Quadrilaterals
Why This Matters
Look around you right now. A door. A book. A window. A mobile phone screen. A cricket field. A kite in the sky.
Almost all of them are four-sided shapes. We call a four-sided shape a quadrilateral.
These shapes are everywhere because they are useful. A rectangle packs neatly into a wall with no wasted space. A square tile fits perfectly next to another square tile. A carpenter making a window frame, a tailor cutting cloth, a mason laying floor tiles — all of them are working with quadrilaterals every single day.
But here is the interesting part. Not all four-sided shapes are the same. Some have equal sides. Some have right-angle corners. Some have one pair of parallel sides, some have two. Each kind has its own name and its own special powers.
In this chapter you will learn what makes a shape a quadrilateral, you will prove a surprising fact about its angles, and you will meet the six famous types and learn exactly how to tell them apart. By the end, you will never look at a door or a kite the same way again.
The Big Idea
A quadrilateral is just four straight sides joined up into a closed shape — that is all. But the moment you start adding rules, like “make the sides equal” or “make the corners square”, you get special, more powerful shapes. The whole chapter is one big family tree. The plain quadrilateral is the grandparent. As you add rules one by one, you travel down to parallelograms, then to rectangles and rhombuses, and finally to the square, which obeys all the rules at once. Knowing which rules each shape follows is knowing everything about it.
Let’s Break It Down
We will go in order. First, what counts as a quadrilateral. Then the one rule that every quadrilateral obeys — its angles add to 360°, and we will prove it. Then the six special types, one by one, with their properties. Then how they all relate. Finally, the secret lives of their diagonals.
Before we start, let us quickly remember two facts from earlier classes that we will lean on again and again.
What exactly is a quadrilateral?
The word quadrilateral comes from two Latin words: quadri meaning four, and latus meaning sides. So it literally means “four sides”.
But not every shape with four lines counts. A quadrilateral must follow three small rules:
- It has exactly four straight sides.
- It is closed — the sides join up with no gaps left open.
- The sides do not cross each other.
A quadrilateral has four vertices (the corners where two sides meet) and four angles (the angles between the sides at each corner).
The picture below shows a proper quadrilateral on the left, and two shapes that fail the rules on the right.
We usually name a quadrilateral by its four corners, going around in order — for example ABCD. The angle at corner A is written ∠A, and so on.
The angle-sum rule: all four angles add to 360°
Here is a question. Can you draw a quadrilateral with three right-angle corners (90° each) and a fourth corner that is not 90°?
Try it. You will find it is impossible. The fourth corner is forced to be 90° as well. Why? Because of a rule that every single quadrilateral obeys.
The four angles of any quadrilateral always add up to 360°.
∠A + ∠B + ∠C + ∠D = 360°
This is not magic. We can prove it, using the triangle fact from the recap. The trick is to cut the quadrilateral into two triangles by drawing one diagonal (a line joining two opposite corners).
The figure below shows the proof on a quadrilateral named SOME.
Let us walk through the proof carefully.
Prove that the four angles of quadrilateral SOME add up to 360°.
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Draw the diagonal SM. This joins corner S to the opposite corner M. It cuts the quadrilateral into two triangles: triangle SEM and triangle SOM. Look at Figure 4.2.
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Use the triangle fact on each triangle. In triangle SEM, the three angles add to 180°. In triangle SOM, the three angles also add to 180°.
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Add the two totals together. All six small angles together add to 180° + 180° = 360°.
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Now notice: these six small angles regroup into exactly the four corner angles of SOME. The angle at S is split into two parts by the diagonal — add them back and you get ∠S. The angle at M is split too — add them back to get ∠M. The angles at E and O are not split at all. So the six small angles ARE the four angles of the quadrilateral. Therefore ∠S + ∠O + ∠M + ∠E = 360°. Proved.
This is why three right angles force the fourth to be 90°: three of them already use up 270°, leaving exactly 360° − 270° = 90° for the last corner.
Let us use the rule to find a missing angle.
Three angles of a quadrilateral are 80°, 95° and 110°. Find the fourth angle.
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All four angles add to 360°. So the fourth angle = 360° minus the other three.
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Add the three known angles: 80° + 95° + 110° = 285°.
