MathsCBSE Class 9 · Quadrilaterals

Properties of a Parallelogram and Tests for Special Quadrilaterals

Updated 17 October 2026 · Fact-checked

In a parallelogram opposite sides are equal and parallel, opposite angles are equal, adjacent angles add up to $180^\circ$ and the diagonals bisect each other. A quadrilateral is a parallelogram if opposite sides (or angles) are equal, or one pair of opposite sides is equal and parallel, or the diagonals bisect each other. A rectangle has equal diagonals, a rhombus has diagonals that are perpendicular bisectors, and a square has both.

Diagonals of any parallelogram bisect each other at O; equal diagonals make a rectangle, perpendicular diagonals a rhombus, and a square has both.

Illustration drawn for QuestionBuddy.

Understand Properties of a Parallelogram and Tests for Special Quadrilaterals

Parallelogram. A quadrilateral with both pairs of opposite sides parallel. Its basic properties follow from a diagonal splitting it into two congruent triangles (ASA, since alternate angles are equal):
1. Opposite sides are equal ( $A B = C D$ , $A D = B C$ ).
2. Opposite angles are equal ( $∠ A = ∠ C$ , $∠ B = ∠ D$ ).
3. Adjacent angles are supplementary ( $∠ A + ∠ B = 18 0^{\circ}$ ), because $A D ∥ B C$ and they are co-interior angles.
4. The diagonals bisect each other: $A O = O C$ and $B O = O D$ .

Tests (converses). A quadrilateral is a parallelogram if any one of these holds: (a) both pairs of opposite sides are equal; (b) both pairs of opposite angles are equal; (c) the diagonals bisect each other; (d) one pair of opposite sides is both equal and parallel. For (b): $2∠ A + 2∠ B = 36 0^{\circ}$ gives $∠ A + ∠ B = 18 0^{\circ}$ , so $A D ∥ B C$ , and similarly $A B ∥ D C$ .

Special parallelograms.

Rectangle: a parallelogram with a right angle (so all four angles are $9 0^{\circ}$ ). Its diagonals are equal. A parallelogram with equal diagonals is a rectangle: with $A C = B D$ , $A B = D C$ and $A D$ common, $△ A B D ≅ △ D C A$ (SSS), so $∠ B A D = ∠ C D A$ ; these are co-interior angles with $A B ∥ D C$ , hence $2∠ A = 18 0^{\circ}$ and $∠ A = 9 0^{\circ}$ .
Rhombus: a parallelogram with all sides equal. Its diagonals bisect each other at right angles and bisect the angles at the vertices. Proof: $△ A O B ≅ △ A O D$ by SSS ( $A B = A D$ , $O B = O D$ , $A O$ common), so $∠ A O B = ∠ A O D$ ; they form a linear pair, so each is $9 0^{\circ}$ .
Square: a rectangle that is also a rhombus: equal sides, right angles, equal diagonals that bisect each other at right angles.

Calculations. In a parallelogram with $∠ A = (4 x + 5)^{\circ}$ and $∠ B = (2 x + 25)^{\circ}$ , adjacent angles add to $18 0^{\circ}$ : $6 x + 30 = 180$ and $x = 25$ , giving $10 5^{\circ}$ and $7 5^{\circ}$ . If diagonals meet at $O$ and $A O = 2 x + 3$ , $O C = 5 x - 6$ , then $2 x + 3 = 5 x - 6$ and $x = 3$ , so each half is $9$ and $A C = 18$ . In a rhombus with diagonals $16$ and $12$ , the half-diagonals $8$ and $6$ form a right triangle with hypotenuse the side: side $= 64 + 36 = 10$ , perimeter $40$ , area $\frac{1}{2} \times 16 \times 12 = 96$ . In a rectangle with diagonals meeting at $O$ and $∠ A O B = 7 0^{\circ}$ , $O A = O B$ gives $∠ O A B = 5 5^{\circ}$ and then $∠ A D B = 9 0^{\circ} - 5 5^{\circ} = 3 5^{\circ}$ .

Using coordinates. $A (0, 0)$ , $B (8, 0)$ , $C (11, 6)$ , $D (3, 6)$ : $A B = D C = (8, 0)$ and $A D = B C = (3, 6)$ , so $A B C D$ is a parallelogram; the diagonals have the same midpoint $(5.5, 3)$ . The area is base $\times$ height $= 8 \times 6 = 48$ . It is neither a rectangle (the sides $A B$ and $A D$ are not perpendicular) nor a rhombus ( $A B = 8 \neq = A D = 35$ ).

Writing a proof. State the given and the claim, draw the figure with equal marks, join a diagonal or use both, choose the congruence rule, and conclude with CPCT. Name each test you use.

