Geometry | ICSE Class 8 Maths Notes

This note covers quadrilaterals and their angles, parallelogram properties, rectangles, rhombuses and squares, quadrilateral constructions, solid shapes, nets and views, Euler’s relation, reflection symmetry, parts of a circle, coordinates and linear graphs.

What are quadrilaterals and how are their angles related?

How do we name the parts of a figure?

A plane is a flat surface. A polygon is a simple closed figure made from straight line segments: its boundary closes without crossing itself. A quadrilateral is a polygon with four sides, four corners and four interior angles.

A corner is a vertex; the plural is vertices. An interior angle lies inside the figure between two sides meeting at a vertex. A diagonal joins two vertices that are not next to each other. It is not a side.

In quadrilateral ABCD, the capital letters name consecutive vertices around the boundary. AB means the segment joining A and B, or its length when used in a calculation. The diagonals are AC and BD. Adjacent sides share a vertex; opposite sides do not.

The symbol ∠ means angle. In ∠ABC, the middle letter B names the vertex. The symbol ° means degrees, the unit used to measure angles. A right angle measures 90°. Two angles are supplementary when their measures add to 180°.

Property: the angle sum of a quadrilateral

The four interior angles of a quadrilateral add to 360°. For a convex quadrilateral, a diagonal separates it into two triangles. Convex means that no part of a diagonal lies outside the polygon. Each triangle has an angle sum of 180°.

Adding the two triangle sums gives 180° + 180° = 360°. At the endpoints of the diagonal, the smaller triangle angles combine to form the original quadrilateral angles. No additional interior angle is introduced by this division.

Draw convex quadrilateral ABCD and join A to C. Mark the angles in the two triangles. Show how the angles at A and C combine when the two triangle angle sums are added.

To find a missing angle, add the three known interior angles and subtract their sum from 360°. First check that the stated angles are interior angles. An exterior angle, formed by extending a side, must not be substituted as though it were an interior angle.

How does a geometric reason differ from a measurement?

A definition specifies what a term means. An axiom is a statement accepted as a starting point for reasoning. A proof is a sequence of justified steps showing why a conclusion follows from the starting information and accepted results.

A necessary condition is something a figure must satisfy. A sufficient condition provides enough information to establish a conclusion. Four equal sides are necessary for a square, but they are not sufficient: a rhombus need not have four right angles.

Verification by measuring or folding helps reveal a property. A reasoned argument explains the connection between the conditions and the result. When explaining a step, name the property being used rather than relying on how a particular drawing looks.

What makes a quadrilateral a parallelogram?

How are parallel sides recognised?

Parallel lines lie in the same plane and do not meet however far they are extended. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. In parallelogram ABCD, AB is parallel to CD, and AD is parallel to BC.

The definition concerns both pairs. Recognising one pair of parallel sides alone does not establish that a quadrilateral is a parallelogram. A trapezium is a quadrilateral with a pair of parallel sides. Keep the properties of the named figure separate from what its outline merely suggests.

Property: opposite sides of a parallelogram are equal

AB = CD and AD = BC in parallelogram ABCD. The equals sign means that the quantities on either side have the same value. Verification can be made by measurement or by matching opposite sides of identical cut-outs.

A logical explanation uses diagonal AC. It produces two triangles whose matching angles are equal because the opposite sides are parallel. The diagonal belongs to both triangles. The triangles match in size and shape, so their corresponding sides have equal lengths.

Triangles that match exactly in size and shape are called congruent. Corresponding parts are the parts that match. This reasoning explains why measuring opposite sides gives equal lengths; measurement itself is a check on the particular drawing.

Worked example 1. Parallelogram PQRS has consecutive sides PQ = 12 cm and QR = 7 cm. Here cm means centimetres. Find its perimeter, the total distance around its boundary.

Answer: Opposite sides are equal, so RS = 12 cm and SP = 7 cm. The perimeter is PQ + QR + RS + SP = 12 + 7 + 12 + 7 = 38 cm.

