MathsCBSE Class 8 · Quadrilaterals

Rectangles, Squares and the Angle Sum of a Quadrilateral for Class 8

Updated 17 October 2026 · Fact-checked

A rectangle is a quadrilateral with all angles 90 degrees; its opposite sides are equal and parallel, and its diagonals are equal and bisect each other. A square is a rectangle with all sides equal, and its diagonals also meet at 90 degrees and bisect its angles. The four angles of any quadrilateral add to 360 degrees. Exams ask for angle chases, proofs by congruence, and true-or-false statements.

Matching tick marks show diagonals cut each other in half; only the square's diagonals cross at 90° and split its corners into equal 45° angles.

Illustration drawn for QuestionBuddy.

Understand Rectangles, Squares and the Angle Sum of a Quadrilateral

A quadrilateral is a closed four-sided figure (the word comes from the Latin *quadri*, four, and *latus*, side). Its angles are the angles between neighbouring sides.

Rectangle. The chapter defines a rectangle as a quadrilateral in which all the angles are $9 0^{\circ}$ . This one condition is enough: it forces the opposite sides to be equal and parallel (join a diagonal and use congruent triangles). Properties:
1. Opposite sides are equal.
2. Opposite sides are parallel.
3. The diagonals are equal in length.
4. The diagonals bisect each other (they cut at their midpoints).

*Why the diagonals are equal:* triangles $A B C$ and $B A D$ have $A B$ common, $B C = A D$ and the included angles both $9 0^{\circ}$ , so they are congruent by SAS and $A C = B D$ . *Why they bisect each other:* in a rectangle $A B C D$ with diagonals meeting at $O$ , triangles $A O B$ and $C O D$ have $∠ O A B = ∠ O C D$ (alternate angles, $A B ∥ D C$ ), $∠ O B A = ∠ O D C$ (alternate angles) and $A B = C D$ , so they are congruent (ASA), and $O A = O C$ , $O B = O D$ .

The carpenter's problem. A rectangular frame can be made from two equal strips of wood pinned at their midpoints, whatever the angle between them. So another definition is: *a rectangle is a quadrilateral whose diagonals are equal and bisect each other.* The chapter notes that carpenters in Europe and farmers in Mozambique use this method.

Square. A square is a quadrilateral with all angles $9 0^{\circ}$ and all sides equal. Hence every square is a rectangle, but not every rectangle is a square (a Venn diagram shows squares inside rectangles). A square has every rectangle property plus:

the diagonals meet at $9 0^{\circ}$ (they bisect each other at right angles);
the diagonals bisect the angles of the square, so each makes $4 5^{\circ}$ with the sides.

*Why the diagonals are perpendicular:* in a square, triangles $A O B$ and $C O B$ are congruent by SSS ( $O A = O C$ , $O B$ common, $A B = C B$ ), so $∠ A O B = ∠ C O B$ ; they form a linear pair, so each is $9 0^{\circ}$ . To make a square from the wooden strips, use equal strips crossing at their midpoints at a right angle.

Angle sum. The four angles of any quadrilateral add to $36 0^{\circ}$ , because a diagonal splits it into two triangles of $18 0^{\circ}$ each. This holds even for a quadrilateral that dents inward. It also explains why a quadrilateral cannot have three right angles and a fourth angle that is not $9 0^{\circ}$ : the fourth is $36 0^{\circ} - 27 0^{\circ} = 9 0^{\circ}$ .

Three ways to recognise a rectangle: all four angles are $9 0^{\circ}$ ; or three angles are $9 0^{\circ}$ (the fourth then is too); or the diagonals are equal and bisect each other.

Parallelograms, rhombuses, kites and trapeziums are on the next topic page of this chapter.

