plane figure with four straight sides, four vertices and four angles. Its two diagonals join opposite vertices. Quadrilateral $A B C D$ is named by going round the vertices in order.

The CISCE Class VIII curriculum asks you to understand:

the angle sum property: the angles of a quadrilateral add to $360°$ ;
the properties of a parallelogram, by verification (measuring and checking): opposite sides are equal, opposite angles are equal, and the diagonals bisect each other;
the diagonals of special parallelograms: in a rectangle they are equal and bisect each other; in a rhombus they bisect each other at right angles; in a square they are equal and bisect each other at right angles.

In a school exam you will find missing angles and lengths using these properties, often by forming and solving a simple equation, and you may be asked to verify a property by drawing and measuring.

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Understanding Quadrilaterals sits in the Geometry theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Angle sum property

Draw one diagonal, say $A C$ , of quadrilateral $A B C D$ . It splits the quadrilateral into two triangles, $A B C$ and $A C D$ . Each triangle's angles add to $180°$ , so the four angles of the quadrilateral add to $2 \times 180° = 360°$ :
$∠ A + ∠ B + ∠ C + ∠ D = 360°$ .

This is true for every quadrilateral, whatever its shape (as long as it does not cross itself).

If angles are given in a ratio, let them be $3 x, 4 x, \dots$ and use the sum.

Worked example

Three angles of a quadrilateral are $75°$ , $95°$ and $110°$ . Find the fourth.

Try it first, then show the solutionSolution

Sum of the three $= 280°$ .
Fourth angle $= 360° - 280° = 80°$ .

Answer: $80°$

Types of quadrilaterals

Shape	Defining feature
Trapezium	one pair of parallel sides
Parallelogram	both pairs of opposite sides parallel
Rectangle	parallelogram with a right angle
Rhombus	parallelogram with all sides equal
Square	rectangle with all sides equal (also a rhombus with a right angle)

So every square is a rectangle and a rhombus; every rectangle and every rhombus is a parallelogram. Each special shape has all the properties of the shapes above it.

In a trapezium with $A B ∥ D C$ , the angles between the parallel sides on the same side are co-interior, so $∠ A + ∠ D = 180°$ and $∠ B + ∠ C = 180°$ .

Worked example

The angles of a quadrilateral are in the ratio $3 : 4 : 5 : 6$ . Find them.

Try it first, then show the solutionSolution

Let the angles be $3 x, 4 x, 5 x, 6 x$ .
$18 x = 360°$ , so $x = 20°$ .
Angles: $60°, 80°, 100°, 120°$ .

Answer: $60°, 80°, 100°, 120°$

Sides and angles of a parallelogram

In parallelogram $A B C D$ (verified by measuring several parallelograms):
1. Opposite sides are equal: $A B = D C$ and $A D = B C$ .
2. Opposite angles are equal: $∠ A = ∠ C$ and $∠ B = ∠ D$ .
3. Adjacent angles are supplementary: $∠ A + ∠ B = 180°$ (co-interior angles between parallel lines).

So if you know one angle of a parallelogram, you know all four.

To verify by measurement: draw a parallelogram with a ruler and set square, measure the sides with a ruler and the angles with a protractor, and record them in a table.

Worked example

One angle of a parallelogram is $65°$ . Find the other three.

Try it first, then show the solutionSolution

The adjacent angle is $180° - 65° = 115°$ .
Opposite angles are equal: the angles are $65°, 115°, 65°, 115°$ .
Check: $65 + 115 + 65 + 115 = 360$ ✓.

Answer: $65°, 115°, 65°, 115°$

Diagonals of a parallelogram bisect each other

If the diagonals $A C$ and $B D$ of parallelogram $A B C D$ meet at $O$ , then
$O A = O C$ and $O B = O D$ .

So $O$ is the mid-point of both diagonals. (The two diagonals are not usually equal to each other.)

This gives an equation whenever $O A$ and $O C$ are written as expressions.

Worked example

In parallelogram $A B C D$ , the diagonals meet at $O$ . $O A = x + 2$ and $O C = 3 x - 4$ (in cm). Find $x$ and $A C$ .

