Quadrilaterals | CBSE Class 8 Maths Notes

This note covers quadrilaterals, rectangles, squares, angle sums, parallelograms, rhombuses, kites, trapeziums, diagonal properties, geometric reasoning, constructions and relationships between families of four-sided figures.

What is a quadrilateral, and how do we describe one?

Definition: A quadrilateral is a closed figure in a plane, or flat surface, bounded by four straight sides. Its vertices are the corners where neighbouring sides meet, and its interior angles are the angles inside the figure between those sides.

A line segment is the straight part joining two endpoints. In a quadrilateral named ABCD, the capital letters A, B, C and D label consecutive vertices around its boundary. Thus AB, BC, CD and DA name its sides.

Adjacent sides share a vertex, while opposite sides do not. AB and BC are adjacent; AB and CD are opposite. Adjacent angles occur at the endpoints of one side. Opposite angles occur at vertices that are not neighbours.

A diagonal joins opposite vertices. AC and BD are the diagonals of ABCD. A midpoint divides a segment into two equal lengths. To bisect means to divide into two equal parts, so diagonals bisect each other when their intersection is the midpoint of both.

How should geometric notation be read?

The symbol ∠ means angle. In ∠ABC, the middle letter B identifies its vertex; ∠B is a shorter form when the angle is clear. The symbol ° means degrees, the unit used for angle measure. A right angle measures 90°, and a straight angle measures 180°.

Parallel lines lie in the same plane and do not meet when extended. The symbol ∥ means “is parallel to”. Perpendicular lines meet at 90°. The unit cm means centimetres. AB = CD means that the two segments have equal lengths; = means “equals”.

Congruent triangles have the same size and shape, with matching sides and angles equal. The symbol △ means triangle and ≅ means “is congruent to”. Corresponding parts are the parts that match. Vertex order in a congruence statement must preserve these matches.

The congruence conditions used below are SAS, two sides and their included angle; SSS, three sides; ASA, two angles and their included side; and AAS, two angles and a corresponding non-included side. “Included” means lying between the two specified parts.

What makes a quadrilateral a rectangle?

Definition: A rectangle is a quadrilateral with four right angles. Its opposite sides are equal and parallel, and its diagonals are equal and bisect each other.

A definition gives conditions that identify a shape. A property is a fact established about that shape. Once all four angles are known to be right angles, opposite-side equality follows by reasoning; it need not be added as a separate requirement.

Property: The diagonals of a rectangle are equal

Consider rectangle ABCD. Compare triangles ADC and DAB. AD is common to both, DC = AB because opposite sides are equal, and ∠ADC = ∠DAB = 90°. The two equal sides enclose equal angles.

Therefore △ADC ≅ △DAB by SAS. Their corresponding sides AC and DB are equal. This establishes diagonal equality from rectangle properties, rather than from how accurately a particular picture has been drawn.

Property: Rectangle diagonals bisect each other

Let O be the intersection of AC and BD. Comparing triangles AOB and COD gives congruent triangles by AAS. Their corresponding parts give OA = OC and OB = OD. Thus O is the midpoint of each diagonal.

Here the angles at O are vertically opposite angles, the opposite angles formed by intersecting lines, which are equal. The other equal angles can be obtained by subtracting the same angle from right angles. Together with AB = CD, these establish the required correspondence.

Two wooden strips join opposite corners of a rectangular outline. They cross at O; the corners are labelled A, B, C and D. The strips represent the diagonals, while the outline represents the thread joining their endpoints.

See Fig. 1 in your NCERT textbook

Worked example 1. A carpenter has one strip 8 cm long. What length and joining point should the second strip have so that thread around the four endpoints forms a rectangle?

Answer: Use another 8 cm strip and join their midpoints. Each midpoint is 4 cm from either end. The resulting diagonals are equal and bisect each other, which identifies a rectangle.

Why do equal diagonals that bisect each other form a rectangle?