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Subtract from 360°: 360° − 285° = 75°. So the fourth angle is 75°.
Can a quadrilateral have all four angles equal to 100°?
No. If all four were 100°, they would add up to 4 × 100° = 400°. But the angles of a quadrilateral must add to exactly 360°. Since 400° is too much, this is impossible. (If all four angles are equal, each must be 360° ÷ 4 = 90° — which gives a rectangle.)
The six special types
Now meet the famous family. Each type is just a quadrilateral with extra rules added. We will go from the loosest rules to the strictest.
The gallery below shows all six, drawn to scale, with the one rule that defines each. Study it before reading the details.
Here is each type with its definition.
- Parallelogram — a quadrilateral in which both pairs of opposite sides are parallel.
- Rectangle — a quadrilateral in which all four angles are 90°.
- Rhombus — a quadrilateral in which all four sides are equal in length.
- Square — a quadrilateral in which all four sides are equal AND all four angles are 90°.
- Trapezium — a quadrilateral with at least one pair of parallel sides.
- Kite — a quadrilateral with two pairs of equal sides that are next to each other (adjacent), labelled so that AB = BC and CD = DA.
Let us now look at the properties that flow out of these definitions.
Properties of a parallelogram
A parallelogram is the most important type, because rectangles, rhombuses and squares are all special parallelograms. Its defining rule is “both pairs of opposite sides parallel”. From that one rule, three more facts follow:
- Opposite sides are equal. (AB = DC and AD = BC.)
- Opposite angles are equal. (∠A = ∠C and ∠B = ∠D.)
- Adjacent angles add up to 180°. (∠A + ∠B = 180°, and so on.)
Why do the angle facts hold? Because the sides are parallel, and we can use the co-interior angle fact from the recap. The figure below shows it.
Let us use these to find all the angles of a parallelogram.
In parallelogram ABCD, ∠A = 70°. Find ∠B, ∠C and ∠D.
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Opposite angles are equal, so ∠C = ∠A = 70°.
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Adjacent angles add to 180°. So ∠B = 180° − ∠A = 180° − 70° = 110°.
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∠D is opposite ∠B, so ∠D = ∠B = 110°.
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So the four angles are ∠A = 70°, ∠B = 110°, ∠C = 70°, ∠D = 110°. Check: 70 + 110 + 70 + 110 = 360°. Correct.
Properties of a rectangle
A rectangle is defined by having all four angles 90°. Because its opposite angles are equal (90° = 90°) and adjacent angles add to 180° (90° + 90°), a rectangle is automatically a parallelogram. So it gets all the parallelogram properties, plus its own:
- All four angles are 90°.
- Opposite sides are equal and parallel (from being a parallelogram).
- The diagonals are equal in length and they bisect each other.
There is a neat way to see why opposite sides must be equal once all angles are 90°. Drop a diagonal and you get two triangles that are congruent (a perfect match), which forces the opposite sides to be the same length. This is the “Carpenter’s Problem” from your textbook: to build a true rectangle, two equal sticks joined at their midpoints, with a thread around the ends, will always form a rectangle.
Properties of a rhombus
A rhombus is defined by having all four sides equal. Because of this, its opposite sides also turn out to be parallel, so a rhombus is also a parallelogram. It gets the parallelogram properties, plus:
- All four sides are equal.
- Opposite sides are parallel; opposite angles are equal.
- The diagonals bisect each other at right angles (90°).
- The diagonals bisect the angles of the rhombus.
A rhombus looks like a “pushed-over square” — a diamond shape. Its corners are usually not right angles, but all its sides are the same length.
Properties of a square
A square is the all-rounder. It has all four sides equal AND all four angles 90°. So it is a rectangle (all angles 90°) and a rhombus (all sides equal) at the same time. It inherits every property of both:
- All sides equal, all angles 90°.
- Opposite sides parallel.
- Diagonals are equal, bisect each other, and meet at 90°.
- Diagonals bisect the angles (each 90° corner is split into two 45° halves).
Trapezium and kite
These two do not have to be parallelograms.
A trapezium has at least one pair of parallel sides. The two parallel sides are usually different lengths, so the other pair is not parallel. Because one pair of sides is parallel, the two angles along each non-parallel side are co-interior angles and add to 180°. When the two non-parallel sides are equal in length, it is called an isosceles trapezium, and then the two angles at the ends of a parallel side are equal.