Rectangles, Squares and the Angle Sum of a Quadrilateral: formulas to remember

Angle sum of a quadrilateral: $∠ A + ∠ B + ∠ C + ∠ D = 36 0^{\circ}$; Two triangles of 180 degrees each.
Rectangle diagonals: $A C = B D, O A = O B = O C = O D$; Equal and bisect each other, so all four half-diagonals are equal.
Square diagonals: $A C = B D, A C ⊥ B D$; They also bisect the angles (45 degrees each).
Base angle in the diagonal triangle: $∠ O A B = \frac{18 0 ^{\circ} - ∠ A O B}{2}$; Triangle AOB is isosceles because OA = OB in a rectangle.
Missing fourth angle: $∠ D = 36 0^{\circ} - (∠ A + ∠ B + ∠ C)$; Use when three angles are given.

How to solve Rectangles, Squares and the Angle Sum of a Quadrilateral questions

A routine for angle and property questions on rectangles and squares.

1Draw the figure and mark every given equal side, right angle and diagonal property.
2If the diagonals meet at O in a rectangle or square, mark OA = OB = OC = OD; each triangle with two such sides is isosceles.
3Use the angle sum of a triangle (180 degrees) and of a quadrilateral (360 degrees).
4For proofs, choose two triangles that share a side or an angle and state the congruence condition (SSS, SAS, ASA or AAS) in full.
5Conclude using "corresponding parts of congruent triangles" and name the equal parts.
6For classification, check the minimum condition: for example, three right angles are enough for a rectangle.

Rectangles, Squares and the Angle Sum of a Quadrilateral 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.

Diagonals make a rectangle

Suppose a carpenter builds a rectangular door frame from four planks and then checks it by measuring the two diagonals.

Each diagonal measures 1.3 m and the carpenter has also checked that they cross at their midpoints. Is the frame a rectangle? How far is the crossing point from each corner?

Try it first, then show the solutionSolution

Diagonals that are equal and bisect each other give a rectangle, whatever the angle between them.
The crossing point is the midpoint of each diagonal, so its distance from each corner is $1.3 \div 2 = 0.65$ m.

Answer: Yes, the frame is a rectangle; the crossing point is 0.65 m from each corner.

Memory hook: Equal diagonals sharing a midpoint force four right angles.

Angle sum of a quadrilateral

Imagine a plot of land in the shape of a quadrilateral with three corner angles measured as 95 degrees, 80 degrees and 100 degrees.

What is the fourth angle?

Try it first, then show the solutionSolution

Sum of the three angles $= 95 + 80 + 100 = 275$ .
Fourth angle $= 360 - 275 = 85$ degrees.

Answer: 85 degrees

Memory hook: Four angles, 360 degrees: subtract what you know.

Tricks to remember Rectangles, Squares and the Angle Sum of a Quadrilateral

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

Two triangles in a quadrilateral
Draw one diagonal and you see two triangles, so the angle sum is 2 x 180 = 360 degrees.
Use it when: Recalling and justifying the angle sum.
Rectangle diagonals: equal and share a midpoint
Say "equal, bisect": E for equal, B for bisect. Square adds P for perpendicular.
Use it when: Listing diagonal properties quickly.
Square equals rectangle plus equal sides
A square is a rectangle that has been made all-sides-equal, so it inherits everything a rectangle has.
Use it when: Questions on classification and Venn diagrams.
Three right angles force the fourth
360 minus 270 is 90, so three right angles always mean a rectangle.
Use it when: True or false questions.

Rectangles, Squares and the Angle Sum of a Quadrilateral: solved examples

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

Example 1: Angles in a ratio · 1 mark · MCQBoard level

The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. The largest angle is: (a) 108 degrees (b) 120 degrees (c) 144 degrees (d) 160 degrees

Try it first, then show the solutionSolution

Sum of ratio parts $= 10$ , and the angle sum is $36 0^{\circ}$ , so one part is $3 6^{\circ}$ .
Angles: $3 6^{\circ}, 7 2^{\circ}, 10 8^{\circ}, 14 4^{\circ}$ (sum $36 0^{\circ}$ ). Largest is $14 4^{\circ}$ .