The property supplies the two missing side lengths. Adding only the given adjacent sides would leave out half the boundary. Write all four sides before simplifying the addition, especially when the diagram is drawn in an unfamiliar orientation.

How do we find angles in a parallelogram?

Property: opposite angles are equal and adjacent angles are supplementary

In parallelogram ABCD, ∠A = ∠C and ∠B = ∠D. Here the single-letter angle names refer to the interior angles at those vertices. Adjacent angles occur at the ends of the same side, and their sum is 180°.

These properties work together. Once one angle is known, its opposite is equal to it. Either adjacent angle is found by subtracting the known angle from 180°. The remaining angle equals its opposite. The four answers must also add to 360°.

Worked example 2. RING is a parallelogram, with its vertices named in boundary order, and ∠R = 70°. Find the other interior angles.

Answer: ∠N = 70° because it is opposite ∠R. Next, ∠I = 180° − 70° = 110° because ∠R and ∠I are adjacent. Finally, ∠G = 110° because it is opposite ∠I.

How does an angle ratio help?

A ratio compares quantities by division. In a ratio of 3 : 2, the quantities contain three and two equal shares respectively. When the quantities are adjacent parallelogram angles, those five shares together represent 180°, not 360°.

Worked example 3. Two adjacent angles of a parallelogram are in the ratio 3 : 2. Find all four angles.

Answer: Let one share be x degrees. The adjacent angles are 3x and 2x degrees. Thus 3x + 2x = 180, giving x = 36. The angles are 108°, 72°, 108° and 72° in boundary order.

Worked example 4. Two adjacent angles of a parallelogram have equal measures. Find every angle.

Answer: The adjacent angles add to 180° and are equal, so each is 180° ÷ 2 = 90°. Each opposite angle has the same measure as its partner. All four angles therefore measure 90°.

The second calculation identifies the extra angle condition of a rectangle. An equal pair of opposite angles alone would not have the same effect, because opposite angles are already equal in every parallelogram.

What does it mean for diagonals to bisect each other?

Property: parallelogram diagonals bisect one another

To bisect a segment means to divide it into two equal lengths. A midpoint is the point making that division. The intersection of two diagonals is their meeting point. These terms describe lengths and positions, not an angle of intersection.

If diagonals AC and BD of parallelogram ABCD meet at O, then AO = OC and BO = OD. Each diagonal cuts the other into equal halves. This does not say that AO equals BO, or that AC equals BD.

The diagonals of a parallelogram, in general, are not of equal length. They can have different lengths and still bisect each other. Equal diagonals are an additional property of a rectangle, which is a special parallelogram.

How can the property be checked?

Draw both diagonals on a parallelogram cut-out. Fold so that the endpoints of one diagonal coincide, meaning that they lie exactly on each other. Its midpoint lies on the fold. Repeat for the other diagonal and compare the midpoints with the intersection.

The parallelogram is labelled A, B, C and D around its boundary. Its diagonals AC and DB are drawn as dotted segments meeting at O inside the figure.

See Fig. 3.23 in your NCERT textbook

Worked example 5. HELP is a parallelogram. Diagonals HL and PE meet at O. Given OE = 4 cm and HL is 5 cm longer than PE, find OH.

Answer: OP = OE = 4 cm because the diagonals bisect each other. Therefore PE = 8 cm and HL = 8 + 5 = 13 cm. Since O also bisects HL, OH = 13 ÷ 2 = 6.5 cm.

Keep the two diagonals distinct throughout this calculation. First reconstruct PE from its half, then use the stated difference to obtain HL. Only after that should HL be halved. The order follows the relationships supplied by the question.

How do rectangles, rhombuses and squares compare?

What extra conditions define these shapes?

A rectangle is a parallelogram with every angle a right angle. A rhombus is a quadrilateral with all four sides equal; it is also a parallelogram. A square is a rectangle with all four sides equal.