The diagonal description works in both directions. Rectangles have equal diagonals that bisect each other, and a quadrilateral with equal, mutually bisecting diagonals is a rectangle. Both equality and bisection matter; neither statement should be dropped from this test.

Result: The crossing angle need not be a right angle

Take diagonals AC and BD meeting at O, with OA = OC, OB = OD and AC = BD. These conditions make all four half-diagonals equal. Consequently, each small triangle around O is isosceles, meaning it has two equal sides.

In an isosceles triangle, the angles opposite the equal sides are equal. These are its base angles. A triangle’s three interior angles total 180°. A ray starts at one endpoint and extends indefinitely in one direction. An angle’s arms are rays. A linear pair consists of adjacent angles whose other arms form a straight line, so its angles total 180°.

Let x be the numerical measure in degrees of ∠AOB. Let a and b be the numerical measures of the base angles in triangles AOB and AOD respectively. The other crossing angles have measures x, 180 − x and 180 − x degrees.

Triangle AOB gives a + a + x = 180, so a = 90 − x/2. Here / denotes division. Triangle AOD gives b + b + 180 − x = 180, so b = x/2.

At each vertex of the quadrilateral, the two small angles combine to give a + b = 90. Congruence of opposite small triangles also gives opposite-side equality. Hence the quadrilateral satisfies the rectangle definition for the crossing angles considered.

Worked example 2. Equal diagonals AC and BD bisect each other at O, with ∠AOB = 60°. Find the small triangle angles and the quadrilateral’s corner angles.

Answer: The crossing angles are 60°, 120°, 60° and 120°. The base angles in a triangle with vertex angle 60° are 60° each; those with vertex angle 120° are 30° each. Every quadrilateral corner is 60° + 30° = 90°.

This reasoning is a deduction, a conclusion justified from established facts. Measurements can suggest a conjecture, a statement we are highly confident about but have not yet proved. Checking several drawings does not establish that a property holds for every rectangle.

How is a square a special rectangle?

Definition: A square is a quadrilateral whose four sides are equal and whose four angles are right angles. It therefore satisfies the rectangle definition as well.

Every square is a rectangle, but every rectangle is not a square. The additional condition is equality of all four sides. A square inherits the rectangle’s equal opposite sides, parallel opposite sides, equal diagonals and diagonals that bisect each other.

Property: Square diagonals meet at right angles

Let O be the intersection of diagonals AC and BD of square ABCD. In triangles BOA and BOC, BO is common, OA = OC, and BA = BC. Therefore △BOA ≅ △BOC by SSS.

The corresponding angles BOA and BOC are equal. Since A, O and C lie on one straight line, these angles total 180°. Each therefore measures 90°. Thus square diagonals are equal and bisect each other at right angles.

Property: Square diagonals bisect the corner angles

Triangle ADC has AD = DC and ∠ADC = 90°. Its other two angles are equal and total 90°, so each is 45°. Applying the same reasoning to the remaining parts shows that the diagonals bisect the square’s corner angles.

Worked example 3. Construct a square with a diagonal of length 8 cm using its diagonal properties.

Answer: Draw diagonal AC = 8 cm and mark its midpoint O, 4 cm from each end. Through O draw a perpendicular line. Mark B and D on opposite sides of O with OB = OD = 4 cm. Join A, B, C and D in order.

The construction produces equal diagonals that bisect each other at 90°. It combines the rectangle conditions with the additional crossing-angle condition needed for a square. Drawing two perpendicular segments without checking their lengths and midpoints would not establish these conditions.

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

Result: The quadrilateral angle sum is 360°

Consider quadrilateral SOME, whose consecutive vertices are S, O, M and E. Draw diagonal SM inside it. This divides the quadrilateral into triangles SEM and SOM. Each triangle has an interior-angle total of 180°.

At S and M, the diagonal splits each quadrilateral angle into two smaller angles. At E and O, the triangle angles are already whole quadrilateral angles. Adding the six triangle angles therefore counts each part of the quadrilateral’s four angles exactly once.