Try it first, then show the solutionSolution

Diagonals bisect each other: $x + 2 = 3 x - 4$ .
$2 x = 6$ , so $x = 3$ .
$O A = 5$ cm, so $A C = 2 \times 5 = 10$ cm.

Answer: $x = 3$ ; $A C = 10$ cm

Diagonals of a rectangle

A rectangle is a parallelogram, so its diagonals bisect each other. In addition, the diagonals of a rectangle are equal: $A C = B D$ .

Together these mean $O A = O B = O C = O D$ : all four half-diagonals are equal. So triangles $A O B$ , $B O C$ , $C O D$ and $D O A$ are all isosceles, which helps to find angles.

Worked example

In rectangle $A B C D$ , the diagonals meet at $O$ . $O A = 2 x + 4$ and $O B = 3 x + 1$ . Find $x$ and the length of each diagonal.

Try it first, then show the solutionSolution

In a rectangle $O A = O B$ : $2 x + 4 = 3 x + 1$ .
$x = 3$ , so $O A = 10$ .
Each diagonal $= 2 \times 10 = 20$ .

Answer: $x = 3$ ; diagonals 20 units each

Diagonals of a rhombus and a square

Rhombus: the diagonals bisect each other at right angles, so $∠ A O B = 90°$ . They also bisect the angles of the rhombus. They are usually not equal.
Square: the diagonals are equal and bisect each other at right angles. Each diagonal makes $45°$ with the sides.

Draw-and-label checklist (verification): draw the shape, draw both diagonals, label their meeting point $O$ , measure $O A$ , $O B$ , $O C$ , $O D$ and $∠ A O B$ , and record them in a table.

Worked example

In rhombus $A B C D$ , $∠ B A D = 70°$ and the diagonals meet at $O$ . Find $∠ O A B$ and $∠ O B A$ .

Try it first, then show the solutionSolution

The diagonal $A C$ bisects $∠ A$ : $∠ O A B = 35°$ .
Diagonals of a rhombus meet at $90°$ : $∠ A O B = 90°$ .
In triangle $A O B$ : $∠ O B A = 180° - 90° - 35° = 55°$ .

Answer: $∠ O A B = 35°$ , $∠ O B A = 55°$

Maths in real life: Understanding 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.

Rectangle test

Suppose a carpenter makes a door frame and measures both diagonals as 150 cm, and checks that they cut each other in half.

Is the frame a rectangle? Why does this check work?

Try it first, then show the solutionSolution

Diagonals that bisect each other make the frame a parallelogram.
Equal diagonals in a parallelogram mean it is a rectangle, so every corner is $90°$ .

Answer: Yes, it is a rectangle.

Memory hook: Equal, bisecting diagonals mean a rectangle.

Parallelogram angles

Imagine a folding gate made of parallelogram-shaped links, and one angle of a link is $50°$ when the gate is half open.

What are the other angles of that link?

Try it first, then show the solutionSolution

Adjacent angle $= 180° - 50° = 130°$ .
Angles: $50°, 130°, 50°, 130°$ .

Answer: $50°, 130°, 50°, 130°$

Memory hook: Neighbouring angles in a parallelogram add to 180°.

Angle sum

Suppose a four-sided garden plot has three corners measured as $90°$ , $90°$ and $110°$ .

What is the fourth corner?

Try it first, then show the solutionSolution

Sum of three $= 290°$ .
Fourth $= 360° - 290° = 70°$ .

Answer: $70°$

Memory hook: Four angles always make 360°.

Topics in this chapter

Understanding Quadrilaterals formula sheet

Angle sum: $∠ A + ∠ B + ∠ C + ∠ D = 360°$; Any quadrilateral.
Parallelogram angles: $∠ A = ∠ C; ∠ A + ∠ B = 180°$; Opposite equal, adjacent supplementary.
Parallelogram diagonals: $O A = O C; O B = O D$; Bisect each other.
Rectangle diagonals: $A C = B D$; So OA = OB = OC = OD.
Rhombus diagonals: $∠ A O B = 90°$; They also bisect the angles.