Answer: (c) 144 degrees

Example 2: Angles at the diagonal intersection · 2 marks · Short answerBoard level

In rectangle ABCD the diagonals meet at O and the angle AOD is 112 degrees. Find angle OAD and angle OAB.

Try it first, then show the solutionSolution

The diagonals of a rectangle are equal and bisect each other, so $O A = O D$ and triangle $A O D$ is isosceles. $∠ O A D = \frac{18 0 ^{\circ} - 11 2 ^{\circ}}{2} = 3 4^{\circ}$ . (1 mark)
$∠ D A B = 9 0^{\circ}$ , so $∠ O A B = 9 0^{\circ} - 3 4^{\circ} = 5 6^{\circ}$ . (1 mark)

Answer: Angle OAD = 34 degrees, angle OAB = 56 degrees

Example 3: Equal diagonals at 60 degrees · 3 marks · Short answerHarder board level

The diagonals of a quadrilateral PQRS are both 12 cm, bisect each other at O and meet at 60 degrees (angle POQ = 60 degrees). Show that PQRS is a rectangle, find PQ and the angles OPQ and OPS.

Try it first, then show the solutionSolution

Equal diagonals that bisect each other give a rectangle. $O P = O Q = 6$ cm. (1 mark)
In triangle $P O Q$ , $O P = O Q$ and the apex angle is $6 0^{\circ}$ , so the base angles are each $6 0^{\circ}$ and the triangle is equilateral: $P Q = 6$ cm and $∠ O P Q = 6 0^{\circ}$ . (1 mark)
$∠ P O S = 18 0^{\circ} - 6 0^{\circ} = 12 0^{\circ}$ (linear pair), so in isosceles triangle $P O S$ , $∠ O P S = \frac{18 0 ^{\circ} - 12 0 ^{\circ}}{2} = 3 0^{\circ}$ . Check: $∠ S P Q = 6 0^{\circ} + 3 0^{\circ} = 9 0^{\circ}$ . (1 mark)

Answer: PQRS is a rectangle; PQ = 6 cm; angle OPQ = 60 degrees; angle OPS = 30 degrees.

Example 4: Unknown angles by the angle sum · 3 marks · Short answerHarder board level

The angles of a quadrilateral are $(2 x + 5)^{\circ}$ , $(3 x - 10)^{\circ}$ , $(x + 25)^{\circ}$ and $(4 x - 20)^{\circ}$ . Find $x$ and each angle. Can this quadrilateral be a rectangle?

Try it first, then show the solutionSolution

Sum $= (2 x + 5) + (3 x - 10) + (x + 25) + (4 x - 20) = 10 x + 0 = 360$ , so $x = 36$ . (1 mark)
Angles: $7 7^{\circ}, 9 8^{\circ}, 6 1^{\circ}, 12 4^{\circ}$ ; check $77 + 98 + 61 + 124 = 360$ . (1 mark)
A rectangle needs all four angles to be $9 0^{\circ}$ ; these are not, so it is not a rectangle. (1 mark)

Answer: x = 36; angles 77, 98, 61, 124 degrees; not a rectangle.

Example 5: Diagonals of a square · 4 marks · Long answerHarder board level

ABCD is a square with diagonals AC and BD meeting at O. Prove that AC is perpendicular to BD.

Try it first, then show the solutionSolution

A square is a rectangle, so its diagonals bisect each other: $O A = O C$ . (1 mark)
Consider triangles $A O B$ and $C O B$ : $A B = C B$ (sides of a square), $O A = O C$ , and $O B$ is common. (1 mark)
So $△ A O B ≅ △ C O B$ by SSS, hence $∠ A O B = ∠ C O B$ . (1 mark)
These angles form a linear pair, so $∠ A O B + ∠ C O B = 18 0^{\circ}$ ; each is $9 0^{\circ}$ , so $A C ⊥ B D$ . (1 mark)

Answer: AC is perpendicular to BD (proved by SSS congruence of triangles AOB and COB).