1. Draw the diagonal SM to identify triangles SEM and SOM.
2. Write that the three angles of triangle SEM add up to 180°.
3. Write that the three angles of triangle SOM also add up to 180°.
4. Add the equations and recombine the split angles at S and M.

The result is ∠S + ∠O + ∠M + ∠E = 360°. The shape does not have to be a rectangle or have equal sides. The angle sum is a general quadrilateral property, rather than a special feature of one family.

Quadrilateral SOME contains diagonal SM. The angles at S are labelled 1 and 4, those at M are 3 and 6, and the angles at E and O are 2 and 5.

Reference: NCERT Class 8, page 95, unnumbered figure

Worked example 4. Three angles of a quadrilateral are right angles. Can its fourth angle have a different measure?

Answer: The three known angles total 90° + 90° + 90° = 270°. The fourth is 360° − 270° = 90°. Hence it cannot have a different measure, and the quadrilateral is a rectangle.

The same total identifies a quadrilateral with four equal angles: each measures 360°/4 = 90°. However, equal angles do not establish equal sides. Such a quadrilateral is a rectangle; further side information is needed to identify a square.

What are the side and angle properties of a parallelogram?

Definition: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. A rectangle is a special parallelogram with four right angles.

To construct a parallelogram ABCD, begin with adjacent sides AB and AD. Draw through D a line parallel to AB, and through B a line parallel to AD. Name their intersection C. This supplies the two pairs of parallel opposite sides.

Property: Adjacent angles are supplementary

Supplementary angles are angles whose measures total 180°. A transversal is a line that crosses two lines at distinct points. When it crosses parallel lines, the interior angles on the same side of it are supplementary.

In parallelogram ABCD, AD crosses the parallel lines AB and CD. Thus ∠A + ∠D = 180°. Similarly, ∠A + ∠B = 180°, ∠B + ∠C = 180° and ∠C + ∠D = 180°.

Both ∠B and ∠D equal 180° − ∠A, so they are equal. Also, ∠C = 180° − ∠B = ∠A. Hence opposite angles are equal. These conclusions follow from parallel sides, without assuming right angles.

Property: Opposite sides are equal

Draw diagonal BD. Triangles ABD and CDB have equal opposite angles of the parallelogram, a common side BD, and equal alternate angles. Alternate interior angles lie between two lines on opposite sides of a transversal; they are equal when those lines are parallel.

AAS congruence gives △ABD ≅ △CDB. Corresponding sides then give AB = CD and AD = CB. Pay attention to the vertex order: it records which triangle parts have been matched.

Worked example 5. In parallelogram ABCD, AB = 4 cm, AD = 5 cm and ∠A = 30°. Find the other sides and angles.

Answer: CD = 4 cm and BC = 5 cm because opposite sides are equal. The adjacent angles B and D each equal 180° − 30° = 150°. The opposite angle C equals 30°.

This example also shows why parallel opposite sides cannot define a rectangle. The figure satisfies the parallelogram definition while its angles are 30° and 150°, rather than right angles.

How do the diagonals of a parallelogram behave?

A parallelogram’s diagonals bisect each other. They need not have equal lengths. This distinction separates the general parallelogram diagonal property from the stronger combination of properties used to identify rectangles.

Property: Both diagonals have the same midpoint

Consider parallelogram EASY, whose consecutive vertices are E, A, S and Y. Let O be the intersection of its diagonals ES and AY. Compare triangles AOE and YOS, which lie on opposite sides of O.

AE = YS because opposite sides of the parallelogram are equal. Since AE ∥ YS, the diagonals provide two pairs of equal alternate angles. ASA gives △AOE ≅ △YOS.

The corresponding sides give OA = OY and OE = OS. O therefore divides both diagonals into equal halves. These equations compare the two halves of each diagonal; they do not say that one entire diagonal equals the other.

Worked example 6. Construct a parallelogram with diagonals 7 cm and 5 cm that intersect at an angle of 140°.