Therefore, a square belongs to both the rectangle and rhombus families. It inherits their properties. A statement true of every parallelogram applies to a square, but a special property of a square need not apply to every parallelogram.

Property: special diagonals

Perpendicular lines meet at a right angle. A perpendicular bisector both passes through a segment’s midpoint and meets it at 90°. The diagonals of a rhombus are perpendicular bisectors of one another. A rectangle’s diagonals are equal and bisect each other.

Worked example 6. Rhombus RICE has diagonals RC and IE meeting at O. Given OI = 5, OC = 12 and ER = 13, all in the same length unit, find OE, OR and side IR.

Answer: OE = OI = 5 units and OR = OC = 12 units, since diagonals bisect one another. IR = ER = 13 units because all sides of a rhombus are equal.

Worked example 7. In rectangle RENT, diagonals RN and ET meet at O. Their half-lengths are OR = 2x + 4 and OT = 3x + 1, in the same length unit; x is an unknown number. Find x.

Answer: Equal diagonals have equal halves, so OR = OT. Therefore 2x + 4 = 3x + 1. Subtracting 2x and then 1 from both sides gives x = 3.

A rectangle’s diagonals are not required to be perpendicular. A rhombus’s diagonals are not required to be equal. In a square, both conditions hold. Naming the figure before choosing a diagonal property prevents transferring a special condition to a more general shape.

How are quadrilaterals constructed from sides and diagonals?

Why do compass arcs locate a vertex?

A construction produces a figure from stated geometric measurements. A pair of compasses transfers a length and draws an arc at a fixed distance from a centre. A straight edge joins points with straight segments. An arc is part of a circle.

Make a rough labelled sketch first. It records which lengths are sides and which are diagonals. A compass arc centred at a known vertex identifies possible positions at the required distance. Intersecting two arcs imposes two distance conditions together.

How do four sides and one diagonal determine the drawing?

Suppose the four sides AB, BC, CD and DA and diagonal AC of convex quadrilateral ABCD are given. The diagonal separates the construction into two triangles whose side lengths are specified. Use the supplied measurements without estimating missing positions by eye.

1. Draw AC with its given length, establishing the common side of the two triangles.
2. Draw arcs centred at A and C with the given lengths AB and CB. Their intersection locates B.
3. On the other side of AC, draw arcs centred at A and C with the given lengths AD and CD. Their intersection locates D.
4. Join AB, BC, CD and DA. Check the four side lengths and the diagonal against the supplied measurements.

What changes with three sides and two diagonals?

For given AB, BC, CD, AC and BD, first construct triangle ABC using AB, BC and AC. Then locate D using an arc centred at B with length BD and another centred at C with length CD. Join AD and CD.

Choose the intersection that gives the required simple convex quadrilateral in the labelled order. The measurements must be compatible: if necessary arcs do not meet, the proposed triangle cannot be completed. A construction is justified by its distance conditions, not merely by looking like the rough sketch.

How are quadrilaterals constructed when angles are given?

What is an included angle?

An included angle is the angle between two specified sides that meet at a vertex. A ray starts at a point and extends in one direction. Drawing an angle fixes a ray’s direction; a given length then fixes a point on that ray.

For three sides and two included angles, take AB, BC and CD as the given consecutive sides, with ∠ABC and ∠BCD given. This arrangement supplies a chain of sides. The angles control how the chain turns at B and C.

1. Draw BC to the given length and construct the given angle at B with BC as one arm.
2. On the ray forming that angle, mark A so that BA has its given length.
3. At C, construct the given angle with CB as one arm. On its ray, mark D so that CD has its given length.
4. Join A to D to close the quadrilateral. Check the angles, lengths and boundary order against the rough sketch.

How do two adjacent sides and three angles work?

Suppose adjacent sides AB and BC and interior angles A, B and C are supplied. Draw AB, construct the angle at B and mark C at the given distance from B. The first three vertices have now been located.