Properties of a Parallelogram and Tests for Special Quadrilaterals: formulas to remember

Opposite sides and angles: $A B = C D, A D = B C; ∠ A = ∠ C, ∠ B = ∠ D$; Parallelogram.
Adjacent angles: $∠ A + ∠ B = 18 0^{\circ}$; Co-interior angles.
Diagonals of a parallelogram: $A O = O C, B O = O D$; They bisect each other.
Rectangle: $A C = B D, all angles 9 0^{\circ}$; Equal diagonals.
Rhombus: $A C ⊥ B D, side = (\frac{d _{1}}{2})^{2} + (\frac{d _{2}}{2})^{2}$; Area $= \frac{1}{2} d_{1} d_{2}$ .
Parallelogram tests: $opp. sides equal, opp. angles equal, diagonals bisect, or one pair equal and parallel$; Any one suffices.

How to solve Properties of a Parallelogram and Tests for Special Quadrilaterals questions

Match the given information to a property or a test, then calculate or prove.

1Identify the quadrilateral and list its properties (sides, angles, diagonals).
2For calculations, translate each property into an equation, such as "adjacent angles sum to $18 0^{\circ}$ " or "half-diagonals are equal".
3For proofs of a property, join a diagonal and prove two triangles congruent.
4To prove a quadrilateral is a parallelogram, apply one of the four tests and state which.
5To prove a rectangle or rhombus, first prove a parallelogram, then add the extra condition.
6Verify numbers by checking that angle sums and side relations hold.

Properties of a Parallelogram and Tests for Special Quadrilaterals 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 bisect each other

Suppose a folding gate has four equal bars joined at their ends, forming a rhombus, and the two diagonals measure 16 cm and 12 cm when the gate is opened a little.

Find the length of each bar.

Try it first, then show the solutionSolution

The diagonals of a rhombus bisect each other at right angles, so half-diagonals are 8 cm and 6 cm.
Each bar is the hypotenuse: $8^{2} + 6^{2} = 10$ cm.

Answer: Each bar is 10 cm long.

Memory hook: Four right triangles make up a rhombus.

Tricks to remember Properties of a Parallelogram and Tests for Special Quadrilaterals

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

Bisect each other
In any parallelogram the diagonals cut each other in half; they are equal only in a rectangle and perpendicular only in a rhombus.
Use it when: Choosing the correct diagonal property.
Rectangle equal, rhombus perpendicular
Rectangle: equal diagonals. Rhombus: perpendicular diagonals. Square: both.
Use it when: Classifying quadrilaterals.
Co-interior adds to 180
Neighbouring angles of a parallelogram sit between parallel lines, so they sum to 180 degrees.
Use it when: Angle problems.

Properties of a Parallelogram and Tests for Special Quadrilaterals: solved examples

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

Example 1: Adjacent angles · 3 marks · Short answerBoard level

In parallelogram $A B C D$ , $∠ A = (4 x + 5)^{\circ}$ and $∠ B = (2 x + 25)^{\circ}$ . Find $x$ and all four angles.

Try it first, then show the solutionSolution

Adjacent angles of a parallelogram are supplementary: $4 x + 5 + 2 x + 25 = 180$ . (1)
$6 x = 150$ , so $x = 25$ . (½)
$∠ A = 10 5^{\circ}$ , $∠ B = 7 5^{\circ}$ ; opposite angles are equal, so $∠ C = 10 5^{\circ}$ and $∠ D = 7 5^{\circ}$ . (1½)

Answer: $x = 25$ ; angles $10 5^{\circ}, 7 5^{\circ}, 10 5^{\circ}, 7 5^{\circ}$ .

Example 2: Diagonals · 3 marks · Short answerBoard level

The diagonals of a parallelogram $A B C D$ meet at $O$ . $A O = (2 x + 3)$ cm, $O C = (5 x - 6)$ cm, $B O = (3 y - 1)$ cm and $O D = (y + 7)$ cm. Find $x$ , $y$ , $A C$ and $B D$ .

Try it first, then show the solutionSolution

Diagonals bisect each other: $2 x + 3 = 5 x - 6$ , so $x = 3$ and $A O = 9$ . (1)
$3 y - 1 = y + 7$ , so $y = 4$ and $B O = 11$ . (1)
$A C = 2 \times 9 = 18$ cm and $B D = 2 \times 11 = 22$ cm. (1)

Answer: $x = 3$ , $y = 4$ , $A C = 18$ cm, $B D = 22$ cm.

Example 3: Equal diagonals give a rectangle · 5 marks · Long answerHarder board level

Prove that a parallelogram with equal diagonals is a rectangle. Hence find $∠ A D B$ in a rectangle $A B C D$ if its diagonals meet at $O$ with $∠ A O B = 7 0^{\circ}$ .