Answer: Draw a 7 cm segment and mark its midpoint. Through that midpoint draw a line making 140° with one ray of the segment. On opposite rays of the new line, mark points 2.5 cm from the midpoint. Join the four endpoints successively around the boundary.

The second diagonal has total length 5 cm. The first midpoint is 3.5 cm from each end of the 7 cm diagonal. Thus both diagonals are bisected, while their unequal lengths remain exactly as required.

The reverse test also holds: a quadrilateral whose diagonals bisect each other is a parallelogram. For a construction check, verify the midpoint of each diagonal separately. Merely making the diagonals cross does not establish bisection.

Equal diagonals would add the rectangle condition. Perpendicular diagonals would add the condition used for a rhombus. These additional tests must be combined with mutual bisection; crossing angle alone does not identify the family.

What makes a rhombus different from a general parallelogram?

Definition: A rhombus is a quadrilateral with four equal sides. Every rhombus is a parallelogram, but its angles need not be right angles.

Begin with two equal adjacent sides that are not perpendicular. A compass can locate the remaining vertex at the same distance from the two free endpoints. This produces four equal sides without requiring the corner angles to be 90°.

Property: Rhombus diagonals bisect its angles

In rhombus GAME, whose consecutive vertices are G, A, M and E, draw diagonal AE. Triangles GAE and MAE have three corresponding equal sides and are congruent by SSS. Each also has equal base angles because it is isosceles.

Together these facts show that AE bisects the angles at A and E. The other diagonal similarly bisects the angles at G and M. Equal alternate angles also establish parallel opposite sides, so the rhombus has the parallelogram properties.

Worked example 7. Rhombus ABCD has ∠A = 50°. Find its remaining angles and the two parts of ∠B made by diagonal BD.

Answer: ∠C = 50°. Angles B and D each equal 180° − 50° = 130°. Since BD bisects ∠B, each part at B measures 130°/2 = 65°. It also divides ∠D into two 65° angles.

Property: Rhombus diagonals are perpendicular

Let O be the intersection of the diagonals of GAME. The triangles GEO and MEO are congruent: GE = ME, GO = MO, and EO is common. Their equal angles GOE and MOE form a linear pair, so each is 90°.

Thus a rhombus has diagonals that bisect each other at right angles and bisect its corner angles. The diagonals need not be equal. A square satisfies the rhombus definition and has the additional rectangle properties.

Worked example 8. Construct a rhombus whose diagonals have lengths 4 cm and 5 cm.

Answer: Draw the 5 cm diagonal and mark its midpoint. Through that point draw a perpendicular line. Mark an endpoint 2 cm away on each side along this line. Join successive endpoints of the two diagonals to form the rhombus.

What is a kite, and which diagonal has special properties?

Definition: A kite is a quadrilateral with two non-overlapping pairs of equal adjacent sides. It can be labelled ABCD so that AB = BC and CD = DA.

The words non-overlapping pairs mean that each pair uses different sides: AB with BC, and CD with DA. The definition does not require the length in the first pair to differ from the length in the second pair. A rhombus therefore satisfies it.

How can two triangles form a kite?

Two copies of a scalene triangle, a triangle with three unequal side lengths, can be joined along a matching side. With the copies placed as reflections, or mirror images, across the shared side, the outer boundary has two equal adjacent pairs.

For triangles with sides 6 cm, 9 cm and 12 cm joined along the 12 cm side in this arrangement, the outer boundary contains two 6 cm sides and two 9 cm sides. The shared side becomes a diagonal inside the kite.

The labelled kite has D above B, A to the left and C to the right. Diagonals BD and AC meet at O. Side markings pair DA with DC and AB with BC.

Reference: NCERT Class 8, page 105, unnumbered figure

Property: One diagonal bisects the other at right angles

For kite ABCD with AB = BC and CD = DA, diagonal BD bisects ∠ABC and ∠ADC. It also bisects AC at O, giving AO = OC, and is perpendicular to AC.