At A, construct the supplied angle with AB as an arm. At C, construct the supplied angle with CB as an arm. The appropriately directed rays meet at D. Joining the boundary completes the quadrilateral, provided the given conditions permit it.

The fourth angle can be checked using the angle sum of 360°. This check does not replace the construction: it verifies that the completed figure fits all the stated angles. Retain the arcs and labels so that each geometric step is visible.

Note: Compass and straight-edge constructions of specified angles require those angles to be constructible or supplied for copying. If a task explicitly calls for measuring an angle, a protractor is the angle-measuring instrument. Follow the stated instrument conditions.

How are solid shapes described and Euler’s relation verified?

What are faces, edges and vertices?

A three-dimensional object has length, breadth and height; a two-dimensional representation is drawn on a flat surface. The abbreviations 3-D and 2-D mean three-dimensional and two-dimensional. A solid bounded by flat polygonal faces is called a polyhedron.

A face is one flat surface of a polyhedron. An edge is a segment where two faces meet. A vertex is a corner where edges meet. Count hidden parts as well as visible parts when studying a drawing of a solid.

A cube has six square faces. A cuboid has six rectangular faces. A prism has two congruent parallel polygonal bases joined by lateral faces, meaning faces along its sides. A pyramid has one polygonal base and triangular lateral faces meeting at an apex, its top vertex.

Property: Euler’s relation for these polyhedra

Let F be the number of faces, V the number of vertices and E the number of edges. A solid is convex if the segment joining any two of its points stays within the solid. For the convex polyhedra considered here, F + V − E = 2. This is Euler’s relation. Verify it by counting each kind of part separately.

A tetrahedron is a triangular pyramid with four triangular faces. A triangular prism is named after its triangular bases; a square pyramid is named after its square base. Distinguishing the bases helps prevent confusing the number of faces with the number of vertices.

Euler’s relation here is being used for the listed flat-faced solids. Do not count a curved surface as a polygonal face and assume the same counting method applies. Establish the kind of solid before substituting any numbers.

How do nets and views represent three-dimensional objects?

What is a net?

A net is a flat arrangement of faces that can be folded into the surface of a solid. Fold lines correspond to edges. When making a model, check which boundary edges come together and whether the faces close the solid without overlapping.

A cube’s net contains its six square faces. A cuboid’s net contains its six rectangular faces. The shapes and sizes of adjoining faces must allow the corresponding edges to meet. Merely drawing the correct number of faces does not establish that the arrangement will fold properly.

For a prism, identify the two matching bases and the faces that join their edges. For a pyramid, identify the single base and the triangular faces that meet at the apex. This comparison links the flat arrangement to the solid it forms.

How does a view differ from a net?

A view shows how an object appears from a specified direction, such as the front, side or top. It does not unfold the object. A view can hide some faces and edges; a net represents the complete set of faces needed for the surface.

A flat drawing can also show depth by using slanting edges. Hidden edges may be shown with broken lines. These lines are drawing conventions, meaning agreed ways of representing an object, rather than extra edges belonging to the solid.

Draw a cuboid and indicate its hidden edges with broken lines. Beside it, make a net of its rectangular faces. Match each face in the net to the face it forms after folding.

When matching a picture to an object, compare the arrangement of parts, not just its outside outline. For a nested or joined arrangement, identify each component before considering the whole. Keep such combinations to no more than two shapes, and distinguish visible parts from hidden ones.

How is reflection symmetry identified?

What does a fold test show?

Reflection symmetry means that a figure matches itself when reflected across a line. That line is a line of symmetry. Reflection produces a mirror image, with corresponding points on opposite sides of the line at equal perpendicular distances from it.

Paper folding gives a practical test. Fold along the proposed line and check whether the two parts coincide exactly. Matching area alone is insufficient: the boundaries and corresponding features must match. A fold that merely divides the paper into two pieces is not necessarily a symmetry line.