How the school exam asks Understanding Quadrilaterals

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

Question type	How it is asked	Method
Missing angle	Three angles are given, or the angles are in a ratio; find the angles.	Angles add to 360°.
Parallelogram angles	One angle, or adjacent angles in a ratio, or angles as expressions in x.	Opposite angles equal, adjacent angles add to 180°.
Diagonal lengths	OA = x + 2 and OC = 3x − 4; find x and AC.	Diagonals bisect each other (and are equal in a rectangle).
Angles at the diagonals	In a rectangle or rhombus, given one angle, find angles in the triangles formed by the diagonals.	Isosceles triangles in a rectangle; 90° and bisected angles in a rhombus.
Verify a property	Draw a parallelogram and verify that its diagonals bisect each other.	Draw accurately, measure, tabulate, conclude.

Tricks to remember Understanding Quadrilaterals

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

Four corners, 360
A quadrilateral is two triangles: 2 × 180° = 360°.
Use it when: Any missing angle.
Opposite equal, neighbours make 180
In a parallelogram, opposite angles match; neighbouring angles add to 180°.
Use it when: Parallelogram angles.
R-E-C-tangle: Equal diagonals
Rectangle: Equal diagonals. Rhombus: Right-angle diagonals. Square: both.
Use it when: Diagonal properties.
Family tree
Square → rectangle and rhombus → parallelogram → quadrilateral: a child has all its parents' properties.
Use it when: Deciding which properties apply.

Understanding Quadrilaterals: exam-level solved questions

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

Example 1: Adjacent angle · 1 mark · MCQExam level

One angle of a parallelogram is $70°$ . The angle next to it is:
(a) $70°$
(b) $110°$
(c) $20°$
(d) $290°$

Try it first, then show the solutionSolution

Adjacent angles of a parallelogram are supplementary.
$180° - 70° = 110°$ .

Answer: (b)

Example 2: Angles in a ratio · 3 marks · Short answerExam level

The angles of a quadrilateral are $x$ , $2 x$ , $3 x$ and $4 x$ . Find each angle.

Try it first, then show the solutionSolution

$x + 2 x + 3 x + 4 x = 360°$ .
$10 x = 360°$ , so $x = 36°$ .
Angles: $36°, 72°, 108°, 144°$ .

Answer: $36°, 72°, 108°, 144°$

Example 3: Angles as expressions · 3 marks · Short answerExam level

In parallelogram $A B C D$ , $∠ A = (3 x - 20) °$ and $∠ C = (x + 40) °$ . Find $x$ and all the angles.

Try it first, then show the solutionSolution

Opposite angles are equal: $3 x - 20 = x + 40$ , so $x = 30$ .
$∠ A = ∠ C = 70°$ .
$∠ B = ∠ D = 180° - 70° = 110°$ .

Answer: $x = 30$ ; $70°, 110°, 70°, 110°$

Example 4: Adjacent angles in a ratio · 3 marks · Short answerHarder exam level

Two adjacent angles of a parallelogram are in the ratio $2 : 3$ . Find all its angles.

Try it first, then show the solutionSolution

Adjacent angles: $2 k + 3 k = 180°$ , so $k = 36°$ .
They are $72°$ and $108°$ .
All angles: $72°, 108°, 72°, 108°$ .

Answer: $72°, 108°, 72°, 108°$

Example 5: Angles in a rectangle · 4 marks · Long answerHarder exam level

The diagonals of rectangle $A B C D$ meet at $O$ . If $∠ O A B = 32°$ , find $∠ O B A$ , $∠ A O B$ and $∠ O B C$ .

Try it first, then show the solutionSolution

In a rectangle $O A = O B$ , so triangle $A O B$ is isosceles: $∠ O B A = 32°$ .
$∠ A O B = 180° - 32° - 32° = 116°$ .
$∠ A B C = 90°$ , so $∠ O B C = 90° - 32° = 58°$ .
Answers: $32°$ , $116°$ , $58°$ .