Where students lose marks in Rectangles, Squares and the Angle Sum of a Quadrilateral

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

Mistake: Assuming the diagonals of a rectangle are perpendicular
They are perpendicular only in a square or a rhombus.
Wrong working
"In rectangle ABCD, angle AOB = 90 degrees."
Right working
In a general rectangle, $∠ A O B$ can take any value less than $18 0^{\circ}$ ; only in a square is it $9 0^{\circ}$ .
Do this: Read the question to see whether it says square or rectangle.
Usually costs: 1 to 2 marks.
Mistake: Using the angle sum of 180 degrees for a quadrilateral
The sum is 360 degrees; 180 is for triangles.
Wrong working
"Fourth angle $= 180 - (80 + 90 + 60)$ "
Right working
Fourth angle $= 360 - (80 + 90 + 60) = 13 0^{\circ}$ .
Do this: Write 360 degrees before substituting.
Usually costs: The answer.
Mistake: Writing a proof without naming the congruence condition
Marks go to the condition used (SSS, SAS, ASA, AAS) and the matching parts listed.
Wrong working
"Triangles are congruent, so OA = OC."
Right working
List the three equal pairs, name the condition, then write corresponding parts of congruent triangles.
Do this: Use a three-line template: the equal pairs, the condition, the conclusion.
Usually costs: 1 to 2 marks.
Mistake: Concluding a square from equal diagonals bisecting each other
That condition gives a rectangle. A square needs the diagonals to be perpendicular too.
Wrong working
"Diagonals equal and bisecting, so it is a square."
Right working
Equal diagonals that bisect each other give a rectangle; adding the perpendicular condition gives a square.
Do this: Check each property against the definition.
Usually costs: 1 mark.

Exam tips

Mark right angles, equal sides and equal half-diagonals on the figure before working: it often reveals isosceles triangles.
In proofs, write the reason after each statement: given, property of a rectangle, common side, and so on.
For ratio-of-angles questions, find one part first (360 divided by the sum of the ratio).
Check each angle chase by adding the angles of the triangle or quadrilateral you used.
When an answer is a name (rectangle, square), write the property that decides it.