A kite has two pairs of equal adjacent sides (AB = BC and CD = DA). Think of the kite you fly. Its special property: one diagonal bisects the other at right angles, and that same diagonal also bisects the two angles it passes through.
Let us find missing angles in a trapezium.
In trapezium PQRS, side PQ is parallel to side SR. ∠P = 105°. Find ∠S.
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PQ is parallel to SR. Side PS joins them, acting as a transversal. So ∠P and ∠S are co-interior angles.
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Co-interior angles add to 180°. So ∠P + ∠S = 180°.
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Therefore ∠S = 180° − ∠P = 180° − 105° = 75°.
How the types relate to each other
This is where many students get confused. Is a square a rectangle? Is every parallelogram a trapezium? The answer is a clear “yes” once you see the family tree.
The big idea: adding a rule makes a shape more special, and a more special shape still keeps all the looser rules. A square obeys the rectangle rule (all angles 90°), so every square is a rectangle. But a rectangle need not have equal sides, so not every rectangle is a square.
The figure below shows how the families nest inside one another.
Read it as a chain of “is a”:
- Every square is a rectangle, a rhombus, and a parallelogram.
- Every rectangle is a parallelogram.
- Every rhombus is a parallelogram.
- Every parallelogram is a trapezium (it has parallel sides — in fact two pairs).
But the arrows do not reverse. A parallelogram need not be a rectangle. A rectangle need not be a square.
Is every rectangle a square? Is every square a rectangle?
Every square is a rectangle — yes, because a square has all angles 90°, which is exactly the rectangle rule. But not every rectangle is a square — a rectangle can be long and thin (like a door), with unequal sides, so it fails the “all sides equal” rule. The square is the special case of a rectangle that also happens to have equal sides.
The secret lives of the diagonals
A diagonal joins two opposite corners. Each quadrilateral has two diagonals, and how they behave is a great way to tell the types apart. Let us compare four of them.
The figure below lines up the diagonals of a parallelogram, a rectangle, a rhombus and a square.
Here is the same information as a table you can return to.
| Shape | Diagonals equal? | Bisect each other? | Meet at 90°? |
|---|---|---|---|
| Parallelogram | No | Yes | No |
| Rectangle | Yes | Yes | No |
| Rhombus | No | Yes | Yes |
| Square | Yes | Yes | Yes |
| Kite | No | One bisects the other | Yes |
This table is a handy detective tool. If someone tells you a quadrilateral’s diagonals are equal and bisect each other but do not meet at 90°, you know it must be a rectangle. If they bisect at 90° but are unequal, it is a rhombus.
The diagonals of a quadrilateral are equal in length, bisect each other, and cross at exactly 90°. What is the quadrilateral?
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Equal diagonals that bisect each other: this is the rectangle property. So the shape is at least a rectangle.
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The diagonals also meet at 90°: this is the rhombus property. So the shape is also a rhombus.
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A shape that is both a rectangle and a rhombus has all angles 90° and all sides equal. That is a square.
Common Mistakes
These are the slip-ups students make most with quadrilaterals. Spotting them now will protect your marks.
A square is a totally different shape from a rectangle, so a square is not a rectangle.
In everyday speech we use 'rectangle' for long shapes and 'square' for the even ones, as if they were two separate things, so they feel like rivals rather than relatives.
A rectangle is any quadrilateral with all four angles 90°. A square has all angles 90° too, so it fits the rectangle rule perfectly. Every square IS a rectangle — just a special one with equal sides. The reverse fails: a long rectangle is not a square.
The diagonals of a rectangle cross at right angles, just like a square's do.
A rectangle looks so much like a square, and a square's diagonals do meet at 90°, so the eye assumes the rectangle behaves the same way.
A rectangle's diagonals are equal and bisect each other, but they do NOT meet at 90° (unless it is a square). It is the rhombus and square whose diagonals cross at right angles. Equal length is the rectangle's diagonal property; the right angle is the rhombus's.
A trapezium and a parallelogram are two completely separate types with nothing in common.