1. Choose a possible symmetry line on a cut-out of the figure.
2. Fold carefully along that line without stretching or reshaping the paper.
3. Compare the boundaries and corresponding corners on the two sides.
4. Unfold and mark the line if the matching is exact; test other possible lines separately.

How does symmetry connect with familiar shapes?

A square has four lines of symmetry: its two diagonals and the two lines joining midpoints of opposite sides. Folding a square along each gives matching halves. These are different folds, although they all pass through its centre.

A non-square rectangle has two lines of symmetry through the midpoints of opposite sides. Its diagonals are not symmetry lines. This helps distinguish equal diagonals from reflection symmetry: equality of lengths alone does not tell you whether folding along a segment will match the figure.

Every diameter of a circle is a line of symmetry. When studying a composite drawing, check the entire arrangement. A component may have symmetry even when its placement prevents the whole figure from having that same line of symmetry.

What are the different parts of a circle?

How are centre, radius, diameter and chord related?

A circle is the set of points in a plane at a fixed distance from a fixed point. The fixed point is its centre. A radius is a segment joining the centre to the circle, and the word also names that fixed distance.

A chord joins two points on the circle. A diameter is a chord passing through the centre. It contains two radii along one straight line. Thus d = 2r, where d is the diameter’s length and r is the radius’s length.

The circumference is the circle’s boundary, or the length around that boundary when referring to a measurement. The circular region includes the inside as well as the boundary. Distinguish a line segment drawn inside the circle from a region enclosed by boundaries.

How do arcs, sectors and segments differ?

An arc is a portion of the circumference between two points. A sector is a region bounded by two radii and their joining arc. A segment of a circle is a region bounded by a chord and its corresponding arc.

A diameter is a special chord, but a chord need not pass through the centre. Similarly, the word segment in “line segment” names a straight piece of a line, whereas a segment of a circle names a region. Use the full term when the context might be unclear.

How are coordinates and linear graphs used?

What fixes the position of a point?

The Cartesian plane is a plane with two perpendicular number lines called coordinate axes. Their intersection is the origin. Use the same unit scale on both axes when locating geometric points. Conventionally, the horizontal axis is the x-axis and the vertical axis is the y-axis.

An ordered pair, written (x, y), gives a point’s coordinates: x is its horizontal coordinate and y its vertical coordinate. The order matters. Locate the horizontal position first, then the vertical position. The origin has coordinates (0, 0).

A variable is a quantity that can change. In a relationship, an independent variable is the input being chosen; a dependent variable changes according to that input. A graph displays how the paired values relate. Label both axes and show the scale before plotting.

How do square perimeter and area graphs compare?

The perimeter of a square is four times its side length. The area, the measure of its enclosed region, is the side length multiplied by itself. If s is the side length, P the perimeter and A the area, then P = 4s and A = s². The superscript ² means squared.

Worked example 8. Describe how to plot the side-perimeter pairs (2, 8), (3, 12), (3.5, 14), (5, 20) and (6, 24), with both quantities measured in centimetres. Explain whether the resulting graph is linear.

Answer: Put side length on the horizontal axis and perimeter on the vertical axis. Plot each ordered pair. The points lie on one straight line, representing P = 4s. Therefore the graph is linear.

A linear graph lies along one straight line. For the square area pairs (2, 4), (3, 9), (4, 16), (5, 25) and (6, 36), the first coordinate is in centimetres and the second in square centimetres. These points do not lie on one straight line.

To read a graph, start from a known value on one axis, move to the graph, then across to the other axis. Read the scale rather than counting unlabelled grid spaces as units. A line graph made of successive segments need not be one straight line throughout.