Answer: $∠ O B A = 32°$ , $∠ A O B = 116°$ , $∠ O B C = 58°$

Example 6: Angles in a rhombus · 4 marks · Long answerHarder exam level

In rhombus $A B C D$ , $∠ A B C = 110°$ and the diagonals meet at $O$ . Find $∠ A B D$ , $∠ B A C$ and $∠ A O B$ , and check that triangle $A O B$ 's angles add to $180°$ .

Try it first, then show the solutionSolution

Diagonal $B D$ bisects $∠ A B C$ : $∠ A B D = 55°$ .
$∠ B A D = 180° - 110° = 70°$ ; diagonal $A C$ bisects it: $∠ B A C = 35°$ .
Diagonals of a rhombus are perpendicular: $∠ A O B = 90°$ .
Check: $55° + 35° + 90° = 180°$ ✓.

Answer: $55°$ , $35°$ , $90°$

Example 7: Angle bisectors in a parallelogram · 5 marks · Long answerHOTS / competency

In parallelogram $A B C D$ , the bisectors of $∠ A$ and $∠ B$ meet at $P$ . Prove that $∠ A P B = 90°$ . Check with $∠ A = 80°$ .

Try it first, then show the solutionSolution

Adjacent angles: $∠ A + ∠ B = 180°$ .
So $\frac{1}{2} ∠ A + \frac{1}{2} ∠ B = 90°$ , that is $∠ P A B + ∠ P B A = 90°$ .
In triangle $A P B$ : $∠ A P B = 180° - 90° = 90°$ .
Check: $∠ A = 80°$ gives $∠ B = 100°$ , halves $40°$ and $50°$ .
$∠ A P B = 180° - 40° - 50° = 90°$ ✓.

Answer: $∠ A P B = 90°$

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Designing floor tiles

Suppose a designer makes parallelogram-shaped tiles $A B C D$ with $A B = 12$ cm, $B C = 7$ cm and $∠ A B C = 60°$ .

(i) · 3 marks
Find the other three angles of the tile.
Answer
Answer: $∠ A = 120°$ , $∠ C = 120°$ , $∠ D = 60°$
Adjacent angles add to $180°$ : $∠ A = ∠ C = 120°$ . Opposite angles equal: $∠ D = 60°$ .
(ii) · 3 marks
Find the perimeter of the tile.
Answer
Answer: 38 cm
Opposite sides equal: $2 (12 + 7) = 38$ cm.
(iii) · 4 marks
A second design is a rhombus with every side 12 cm and $∠ A B C = 60°$ . Show that diagonal $A C$ is 12 cm long.
Answer
Answer: $A C = 12$ cm
In triangle $A B C$ , $A B = B C = 12$ cm, so it is isosceles and the base angles are $\frac{180° - 60°}{2} = 60°$ each. All angles are $60°$ , so triangle $A B C$ is equilateral and $A C = 12$ cm.

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.

The angles of a quadrilateral are in the ratio $2 : 3 : 3 : 4$ . Find them. (3 marks)

Step	Marks
Angles $2 x, 3 x, 3 x, 4 x$ with $12 x = 360°$	1
$x = 30°$	1
Angles $60°, 90°, 90°, 120°$	1

In parallelogram $A B C D$ , $∠ A = (2 x + 25) °$ and $∠ B = (3 x - 5) °$ . Find $x$ and all four angles. (4 marks)

Step	Marks
Adjacent angles are supplementary: $2 x + 25 + 3 x - 5 = 180$	1
$5 x = 160$ , $x = 32$	1
$∠ A = 89°$ , $∠ B = 91°$	1
$∠ C = 89°$ , $∠ D = 91°$ (opposite angles equal)	1

The diagonals of rectangle $A B C D$ meet at $O$ , and $∠ A O B = 70°$ . Find $∠ O A B$ , $∠ O A D$ and $∠ O D A$ . (5 marks)