They are usually drawn differently — a trapezium with a slanted side, a parallelogram leaning evenly — so they look unrelated, and a trapezium is often presented as 'the one with only one parallel pair'.
A trapezium needs at least ONE pair of parallel sides. A parallelogram has TWO such pairs, so it also satisfies the trapezium rule. Every parallelogram is a trapezium. A plain trapezium just happens to stop at one parallel pair.
Quick Check
Try these quick questions. The explanation appears after you answer, so read it either way.
The three angles of a quadrilateral are 90°, 90° and 100°. What is the fourth angle?
All four angles add to 360°. The three given add to 90 + 90 + 100 = 280°. So the fourth is 360 − 280 = 80°.
Which quadrilateral has all sides equal but its angles are usually NOT 90°?
A rhombus has all four sides equal, but its corners are usually slanted, not right angles. (A square also has equal sides, but its angles ARE 90°. A rectangle has 90° angles but unequal sides.)
A quadrilateral's diagonals bisect each other at exactly 90°, but the two diagonals are NOT equal in length. What is it?
Bisecting at 90° is the rhombus (and square) property. But a square’s diagonals are equal, and here they are NOT equal. So it must be a rhombus.
Which statement is TRUE?
Every square is a parallelogram — a square has both pairs of opposite sides parallel, so it fits the parallelogram rule. The others reverse the family tree wrongly: a rectangle need not be a square, a trapezium need not have two parallel pairs, and a kite need not have all sides equal.
Practice Problems
Try each one yourself first. Then tap to check your full solution.
Easy
Three angles of a quadrilateral are 60°, 130° and 85°. Find the fourth angle.
All four angles add to 360°.
Add the three known angles: 60° + 130° + 85° = 275°.
Fourth angle = 360° − 275° = 85°.
In a parallelogram, one angle is 65°. Find the other three angles.
Opposite angles are equal, so the angle opposite the 65° one is also 65°.
Adjacent angles add to 180°, so the next angle is 180° − 65° = 115°.
Its opposite is also 115°.
So the four angles are 65°, 115°, 65°, 115°. (Check: they add to 360°.)
Name the quadrilateral whose diagonals are equal and bisect each other, but do NOT cross at 90°.
Equal diagonals that bisect each other is the property of a rectangle.
The diagonals do not cross at 90°, which rules out the square (whose diagonals do meet at 90°).
So the shape is a rectangle.
Medium
In trapezium ABCD, AB is parallel to DC. ∠A = 115° and ∠B = 100°. Find ∠C and ∠D.
AB is parallel to DC. Side AD is a transversal, so ∠A and ∠D are co-interior angles.
∠A + ∠D = 180°, so ∠D = 180° − 115° = 65°.
Side BC is also a transversal, so ∠B and ∠C are co-interior angles.
∠B + ∠C = 180°, so ∠C = 180° − 100° = 80°.
Check all four: 115 + 100 + 80 + 65 = 360°. Correct.
Is a quadrilateral with four equal sides and one angle of 90° always a square? Justify.
Four equal sides means the shape is a rhombus.
In a rhombus, opposite angles are equal and adjacent angles add to 180°.
If one angle is 90°, then its adjacent angle is 180° − 90° = 90°, and its opposite angle is also 90°. So all four angles become 90°.
A shape with four equal sides and all angles 90° is a square.
So yes, it is always a square.
A quadrilateral has its opposite sides equal in length. What type must it be? Justify using a diagonal.
It must be a parallelogram. Here is why.
Call the quadrilateral ABCD, with AB = DC and AD = BC. Draw the diagonal AC.
This makes two triangles, ABC and CDA. In them: AB = CD, BC = DA, and AC = AC (the shared side). So by the SSS condition, the two triangles are congruent.
From the congruence, the matching angles are equal. This makes AB parallel to DC, and AD parallel to BC (the equal angles are alternate angles across the diagonal).
Both pairs of opposite sides are parallel, so ABCD is a parallelogram.
Challenge
In a quadrilateral, the four angles are in the ratio 1 : 2 : 3 : 4. Find each angle. What do you notice about the largest angle?
The four angles share the ratio 1 : 2 : 3 : 4. Let one ‘part’ be x. Then the angles are x, 2x, 3x and 4x.