Glossary

• Quadrilateral — A simple closed polygon with four sides, four vertices and four interior angles.
• Diagonal — A line segment joining two vertices of a polygon that are not adjacent.
• Parallelogram — A quadrilateral in which both pairs of opposite sides are parallel.
• Supplementary angles — Two angles whose measures together add up to one hundred and eighty degrees.
• Bisect — To divide a line segment or an angle into two equal parts.
• Perpendicular lines — Lines that intersect so that the angles formed at their meeting point are right angles.
• Rhombus — A quadrilateral with all four sides equal in length, also having every parallelogram property.
• Polyhedron — A three-dimensional solid whose surface consists of flat polygonal faces.
• Net — A flat arrangement of faces that folds to form the surface of a solid.
• Line of symmetry — A line across which reflection makes a figure coincide exactly with itself.
• Chord — A straight line segment with both endpoints lying on the same circle.
• Sector — A region of a circle bounded by two radii and the arc joining their endpoints.
• Segment of a circle — A region bounded by a chord and the corresponding arc of the circle.
• Ordered pair — Two coordinates written in a specified order to locate a point in a plane.
• Linear graph — A graph whose plotted relationship lies along a single straight line.

Common errors and misconceptions

• Misconception: Adjacent angles of a parallelogram are equal. Correct: They are supplementary. Opposite angles are equal; adjacent angles are equal in the right-angle case.
• Misconception: Bisecting diagonals must have equal total lengths. Correct: Each is divided into its own equal halves; the two totals can differ.
• Misconception: Every rectangle has perpendicular diagonals. Correct: Its diagonals are equal and bisect each other; perpendicularity is not required.
• Misconception: A square cannot be a rhombus or a rectangle. Correct: It satisfies both definitions and therefore has the properties of both.
• Misconception: A net is just a top view. Correct: A net unfolds the faces, while a top view shows the object from one direction.
• Misconception: A sector is bounded by a chord and an arc. Correct: That describes a segment; a sector is bounded by two radii and an arc.
• Misconception: Reversing coordinates leaves the point unchanged. Correct: The first coordinate locates the horizontal position and the second locates the vertical position; order must be preserved.
• Misconception: Every graph formed by joining points is linear. Correct: A linear graph lies on one straight line; successive joined segments can change direction.

Exam-style questions with model answers

Q1. State two properties of the diagonals of a rectangle. [2 marks]

1. The two diagonals of a rectangle are equal in total length.
2. They bisect each other, so their intersection is the midpoint of each diagonal.

Q2. Parallelogram RING has its vertices in boundary order and interior angle R = 70°. Find angles N, I and G, giving reasons. [3 marks]

1. Angle N is opposite angle R. Opposite angles of a parallelogram are equal, so angle N is also 70°.
2. Angles R and I are adjacent. Their measures add to 180°, so angle I = 180° − 70° = 110°.
3. Angle G is opposite angle I. It therefore equals 110°, completing the angles as 70°, 110°, 70° and 110° around the boundary.

Q3. HELP is a parallelogram whose diagonals HL and PE meet at O. If OE = 4 cm and HL is 5 cm longer than PE, find OH with reasons. [4 marks]

1. Diagonals of a parallelogram bisect each other. Thus O divides PE into equal halves, giving OP = OE = 4 cm.
2. The full diagonal PE consists of OP and OE, so PE = 4 + 4 = 8 cm.
3. The stated difference applies to the complete diagonals. Therefore HL = PE + 5 = 8 + 5 = 13 cm.
4. O also bisects HL, so OH is half of 13 cm. Therefore OH = 13 ÷ 2 = 6.5 cm.

Q4. Explain why a square is a quadrilateral, parallelogram, rectangle and rhombus. State its combined diagonal properties. [5 marks]

1. A square is a quadrilateral because it is a simple closed figure with four straight sides. Its four vertices separate the four interior angles.
2. It is a parallelogram because each pair of opposite sides is parallel. Therefore it also has equal opposite sides and equal opposite angles.
3. It is a rectangle because it is a parallelogram with four right angles. Consequently its diagonals are equal in length and bisect each other.
4. It is a rhombus because all four of its sides have equal lengths. Consequently its diagonals meet at right angles as well as bisecting each other.
5. Combining these properties, a square’s diagonals are equal perpendicular bisectors of one another. Their intersection divides each diagonal into two equal parts.