Practice questions on Rectangles, Squares and the Angle Sum of a Quadrilateral

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): Every square is a rectangle.
Reason (R): A square has all angles 90 degrees, which is the definition of a rectangle, and also has all sides equal.
(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
Compare the definition of a rectangle with the angles of a square.
Answer
(a). A rectangle is defined by all angles being $9 0^{\circ}$ . A square meets that condition, so it is a rectangle.
Q2 · 2 marks · Very short answerBoard level
Three angles of a quadrilateral are 90 degrees each. Find the fourth angle and name the quadrilateral.
Hint
Use the 360 degree sum.
Answer
Fourth angle $= 36 0^{\circ} - 3 \times 9 0^{\circ} = 9 0^{\circ}$ . All four angles are $9 0^{\circ}$ , so it is a rectangle.
Q3 · 3 marks · Short answerHarder board level
The diagonals of a quadrilateral ABCD are equal, bisect each other at O, and angle AOB = 50 degrees. Find angle OAB, angle OAD and angle BAD. State what ABCD is.
Hint
OA = OB, and angle AOD is a linear pair with angle AOB.
Answer
Equal diagonals bisecting each other make ABCD a rectangle. In isosceles triangle $A O B$ : $∠ O A B = \frac{18 0 ^{\circ} - 5 0 ^{\circ}}{2} = 6 5^{\circ}$ . $∠ A O D = 18 0^{\circ} - 5 0^{\circ} = 13 0^{\circ}$ , so in isosceles triangle $A O D$ : $∠ O A D = \frac{18 0 ^{\circ} - 13 0 ^{\circ}}{2} = 2 5^{\circ}$ . $∠ B A D = 6 5^{\circ} + 2 5^{\circ} = 9 0^{\circ}$ , which agrees with a rectangle.
Q4 · 4 marks · Long answerHOTS / competency
Decide whether each statement is true or false and justify briefly.
(i) Every rectangle is a square.
(ii) If the diagonals of a quadrilateral are equal, bisect each other and meet at 90 degrees, then it is a square.
(iii) A quadrilateral can have angles 150, 150, 40 and 40 degrees.
(iv) A quadrilateral with all angles equal and unequal diagonals exists.
Hint
For (ii), show that the adjacent sides are equal using congruent triangles. For (iv), find the common angle first.
Answer
(i) False: a rectangle of sides 3 cm and 5 cm has unequal sides. (ii) True: equal diagonals that bisect each other give a rectangle; with the diagonals meeting at $9 0^{\circ}$ , triangles $A O B$ and $B O C$ are congruent (SAS: $O A = O C$ , $∠ A O B = ∠ B O C = 9 0^{\circ}$ , $O B$ common), so $A B = B C$ and the rectangle has equal adjacent sides: a square. (iii) False: $150 + 150 + 40 + 40 = 380 \neq = 360$ . (iv) False: four equal angles are each $9 0^{\circ}$ , which gives a rectangle, whose diagonals are equal.
Q5 · 4 marks · Case-basedHOTS / competency
Case study: a carpenter makes a frame by pinning two wooden strips of equal length 1.4 m at their midpoints, so that the ends of the strips are the corners of the frame.
(i) What shape is the frame, whatever the angle between the strips? (1 mark)
(ii) The strips cross at 60 degrees. Find the angles each strip makes with the shorter side, and the length of the shorter side. (2 marks)
(iii) At what angle must the strips cross for the frame to be square? (1 mark)
Hint
Half of each strip is 0.7 m. The triangle with the 60 degree angle is isosceles.
Answer
(i) A rectangle (equal diagonals that bisect each other). (ii) Half of each strip is $0.7$ m. The triangle formed by the 60 degree angle has two sides of $0.7$ m and apex $6 0^{\circ}$ , so it is equilateral: the shorter side is $0.7$ m and the strip makes $6 0^{\circ}$ with it. In the adjacent triangle, with apex $12 0^{\circ}$ , the strip makes $3 0^{\circ}$ with the longer side. (iii) $9 0^{\circ}$ : the diagonals of a square meet at right angles.

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 sum of the angles of a quadrilateral?Show answer
360 degrees.
2.How is a rectangle defined in the chapter?Show answer
A quadrilateral in which all the angles are 90 degrees.
3.What are the diagonal properties of a rectangle?Show answer
They are equal in length and bisect each other.
4.What extra diagonal property does a square have?Show answer
They bisect each other at 90 degrees and bisect the angles of the square.
5.Is every square a rectangle?Show answer
Yes, but not every rectangle is a square.
6.Why can a quadrilateral not have exactly three right angles?Show answer
The fourth angle would be 360 - 270 = 90 degrees as well.

Rectangles, Squares and the Angle Sum of a Quadrilateral: frequently asked questions

Do I have to prove the angle sum of a quadrilateral?

You should be able to: draw a diagonal, split the quadrilateral into two triangles and add their angle sums. The chapter also asks you to check this for a quadrilateral with an inward dent.

Can I define a rectangle by equal opposite sides?

No. Opposite sides equal and parallel is also true of non-rectangular parallelograms, so it does not define a rectangle.

Are the diagonals of a rectangle angle bisectors?

No. In a rectangle the diagonals do not bisect the angles unless the rectangle is a square. Among rectangles only a square has this property; a rhombus (which is not a rectangle) also has it.

What if the question gives "diagonals bisect each other" only?

That makes the quadrilateral a parallelogram. You also need equal diagonals for a rectangle.