CLASS 8 MATHS · PART 1 · CHAPTER 4 · STUDY NOTES
Quadrilaterals
Understand quadrilaterals through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.
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Four sides must close into one figure
Count the sides, check the endpoints and look for crossings.Read the boundary in order
y (units)
x (units)
- Boundary · Follow the vertices in order.
- Diagonal AC · A and C are not neighbours.
- AA · (0, 0)
- BB · (6, 0)
- CC · (5, 4)
- DD · (1, 5)
This is one irregular convex quadrilateral, drawn with equal unit scales. It need not have equal sides. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 6 | 0 |
| 5 | 4 |
| 1 | 5 |
| 0 | 0 |
| x | y |
|---|---|
| 0 | 0 |
| 5 | 4 |
- Quadrilateral
- A closed simple polygon with four straight sides and four vertices.
- Diagonal
- A segment joining two nonadjacent vertices.
A gap leaves an open figure, and a curved edge is not a polygon side. A crossed bow-tie is not the simple quadrilateral meant here.
Labels A, B, C, D must follow the boundary in order. Then AC and BD are the diagonals.
Take a moment. What do you think?
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Question
For quadrilateral PQRS named around its boundary, name the diagonals.
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One diagonal explains the 360° angle sum
A diagonal inside a convex quadrilateral makes two triangles. Their angle sums add.Split into two triangles
y (units)
x (units)
- Boundary · Follow the vertices in order.
- Triangle ABC · One group has a 180° sum.
- Triangle ACD · The other group has a 180° sum.
- AA · (0, 0)
- BB · (6, 0)
- CC · (5, 4)
- DD · (1, 5)
The internal diagonal AC separates two non-overlapping triangles. Angles at A and C each split into two pieces. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 6 | 0 |
| 5 | 4 |
| 1 | 5 |
| 0 | 0 |
| x | y |
|---|---|
| 0 | 0 |
| 6 | 0 |
| 5 | 4 |
| 0 | 0 |
| x | y |
|---|---|
| 0 | 0 |
| 5 | 4 |
| 1 | 5 |
| 0 | 0 |
- Interior angle
- The angle inside a polygon at a vertex.
Each triangle has angle sum 180°. Together they account for all four quadrilateral angles, so the total is 360°.
Three right angles leave 360° − 270° = 90° for the fourth. A concave simple quadrilateral also has total 360°, counting its reflex interior angle.
- 1
Draw diagonal AC inside the convex figure.
- 2
Use 180° for triangle ABC and 180° for triangle ACD.
- 3
Recombine the split angles at A and C.
The four interior angles total 360°.
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Question
Three angles are 75°, 110° and 95°. Find the fourth.
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A rectangle has four right angles
The right-angle condition brings other properties with it.Check the midpoint, not a right angle between diagonals
y (units)
x (units)
- Boundary · Follow the vertices in order.
- AC · Midpoint is O = (3,2).
- BD · Same length, same midpoint.
- AA · (0, 0)
- BB · (6, 0)
- CC · (6, 4)
- DD · (0, 4)
Both diagonals have length √52 coordinate units and the same midpoint. They are not perpendicular. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 6 | 0 |
| 6 | 4 |
| 0 | 4 |
| 0 | 0 |
| x | y |
|---|---|
| 0 | 0 |
| 6 | 4 |
| x | y |
|---|---|
| 6 | 0 |
| 0 | 4 |
- Rectangle
- A quadrilateral with all four angles equal to 90°.
- Bisect
- Divide into two equal parts.
Opposite sides are parallel and equal. Its diagonals have equal lengths and bisect each other.
The diagonals do not generally meet at 90°. Equal diagonals alone do not prove that a quadrilateral is a rectangle; they must also bisect each other.
Let’s look a little closer.
Go deeper: Why equal diagonals that bisect give a rectangle
If the diagonals bisect one another, the quadrilateral is a parallelogram. In a parallelogram, equal diagonals make two triangles with matching sides; their adjacent corner angles are equal. These adjacent angles also sum to 180°, so each is 90°. Measurements suggest a rule; this reasoning explains why it holds.