They must add to 360°:
x + 2x + 3x + 4x = 360°
10x = 360°, so x = 36°.
So the angles are: 1 × 36° = 36°, 2 × 36° = 72°, 3 × 36° = 108°, 4 × 36° = 144°.
Check: 36 + 72 + 108 + 144 = 360°. Correct.
The largest angle is 144°, which is more than 90° but less than 180°. So this quadrilateral has no right angles and is not any of the special named types — it is just a general quadrilateral. The angle-sum rule still works perfectly.
CASE is a square. Points U, V, W, X are the midpoints of its four sides, joined in order. Show that UVWX is also a square.
Let the side of square CASE be of length 2 units, so each half-side is 1 unit. Each midpoint sits exactly in the middle of a side.
Join U, V, W, X in order. Each side of UVWX is the longest side of a small right-angled triangle cut off at a corner of CASE. For example, at corner C, the two short sides are both 1 unit (half-sides), and they meet at 90° (a corner of the square).
By the Pythagoras relation, each such longest side is √(1² + 1²) = √2 units. Every corner of CASE is identical, so all four sides of UVWX are √2 — they are equal.
Now the angles. At each corner of CASE, the cut-off triangle is right-angled with two equal short sides, so its two base angles are each 45°. At every vertex of the inner shape (say U), two of these 45° angles sit on either side of the inner angle, along a straight side of CASE. A straight angle is 180°, so the inner angle = 180° − 45° − 45° = 90°.
So UVWX has all four sides equal AND all four angles 90°. That is a square.
Summary
- A quadrilateral is a closed figure with four straight sides, four vertices and four angles, where no side crosses another.
- The four angles of any quadrilateral add up to 360°. Proof: one diagonal splits it into two triangles (180° each), and 180° + 180° = 360°.
- A parallelogram has both pairs of opposite sides parallel. So opposite sides are equal, opposite angles are equal, and adjacent angles add to 180°.
- A rectangle has all angles 90°; a rhombus has all sides equal; a square has both at once. All three are special parallelograms.
- A trapezium has at least one pair of parallel sides; a kite has two pairs of equal adjacent sides.
- The family tree: every square is a rectangle, a rhombus and a parallelogram; every rectangle and rhombus is a parallelogram; every parallelogram is a trapezium. The arrows do not reverse.
- Diagonals: a parallelogram’s bisect each other; a rectangle’s are also equal; a rhombus’s also meet at 90°; a square’s do all three; a kite’s one diagonal bisects the other at 90°.
What’s Next
You can now recognise any four-sided shape, find its angles, and tell exactly which special type it is. That is a real superpower for geometry and for exams.
Next, in Chapter 5 — Number Play, you will step away from shapes and explore the playful patterns hidden inside numbers — clever tricks, surprising results, and puzzles that make numbers come alive. The careful, step-by-step reasoning you practised here will help you spot those patterns. Onward!
Frequently Asked Questions
What is a quadrilateral?
A quadrilateral is a closed figure made of four straight sides, with four corners (vertices) and four angles. The sides must not cross each other and there must be no gaps. The word comes from Latin: quadri means four and latus means sides.
Why do the four angles of a quadrilateral always add up to 360 degrees?
Draw one diagonal. It splits the quadrilateral into two triangles. The angles of each triangle add to 180 degrees. The six small angles regroup exactly into the four corner angles of the quadrilateral, so the total is 180 plus 180, which is 360 degrees.
What is the difference between a square, a rectangle and a rhombus?
A rectangle has all four angles equal to 90 degrees. A rhombus has all four sides equal in length. A square has both at once: all sides equal AND all angles 90 degrees. So a square is a special rectangle and a special rhombus at the same time.
What is the difference between a parallelogram and a trapezium?
A parallelogram has BOTH pairs of opposite sides parallel. A trapezium has at least ONE pair of parallel sides. So every parallelogram is also a trapezium, but a plain trapezium has only one parallel pair, not two.
What do the diagonals of these quadrilaterals do?
In a parallelogram the diagonals bisect each other. In a rectangle they are also equal. In a rhombus they bisect each other at 90 degrees. In a square the diagonals are equal AND bisect each other at 90 degrees. In a kite one diagonal bisects the other at right angles.