Try it first, then show the solutionSolution

Given: parallelogram $A B C D$ with $A C = B D$ . (½)
In $△ A B D$ and $△ D C A$ : $A B = D C$ (opposite sides), $A D = D A$ (common), $B D = C A$ (given). (1)
So $△ A B D ≅ △ D C A$ (SSS), giving $∠ B A D = ∠ C D A$ . (1)
$A B ∥ D C$ , so $∠ B A D + ∠ C D A = 18 0^{\circ}$ (co-interior), hence each is $9 0^{\circ}$ ; a parallelogram with one right angle is a rectangle. (1)
In rectangle $A B C D$ : $O A = O B$ , so $∠ O A B = ∠ O B A = \frac{18 0 ^{\circ} - 7 0 ^{\circ}}{2} = 5 5^{\circ}$ ; $∠ A D B = 9 0^{\circ} - ∠ A B D = 9 0^{\circ} - 5 5^{\circ} = 3 5^{\circ}$ . (1½)

Answer: It is a rectangle; $∠ A D B = 3 5^{\circ}$ .

Example 4: Rhombus angles · 3 marks · Short answerHarder board level

In a rhombus $A B C D$ , $∠ A = 5 0^{\circ}$ and the diagonals meet at $O$ . Find $∠ A B C$ , $∠ O A B$ and $∠ O B A$ .

Try it first, then show the solutionSolution

Adjacent angles are supplementary: $∠ A B C = 18 0^{\circ} - 5 0^{\circ} = 13 0^{\circ}$ . (1)
Diagonals bisect the vertex angles: $∠ O A B = 2 5^{\circ}$ . (1)
Diagonals are perpendicular, so in $△ A O B$ : $∠ O B A = 9 0^{\circ} - 2 5^{\circ} = 6 5^{\circ}$ (also half of $13 0^{\circ}$ ). (1)

Answer: $∠ A B C = 13 0^{\circ}$ , $∠ O A B = 2 5^{\circ}$ , $∠ O B A = 6 5^{\circ}$ .

Example 5: A plot on a grid · 4 marks · Case-basedHOTS / competency

Case: a plot has corners $A (0, 0)$ , $B (8, 0)$ , $C (11, 6)$ and $D (3, 6)$ on a grid (units in metres). (i) Show that $A B C D$ is a parallelogram. (ii) Show that the diagonals bisect each other. (iii) Find the area. (iv) Is it a rectangle or a rhombus? Give reasons.

Try it first, then show the solutionSolution

(i) $A B$ and $D C$ are both horizontal of length 8; the horizontal and vertical shifts from $A$ to $D$ $(3, 6)$ and from $B$ to $C$ $(3, 6)$ are equal, so $A D = B C$ and $A D ∥ B C$ : a pair of opposite sides is equal and parallel. (1½)
(ii) Midpoint of $A C = (5.5, 3)$ and midpoint of $B D = (5.5, 3)$ . (½)
(iii) Base $A B = 8$ , height $= 6$ (vertical distance between $y = 0$ and $y = 6$ ): area $= 48$ m $^{2}$ . (1)
(iv) $A B = 8$ but $A D = 3^{2} + 6^{2} = 35 \approx 6.7$ , so not a rhombus; the angle at $A$ is not $9 0^{\circ}$ since $A D$ is slanted, so not a rectangle. (1)

Answer: (i) proved; (ii) proved; (iii) 48 m $^{2}$ ; (iv) neither a rectangle nor a rhombus.

Where students lose marks in Properties of a Parallelogram and Tests for Special Quadrilaterals

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

Mistake: Equating adjacent angles
Using the rule for opposite angles.
Wrong working
$∠ A = ∠ B$ in a parallelogram.
Right working
$∠ A = ∠ C$ and $∠ A + ∠ B = 18 0^{\circ}$ .
Do this: Opposite equal, adjacent supplementary.
Usually costs: The whole answer.
Mistake: Equal diagonals in a rhombus
Confusing rhombus with rectangle.
Wrong working
The diagonals of a rhombus are equal.
Right working
They are perpendicular bisectors but not generally equal.
Do this: Remember: rectangle equal, rhombus perpendicular.
Usually costs: 1 mark.
Mistake: Proving a parallelogram with only one pair of equal sides
Not adding parallel.
Wrong working
$A B = C D$ , so $A B C D$ is a parallelogram.
Right working
One pair of opposite sides must be both equal and parallel.
Do this: State both conditions.
Usually costs: 2 marks.

Exam tips

Mark equal sides and angles on a figure before writing the proof.
Name the congruence rule and the test you use.
Check calculations by testing opposite angles and adjacent sums.
State clearly which special quadrilateral you have proved and why.