To see the angle bisection, compare triangles ABD and CBD. Their paired equal sides and common side BD establish SSS congruence. The resulting equal angles at B, together with AB = CB and the common side BO, establish congruence of triangles AOB and COB.

Consequently their angles at O are equal and supplementary, so both are right angles. This proves the stated property of BD. It does not establish that AC bisects BD: mutual bisection is an additional property in a rhombus.

How do trapeziums and isosceles trapeziums behave?

Definition: A trapezium is a quadrilateral with at least one pair of parallel opposite sides. “At least one pair” allows a quadrilateral with both pairs parallel to belong to this family.

In trapezium PQRS, let PQ ∥ SR. The sides PS and QR act as transversals. Interior angles on the same side of each transversal total 180°, giving ∠S + ∠P = 180° and ∠R + ∠Q = 180°.

These relations let us find the remaining angles once one angle on each transversal is known. They do not require the two base angles to be equal. Here base angles are the angles at the ends of a chosen parallel side.

Worked example 9. Trapezium PQRS has PQ ∥ SR, ∠P = 135° and ∠Q = 105°. Find ∠S and ∠R.

Answer: ∠S = 180° − 135° = 45°, and ∠R = 180° − 105° = 75°. Each calculation uses the pair of angles on one non-parallel side, rather than a pair of opposite angles.

What extra condition gives an isosceles trapezium?

An isosceles trapezium has non-parallel sides of equal length. Consider UVWX with parallel sides UV and XW and equal non-parallel sides UX and VW. Its base angles at U and V are equal.

To justify this, draw XY and WZ perpendicular to UV, with Y and Z on UV. The inner quadrilateral XWZY has right angles and is a rectangle. Its opposite sides XY and WZ are equal.

A right triangle has one right angle. The right triangles UXY and VWZ have equal hypotenuses UX and VW and equal corresponding sides XY and WZ. A hypotenuse is the side opposite a right angle in a right triangle. These triangles are congruent, giving ∠U = ∠V.

The angles at X and W are then also equal because each supplements an equal base angle. The condition concerns equal non-parallel sides; a general trapezium does not automatically have equal base angles.

How are the different quadrilateral families related?

A Venn diagram represents sets, or collections of objects, using closed curves. A point inside a curve represents a member of that set. Typically, these closed curves are ovals or circles. Overlapping regions show objects that satisfy more than one definition.

A square is a rectangle because it has four right angles, and a rhombus because it has four equal sides. Both rectangles and rhombuses are parallelograms. A rhombus is also a kite because its four equal sides form the required two adjacent pairs.

The inclusive trapezium definition places parallelograms within trapeziums. It does not place every kite there. The relationships must be checked against the defining conditions rather than the orientation or familiar appearance of a drawing.

How can properties be compared?

How can a construction be checked?

Start from the property used to construct the shape. If diagonals were used, check their lengths, intersection point and crossing angle as appropriate. If sides were used, check the required equalities and parallel pairs.

Then explain why those observations match a proven condition. A measurement check helps detect a construction error, while geometric reasoning explains why the method works. A shape may legitimately have several names because it belongs to several nested or overlapping families.

Glossary

• Quadrilateral — A closed plane figure bounded by four straight sides, with four vertices and four interior angles.
• Diagonal — A line segment joining two opposite vertices of a quadrilateral.
• Adjacent sides — Two sides of a quadrilateral that share a common vertex.
• Bisect — To divide a line segment, angle or other quantity into two equal parts.
• Congruent triangles — Triangles of the same size and shape, with equal corresponding sides and angles.
• Conjecture — A statement about which we are highly confident but not yet sure that it always holds.
• Rectangle — A quadrilateral whose four interior angles are all right angles.
• Square — A quadrilateral with four equal sides and four right angles.
• Parallelogram — A quadrilateral in which both pairs of opposite sides are parallel.
• Rhombus — A quadrilateral in which all four sides have the same length.
• Kite — A quadrilateral with two non-overlapping pairs of adjacent sides of equal length.
• Trapezium — A quadrilateral having at least one pair of parallel opposite sides.
• Isosceles trapezium — A trapezium whose two non-parallel sides have the same length.
• Supplementary angles — Two angles whose measures add up to a straight angle of 180°.
• Transversal — A line that intersects two other lines at distinct points.