Q5. A convex quadrilateral ABCD has given side lengths AB, BC, CD and DA, together with diagonal AC. The measurements permit the two required triangles. Describe its compass and straight-edge construction and explain the steps. [5 marks]

1. Make a rough sketch with A, B, C and D in boundary order. Draw AC at the supplied length, since it is the shared side of the two triangles.
2. With A as centre and the supplied AB as radius, draw an arc. This restricts B to points at the required distance from A.
3. With C as centre and the supplied BC as radius, draw a second arc. Its intersection with the first locates B, satisfying both required side lengths.
4. On the opposite side of AC, draw arcs centred at A and C using the supplied AD and CD. Their intersection locates D and completes the second triangle.
5. Join AB, BC, CD and DA. Check all four boundary lengths and AC against the given measurements, retaining the arcs to show how the vertices were located.

Q6. A cube has 6 faces, 8 vertices and 12 edges. Verify Euler’s relation, defining the symbols you use. [3 marks]

1. Let F represent the number of faces, V the number of vertices and E the number of edges. Here F = 6, V = 8 and E = 12.
2. Euler’s relation for this convex polyhedron is F + V − E = 2. Substitute the given counts to obtain 6 + 8 − 12.
3. The left side equals 14 − 12 = 2, matching the right side. Therefore the supplied cube counts satisfy Euler’s relation.

Q7. Distinguish a sector from a segment of a circle by describing the boundaries of each region. [2 marks]

1. A sector is bounded by two radii and the arc joining their endpoints on the circle.
2. A segment is bounded by a chord and its corresponding arc; two radii are not its defining boundaries.

Q8. Square side lengths 2, 3, 3.5, 5 and 6 cm have corresponding perimeters 8, 12, 14, 20 and 24 cm. Describe the axes, ordered pairs and resulting graph. [3 marks]

1. Use the horizontal axis for side length and the vertical axis for perimeter. Label both in centimetres and mark a clear, uniform unit scale.
2. Plot the ordered pairs (2, 8), (3, 12), (3.5, 14), (5, 20) and (6, 24), taking the side length as the first coordinate.
3. The points lie on a straight line, so the graph is linear. It represents P = 4s, where P is perimeter and s is side length.

Key takeaways

• A quadrilateral’s four interior angles total 360°, so three known angles determine the fourth.
• A parallelogram has equal opposite sides and opposite angles, while its adjacent angles are supplementary.
• Bisecting diagonals divide each other into equal halves without necessarily having equal total lengths.
• A square combines rectangle and rhombus properties: its diagonals are equal and bisect each other perpendicularly.
• Construct quadrilaterals by locating vertices from given distances and angles, checking each step against the labelled sketch.
• Count faces, vertices and edges separately when checking Euler’s relation for the specified convex polyhedra.
• Reflection symmetry requires exact matching across a line, while a net unfolds the faces of a solid.
• Distinguish circle regions by their boundaries, and preserve coordinate order when plotting or reading a graph.

Test yourself

What is the sum of the interior angles of a quadrilateral?

The interior angles add to 360°. A diagonal divides a convex quadrilateral into two triangles, each with an angle sum of 180°.

What does it mean when two diagonals bisect each other?

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

Which special parallelogram has both equal and perpendicular diagonals?

A square has equal diagonals that bisect each other at right angles.

Why do intersecting compass arcs help locate a vertex?

Each arc imposes a fixed distance from its centre. Their intersection satisfies both distance conditions at once.

How do a net and a view differ?

A net unfolds the solid’s faces into a flat arrangement. A view shows its appearance from a chosen direction.

What is the fold test for a line of symmetry?

Fold the figure along the proposed line and check whether its two parts coincide exactly.

How does a diameter differ from another chord?

A diameter passes through the centre; a chord generally just joins two points on the circle.

What do the two entries of an ordered pair represent?

The first entry gives the horizontal coordinate and the second the vertical coordinate of the point.