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Question
A rectangle has diagonal AC = 10 cm and the diagonals meet at O. Find BD, AO and BO.
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A square passes both the side test and the angle test
Four equal sides alone make a rhombus. Add four right angles to get a square.A tilted square is still a square
y (units)
x (units)
- Boundary · Follow the vertices in order.
- Horizontal diagonal · Length 6; midpoint (3,3).
- Vertical diagonal · Length 6; midpoint (3,3).
- AA · (0, 3)
- BB · (3, 0)
- CC · (6, 3)
- DD · (3, 6)
All four sides have length √18. Orientation on the page does not change the properties. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 3 |
| 3 | 0 |
| 6 | 3 |
| 3 | 6 |
| 0 | 3 |
| x | y |
|---|---|
| 0 | 3 |
| 6 | 3 |
| x | y |
|---|---|
| 3 | 0 |
| 3 | 6 |
- Square
- A quadrilateral with four equal sides and four right angles.
A square is a rectangle, a rhombus and a parallelogram. Its diagonals are equal, bisect one another at right angles, and bisect its corner angles.
Joining the endpoints of two equal perpendicular segments at their shared midpoint produces a square, provided the segments are nonzero.
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Question
A quadrilateral has equal diagonals that bisect one another. What extra diagonal condition makes it a square?
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Parallel opposite sides explain equal sides and angles
A parallelogram may lean. It does not need four right angles.Match triangles across the diagonal
y (units)
x (units)
- Boundary · Follow the vertices in order.
- AC · Shared diagonal joins A to C.
- AA · (0, 0)
- BB · (5, 0)
- CC · (7, 3)
- DD · (2, 3)
AB is parallel to CD, and AD is parallel to BC. The leaning shape is not a rectangle. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 5 | 0 |
| 7 | 3 |
| 2 | 3 |
| 0 | 0 |
| x | y |
|---|---|
| 0 | 0 |
| 7 | 3 |
- Parallelogram
- A quadrilateral with both pairs of opposite sides parallel.
- Supplementary angles
- Angles whose measures add to 180°.
Opposite sides are equal, opposite angles are equal, and adjacent angles add to 180°.
A diagonal divides a parallelogram into two congruent triangles. Use the parallel sides to find equal alternate angles, then use the shared diagonal.
- 1
Draw AC.
- 2
Parallel sides give ∠BAC = ∠DCA and ∠BCA = ∠DAC.
- 3
AC is common, so triangles ABC and CDA are congruent by ASA.
- 4
Matching sides give AB = CD and BC = DA.
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Question
One angle of a parallelogram is 65°. Find the other three.
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Bisecting is different from having equal diagonals
Every parallelogram has diagonals that share a midpoint; their lengths may differ.Same midpoint, different lengths
y (units)
x (units)
- Boundary · Follow the vertices in order.
- AC · Midpoint (3.5,1.5); length √58.
- BD · Midpoint (3.5,1.5); length √18.
- AA · (0, 0)
- BB · (5, 0)
- CC · (7, 3)
- DD · (2, 3)
The midpoint belongs to both diagonals. Their lengths are deliberately different. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 5 | 0 |
| 7 | 3 |
| 2 | 3 |
| 0 | 0 |
| x | y |
|---|---|
| 0 | 0 |
| 7 | 3 |
| x | y |
|---|---|
| 5 | 0 |
| 2 | 3 |
- Midpoint
- A point halfway along a segment.
Where AC and BD meet at O, AO = OC and BO = OD. This does not say AC = BD.
Conversely, a simple nondegenerate quadrilateral whose diagonals bisect each other is a parallelogram.
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Question
If AO = 4 cm and BO = 6 cm in a parallelogram, find AC and BD.
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A rhombus has four equal sides, but may not be square
Equal sides make a useful symmetry along the diagonals.Perpendicular diagonals of unequal lengths
y (units)
x (units)
- Boundary · Follow the vertices in order.
- AC · Horizontal length 8.
- BD · Vertical length 4.