Common errors and misconceptions

• Misconception: A square cannot be a rectangle. Correct: Its four right angles satisfy the rectangle definition; its equal sides make it a special rectangle.
• Misconception: Equal diagonals that bisect each other guarantee a square. Correct: They guarantee a rectangle. Perpendicularity is also needed for the square diagonal test.
• Misconception: Every parallelogram has equal diagonals. Correct: Its diagonals bisect each other but need not be equal.
• Misconception: Every rhombus has four right angles. Correct: Its sides are equal; the 50° example has angles 50°, 130°, 50° and 130°.
• Misconception: Perpendicular diagonals alone prove that a quadrilateral is a rhombus. Correct: A kite also has perpendicular diagonals, without requiring both diagonals to be bisected.
• Misconception: A trapezium must have exactly one parallel pair. Correct: Its definition requires at least one pair, so parallelograms qualify too.
• Misconception: Measurements of several examples prove a property for the whole family. Correct: They support a conjecture; a general justification is needed to establish the property.
• Misconception: Both diagonals of every kite bisect each other. Correct: For AB = BC and CD = DA, BD bisects AC; the reverse does not follow.

Exam-style questions with model answers

Q1. A quadrilateral has three angles of 90°. Find its fourth angle and identify the shape justified by this information. [2 marks]

1. The fourth angle is 360° − (90° + 90° + 90°) = 90°, using the quadrilateral angle sum.
2. All four angles are right angles, so the quadrilateral is a rectangle.

Q2. A carpenter uses two strips as the diagonals of a quadrilateral. One is 8 cm long. State the second strip’s length and the joining condition needed to form a rectangle. [2 marks]

1. The second strip must also be 8 cm long because a rectangle has equal diagonals.
2. Join the strips at their midpoints, 4 cm from every endpoint, so that the diagonals bisect each other.

Q3. Parallelogram ABCD has AB = 4 cm, AD = 5 cm and ∠A = 30°. Find the remaining sides and angles, giving reasons. [3 marks]

1. Opposite sides of a parallelogram are equal. Therefore CD = AB = 4 cm and BC = AD = 5 cm.
2. Adjacent angles are supplementary. Thus ∠B = 180° − 30° = 150° and ∠D = 180° − 30° = 150°.
3. Opposite angles are equal, so ∠C = ∠A = 30°. The angles in boundary order are therefore 30°, 150°, 30° and 150°.

Q4. Rhombus ABCD has ∠A = 50° and diagonal BD. Find ∠C, ∠B, ∠D and the two parts into which BD divides ∠B. Explain your reasoning. [4 marks]

1. A rhombus is a parallelogram, so opposite angles are equal. Therefore ∠C = ∠A = 50°.
2. The adjacent angles A and B total 180°, giving ∠B = 180° − 50° = 130°.
3. Angles B and D are opposite angles of the rhombus. Hence ∠D = ∠B = 130°.
4. A rhombus diagonal bisects its corner angles, so ∠ABD = ∠DBC = 130°/2 = 65°.

Q5. Trapezium PQRS has PQ ∥ SR, ∠P = 135° and ∠Q = 105°. Find ∠S and ∠R and verify the quadrilateral angle sum. [3 marks]

1. PS is a transversal to the parallel sides PQ and SR. Therefore ∠P + ∠S = 180°, giving ∠S = 45°.
2. QR is another transversal to those parallel sides. Therefore ∠Q + ∠R = 180°, giving ∠R = 75°.
3. The four angles total 135° + 105° + 75° + 45° = 360°. This agrees with the angle-sum property of a quadrilateral.