Practice questions on Properties of a Parallelogram and Tests for Special 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 · MCQBoard level
Two adjacent angles of a parallelogram are in the ratio $2 : 3$ . The larger angle is
(a) $7 2^{\circ}$
(b) $9 0^{\circ}$
(c) $10 8^{\circ}$
(d) $12 0^{\circ}$
Hint
Adjacent angles add to $18 0^{\circ}$ .
Answer
(c) $10 8^{\circ}$ . $5 k = 18 0^{\circ}$ gives $k = 3 6^{\circ}$ ; the angles are $7 2^{\circ}$ and $10 8^{\circ}$ .
Q2 · 3 marks · Short answerBoard level
In parallelogram $P QR S$ , $P Q = (2 x + 5)$ cm, $S R = (3 x - 4)$ cm, $P S = (y + 3)$ cm and $QR = (2 y - 1)$ cm. Find $x$ , $y$ and the perimeter.
Hint
Opposite sides are equal.
Answer
$2 x + 5 = 3 x - 4$ gives $x = 9$ , so $P Q = S R = 23$ cm. $y + 3 = 2 y - 1$ gives $y = 4$ , so $P S = QR = 7$ cm. Perimeter $= 2 (23 + 7) = 60$ cm.
Q3 · 3 marks · Short answerHarder board level
In rectangle $A B C D$ the diagonals intersect at $O$ and $∠ A O B = 7 0^{\circ}$ . Find $∠ O A B$ , $∠ O B C$ and $∠ A D B$ .
Hint
The diagonals of a rectangle are equal and bisect each other.
Answer
$O A = O B$ , so $∠ O A B = ∠ O B A = 5 5^{\circ}$ . $∠ O B C = ∠ A B C - ∠ O B A = 9 0^{\circ} - 5 5^{\circ} = 3 5^{\circ}$ . Since $A D ∥ B C$ , $∠ A D B = ∠ D B C = ∠ O B C = 3 5^{\circ}$ (alternate angles). Also $∠ A D B = 9 0^{\circ} - ∠ A B D = 9 0^{\circ} - 5 5^{\circ} = 3 5^{\circ}$ .
Q4 · 5 marks · Long answerHarder board level
Prove that the diagonals of a rhombus bisect each other at right angles. If the diagonals of a rhombus are 16 cm and 12 cm, find its side, perimeter and area.
Hint
A rhombus is a parallelogram, so the diagonals bisect each other; use SSS for the right angle.
Answer
A rhombus is a parallelogram, so the diagonals bisect each other at $O$ . In $△ A O B$ and $△ A O D$ : $A B = A D$ , $O B = O D$ , $A O$ common, so they are congruent (SSS), and $∠ A O B = ∠ A O D$ . These form a linear pair, so each is $9 0^{\circ}$ . Numbers: half-diagonals $8$ and $6$ , side $= 64 + 36 = 10$ cm, perimeter $= 40$ cm, area $= \frac{1}{2} \times 16 \times 12 = 96$ cm $^{2}$ .
Q5 · 4 marks · Case-basedHOTS / competency
Case: a square tile $A B C D$ has side 10 cm and its diagonals meet at $O$ . (i) Find the length of a diagonal in surd form. (ii) Find $∠ O A B$ . (iii) Find the area of $△ A O B$ and compare it with the area of the tile. (iv) Find the perimeter of $△ A O B$ (use $2 = 1.414$ ).
Hint
The diagonals of a square are equal, bisect each other at right angles and bisect the angles.
Answer
(i) Diagonal $= 1 0^{2} + 1 0^{2} = 102$ cm. (ii) $4 5^{\circ}$ . (iii) $O A = O B = 52$ and $∠ A O B = 9 0^{\circ}$ , so the area is $\frac{1}{2} \times 52 \times 52 = 25$ cm $^{2}$ , one quarter of the tile ( $100$ cm $^{2}$ ). (iv) $10 + 52 + 52 = 10 + 102 \approx 24.14$ cm.

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.Diagonals of a parallelogram?Show answer
They bisect each other.
2.Adjacent angles in a parallelogram?Show answer
Supplementary.
3.Which parallelogram has equal diagonals?Show answer
A rectangle.
4.Diagonals of a rhombus?Show answer
Perpendicular bisectors of each other.
5.Test using one pair of sides?Show answer
One pair of opposite sides both equal and parallel.

Properties of a Parallelogram and Tests for Special Quadrilaterals: frequently asked questions

Is a square a rhombus and a rectangle?

Yes. A square has all sides equal and all angles right angles, so it is both.

Is every rectangle a parallelogram?

Yes. Its opposite sides are parallel, so it is a parallelogram with a right angle.

Can a parallelogram have equal diagonals without being a rectangle?

No. A parallelogram with equal diagonals must be a rectangle.