- AA · (0, 0)
- BB · (4, -2)
- CC · (8, 0)
- DD · (4, 2)
Every side has length √20. Both diagonals have midpoint (4,0), but their lengths differ. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 4 | -2 |
| 8 | 0 |
| 4 | 2 |
| 0 | 0 |
| x | y |
|---|---|
| 0 | 0 |
| 8 | 0 |
| x | y |
|---|---|
| 4 | -2 |
| 4 | 2 |
- Rhombus
- A quadrilateral with all four sides equal.
A rhombus is a parallelogram. Its diagonals bisect each other at right angles and bisect the corner angles.
The diagonals need not have equal lengths. Equal diagonals would make the rhombus a square.
Go deeper: Why the diagonals meet at a right angle
The triangles on the two sides of one diagonal have matching equal sides. At the intersection, their corresponding angles are equal and form a straight angle. Each must therefore be 90°.
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Question
A rhombus has an angle of 80°. What are the other angles and the two halves of that 80° corner?
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A kite has two pairs of equal neighbouring sides
Find the symmetry diagonal and check which diagonal it bisects.Only one diagonal needs to bisect the other
y (units)
x (units)
- Boundary · Follow the vertices in order.
- AC · Symmetry diagonal, from (3,6) to (3,0).
- BD · Bisected at (3,2).
- AA · (3, 6)
- BB · (1, 2)
- CC · (3, 0)
- DD · (5, 2)
The upper equal sides have length √20 and the lower equal sides √8. BD does not pass through the midpoint (3,3) of AC. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 3 | 6 |
| 1 | 2 |
| 3 | 0 |
| 5 | 2 |
| 3 | 6 |
| x | y |
|---|---|
| 3 | 6 |
| 3 | 0 |
| x | y |
|---|---|
| 1 | 2 |
| 5 | 2 |
- Kite
- Here, a quadrilateral with two non-overlapping pairs of equal adjacent sides, including a rhombus as a special case.
For the labelled kite, AB = AD and CB = CD. Diagonal AC bisects angles A and C, and is the perpendicular bisector of BD.
BD need not bisect AC. Do not transfer every rhombus property to a general kite.
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Question
In a kite with AB = AD and CB = CD, which diagonal is the symmetry diagonal?
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A trapezium needs at least one parallel pair in this book
Be clear about the definition: this NCERT chapter includes parallelograms among trapeziums.Drop equal heights to compare the end triangles
y (units)
x (units)
- Boundary · Follow the vertices in order.
- Left height · Perpendicular from D to the lower base.
- Right height · Same separation of parallel bases.
- AA · (0, 0)
- BB · (8, 0)
- CC · (6, 3)
- DD · (2, 3)
This one-pair trapezium has equal legs √13. The two end right triangles have matching heights and hypotenuses. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 8 | 0 |
| 6 | 3 |
| 2 | 3 |
| 0 | 0 |
| x | y |
|---|---|
| 2 | 3 |
| 2 | 0 |
| x | y |
|---|---|
| 6 | 3 |
| 6 | 0 |
- Trapezium
- Here, a quadrilateral with at least one pair of parallel opposite sides.
- Isosceles trapezium
- For the one-parallel-pair case, a trapezium with equal nonparallel sides.
Angles at the ends of each nonparallel side add to 180°, because that side crosses the parallel bases.
In a one-pair isosceles trapezium, angles at the same base are equal. This is a more specific property than the general trapezium rule.
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Question
An isosceles trapezium has one base angle 70°. Find all four angles.
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Classify by properties; one shape can have several names
A more specific shape keeps the properties of the wider group it belongs to.The square satisfies two sets of conditions
- Subset
- A group all of whose members also belong to a larger group.
Every square is both a rectangle and a rhombus. Every rectangle and every rhombus is a parallelogram. Under this book’s definitions, every rhombus is also a kite and every parallelogram is a trapezium.
One counterexample is enough to disprove “all”. The nonsquare rectangle earlier disproves “every rectangle is a square”.
Go deeper: Construct, test, then justify
On a dot grid, two equal perpendicular diagonals that bisect one another give a square. Lengthening just one while keeping the midpoint gives a nonsquare rhombus. The construction changes a property; the classification changes for a reason. Measuring one drawing does not prove a rule for every drawing.
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Question
A shape is both a rectangle and a rhombus. What must it be?
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