Step	Marks
Diagonals of a rectangle are equal and bisect each other, so $O A = O B = O D$	1
Triangle $A O B$ is isosceles: $∠ O A B = \frac{180° - 70°}{2} = 55°$	1
$∠ D A B = 90°$ , so $∠ O A D = 90° - 55° = 35°$	1
Triangle $A O D$ is isosceles ( $O A = O D$ ): $∠ O D A = ∠ O A D$	1
$∠ O D A = 35°$	1

Where students lose marks in Understanding Quadrilaterals

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

Mistake: Using 180° for a quadrilateral
180° is the angle sum of a triangle.
Wrong working
Fourth angle $= 180° - 280°$
Right working
Fourth angle $= 360° - 280° = 80°$
Do this: Quadrilateral: 360°. Triangle: 180°.
Usually costs: The answer.
Mistake: Making adjacent angles of a parallelogram equal
Adjacent angles add to 180°; only opposite angles are equal.
Wrong working
$∠ A = ∠ B = 70°$
Right working
$∠ B = 110°$
Do this: Opposite: equal. Adjacent: supplementary.
Usually costs: 1 to 2 marks.
Mistake: Saying the diagonals of every parallelogram are equal
Only a rectangle (and a square) has equal diagonals.
Do this: In a general parallelogram the diagonals only bisect each other.
Usually costs: 1 mark.
Mistake: Giving half the diagonal as the diagonal
$O A$ is half of $A C$ .
Wrong working
$A C = O A = 5$ cm
Right working
$A C = 2 \times O A = 10$ cm
Do this: Double the half-diagonal at the end.
Usually costs: 1 mark.
Mistake: Forgetting the 90° at the centre of a rhombus
The right angle at $O$ is the key to the triangle's angles.
Do this: Mark the right angle on your diagram first.
Usually costs: 1 to 2 marks.

Practice questions on Understanding Quadrilaterals

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 · MCQExam level
The sum of the angles of a quadrilateral is:
(a) $180°$
(b) $270°$
(c) $360°$
(d) $540°$
Hint
Two triangles.
Answer
(c)
Q2 · 1 mark · Very short answerExam level
Three angles of a quadrilateral are $85°$ , $95°$ and $100°$ . Find the fourth angle.
Hint
Subtract from 360°.
Answer
$80°$ .
Q3 · 2 marks · Short answerExam level
In parallelogram $A B C D$ , $A B = 8$ cm and $B C = 5$ cm. Find its perimeter.
Hint
Opposite sides are equal.
Answer
$2 (8 + 5) = 26$ cm.
Q4 · 3 marks · Short answerHarder exam level
The perimeter of a parallelogram is 50 cm, and one side is 4 cm longer than the other. Find the sides.
Hint
$2 (x + x + 4) = 50$ .
Answer
$4 x + 8 = 50$ , $x = 10.5$ cm; sides 10.5 cm and 14.5 cm.
Q5 · 1 mark · Assertion-reasonHarder exam level
Assertion (A): Every square is a rhombus.
Reason (R): All four sides of a square are equal.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
Definition of a rhombus.
Answer
(c)
Q6 · 2 marks · Short answerHarder exam level
The diagonals of parallelogram $P QR S$ meet at $O$ . If $O P = 2 y + 1$ and $O R = y + 5$ , find $y$ and $P R$ .
Hint
Diagonals bisect each other.
Answer
$2 y + 1 = y + 5$ , $y = 4$ ; $O P = 9$ , $P R = 18$ .
Q7 · 3 marks · Short answerHarder exam level
The diagonals of rhombus $A B C D$ meet at $O$ . If $∠ O A B = 28°$ , find $∠ O B A$ , $∠ A B C$ and $∠ B A D$ .
Hint
$∠ A O B = 90°$ .
Answer
$∠ O B A = 62°$ ; $∠ A B C = 124°$ ; $∠ B A D = 56°$ (check $124 + 56 = 180$ ).
Q8 · 3 marks · Short answerHarder exam level
The angles of a quadrilateral are $(x + 10) °$ , $(2 x - 10) °$ , $(x + 30) °$ and $2 x °$ . Find $x$ and the angles.
Hint
Add and equate to 360.
Answer
$6 x + 30 = 360$ , $x = 55$ . Angles $65°, 100°, 85°, 110°$ .
Q9 · 3 marks · Short answerHarder exam level
In rectangle $A B C D$ , the diagonals meet at $O$ . If $O A = 3 x - 2$ and $O D = x + 6$ , find $x$ and the length of diagonal $B D$ .
Hint
In a rectangle all four half-diagonals are equal.
Answer
$3 x - 2 = x + 6$ , $x = 4$ ; $O D = 10$ , $B D = 20$ .
Q10 · 2 marks · Short answerHarder exam level
Can a quadrilateral have angles $100°$ , $120°$ , $80°$ and $70°$ ? Give a reason.
Hint
Add them.
Answer
No. They add to $370°$ , but the angles of a quadrilateral add to $360°$ .
Q11 · 4 marks · Long answerHOTS / competency
In parallelogram $A B C D$ , $∠ A$ is $24°$ less than twice $∠ B$ . Find all the angles.
Hint
$∠ A + ∠ B = 180°$ and $∠ A = 2∠ B - 24°$ .
Answer
$3∠ B - 24 = 180$ , $∠ B = 68°$ , $∠ A = 112°$ . Angles $112°, 68°, 112°, 68°$ .
Q12 · 3 marks · Short answerHOTS / competency
$A B C D$ is a square whose diagonals meet at $O$ . Show that triangle $A O B$ is an isosceles right-angled triangle and find $∠ O A B$ .
Hint
Use both diagonal properties of a square.
Answer
Diagonals are equal and bisect each other, so $O A = O B$ ; they meet at right angles, so $∠ A O B = 90°$ . Hence isosceles right-angled, and $∠ O A B = \frac{90°}{2} = 45°$ .