Q6. Describe how to construct a square with diagonal 8 cm using its diagonal properties. Explain why the construction forms a square. [5 marks]

1. Draw a line segment AC of length 8 cm. This will be one diagonal of the required square, joining opposite vertices.
2. Mark its midpoint O so that AO = OC = 4 cm. This fixes the required intersection point of the two diagonals.
3. Draw a line perpendicular to AC through O. On opposite sides of O, mark points B and D with OB = OD = 4 cm.
4. Join AB, BC, CD and DA to form the boundary. The second diagonal BD measures 8 cm, equal to AC, and both diagonals are bisected at O.
5. The diagonals are equal, bisect each other and meet at 90°. These are the diagonal conditions for a square, so ABCD is the required square.

Q7. Equal diagonals AC and BD of quadrilateral ABCD bisect each other at O. Given ∠AOB = 60°, show that ABCD is a rectangle. [5 marks]

1. Since the equal diagonals are bisected at O, all four half-diagonals OA, OB, OC and OD are equal. Thus each small triangle around O is isosceles.
2. Vertically opposite angles are equal and linear pairs total 180°. The four angles around O therefore alternate between 60° and 120°.
3. In each small triangle with a 60° vertex angle, the two equal base angles each measure (180° − 60°)/2 = 60°.
4. In each small triangle with a 120° vertex angle, the two equal base angles each measure (180° − 120°)/2 = 30°.
5. Each corner of ABCD combines a 60° angle and a 30° angle, making 90°. All four corners are right angles, so ABCD is a rectangle.

Q8. Kite ABCD has AB = BC and CD = DA. Diagonals AC and BD meet at O. State the three special properties of BD, specifying the angles or segments involved. [3 marks]

1. BD bisects the angle at B. Therefore ∠ABD = ∠DBC: the angle between BA and BD equals the angle between BD and BC.
2. BD also bisects the angle at D. Therefore ∠ADB = ∠BDC: it divides that corner angle into two equal parts.
3. BD is the perpendicular bisector of AC. Hence AO = OC, and the angles formed where the diagonals intersect are right angles.

Key takeaways

• A quadrilateral has four straight sides, and its four interior angles have a total measure of 360°.
• A rectangle has four right angles; its diagonals are equal and bisect each other.
• A square is both a rectangle and a rhombus, with equal diagonals that bisect each other at right angles.
• A parallelogram has equal, parallel opposite sides, equal opposite angles, supplementary adjacent angles and mutually bisecting diagonals.
• A rhombus has four equal sides; its diagonals bisect its angles and each other at right angles.
• A kite has two equal adjacent side pairs; the diagonal joining their common vertices bisects the other perpendicularly.
• A trapezium has at least one parallel opposite pair, so the definition includes parallelograms.
• Measurements can support a conjecture, while geometric reasoning establishes why a property holds for a whole family.

Test yourself

What does it mean for diagonals to bisect each other?

Their intersection is the midpoint of each diagonal, dividing each into two equal lengths.

Why does every square satisfy the rectangle definition?

A square has four right angles, which is exactly the condition defining a rectangle.

Do a parallelogram’s diagonals have to be equal?

No. They bisect each other, but their lengths need not be equal.

A rhombus has one angle of 50°. What are its other angles?

The opposite angle is 50°, and each adjacent angle is 130°, because adjacent angles are supplementary.

What extra diagonal condition turns the rectangle construction into a square construction?

The equal diagonals that bisect each other must also intersect at right angles.

Why does the trapezium definition include parallelograms?

It requires at least one parallel opposite pair, and a parallelogram has two such pairs.

In kite ABCD with AB = BC and CD = DA, which diagonal bisects AC?

BD bisects AC at right angles and also bisects the corner angles at B and D.

Does measuring several rectangles prove a property for every rectangle?

No. Measurements can suggest a conjecture, but a general justification is needed to prove it.