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 the angle sum of a quadrilateral?Show answer
360°.
2.What can you say about opposite angles of a parallelogram?Show answer
They are equal.
3.What do adjacent angles of a parallelogram add to?Show answer
180°.
4.Which quadrilaterals have equal diagonals?Show answer
Rectangles and squares.
5.In which quadrilaterals do the diagonals meet at 90°?Show answer
Rhombuses and squares.
6.Is every rhombus a square?Show answer
No; only if it has a right angle.
7.What does 'the diagonals bisect each other' mean?Show answer
They cut each other into two equal halves at their meeting point.
8.A trapezium has how many pairs of parallel sides?Show answer
One.

Best order to study it

Learn the parts of a quadrilateral and prove the angle sum property with a diagonal.
Learn the family of quadrilaterals and their defining features.
Verify and use the side and angle properties of a parallelogram.
Verify that the diagonals of a parallelogram bisect each other.
Finish with the diagonals of rectangles, rhombuses and squares.

Quick revision checklist

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

Did you use 360° for the four angles?
Did you make opposite angles equal and adjacent angles add to 180°?
Did you use OA = OC (and OA = OB only in a rectangle)?
Did you mark 90° at the centre of a rhombus or square?
Did you double the half-diagonal when asked for the diagonal?
Is your diagram labelled in order round the shape?

Understanding Quadrilaterals: frequently asked questions

Why do the angles of a quadrilateral add to 360°?

A diagonal splits it into two triangles, each with angles adding to 180°, so the total is 2 × 180° = 360°.

What does 'by verification' mean in the syllabus?

You check the property by drawing several parallelograms, measuring the sides, angles or diagonals, and seeing that the property holds each time. A formal proof comes in Class IX.

Is a square a rectangle?

Yes. A square has four right angles, so it is a rectangle; it is also a rhombus because all its sides are equal.

Do the diagonals of a rhombus have to be equal?

No. They bisect each other at right angles, but they are equal only when the rhombus is a square.

Is a trapezium a parallelogram?

No. A trapezium has only one pair of parallel sides; a parallelogram has two.

Based on: CISCE Curriculum for Upper Primary Classes (VI-VIII), Mathematics (Class VIII); CISCE RDCD: Curriculum Pre-School to Class VIII