Geometry | ICSE Class 6 Maths Notes

Geometry covers points, lines, rays, angles, open and closed figures, polygons, triangles, quadrilaterals, circles, solid shapes, nets, reflection symmetry and constructions using a ruler, protractor and compasses.

What are points, lines, line segments and rays?

Geometry studies shapes, their sizes and their positions. A plane is a flat surface extending without an end. A plane figure is two-dimensional, meaning that it has length and breadth. A point marks an exact position and has no length, breadth or height.

Capital letters name points. A and B name points, not measurements. A drawn dot has thickness, but the mathematical point it represents does not.

How do the three straight objects differ?

A line segment is the shortest path joining two points, including both points. These are its endpoints. Segment AB joins endpoints A and B. Segment BA names the same segment. Its length is the distance between those endpoints.

A line extends endlessly in both directions. A ray starts at one endpoint and extends endlessly in one direction. In the name ray AP, A is the starting point and P is another point in its direction.

Use arrowheads to show endless extension: one for a ray and two for a line. A segment has no arrowheads. The letters in a ray's name cannot generally be reversed because doing so changes its starting point.

Property: Two distinct points determine a line

Distinct means different. Exactly one line passes through two distinct points. Many lines can pass through a single point. This explains why two marked points are sufficient to position a ruler for drawing their joining line.

To measure a segment, put the ruler's zero mark at one endpoint and read the other endpoint. Keep the ruler along the segment. The abbreviation cm means centimetres, and mm means millimetres. Record the unit with the measurement.

How do open figures, closed figures and polygons differ?

A curve is a path drawn without lifting the pencil; in geometry it can include straight portions. A simple curve does not cross itself. An open figure has an unjoined end, whereas a closed figure returns to its starting point without leaving a gap.

The boundary is the outline of a figure. A linear boundary consists of straight portions. A curvilinear boundary contains curved portions. A circle has a curved boundary, whereas the boundary of a triangle consists of line segments.

What are the interior and exterior?

A simple closed figure separates the plane into its interior, the region inside, and its exterior, the region outside. A point on the boundary belongs to neither of these regions. An open curve does not enclose an interior in this way.

A polygon is a simple closed figure made entirely of line segments. Its segments are sides. The meeting points of neighbouring sides are vertices, the plural of vertex. A closed figure with a curved part is not a polygon.

When is a polygon regular?

An angle is an opening between two rays with a common starting point. A regular polygon has all sides equal and all interior angles equal. Equal-sided means equilateral; equal-angled means equiangular. Both conditions matter. An irregular polygon does not satisfy both conditions. A square is regular, but a rectangle with unequal neighbouring sides is not.

A diagonal joins two vertices that are not neighbours. In a convex polygon, no part of a diagonal lies outside the figure. The quadrilaterals considered here are convex. Trace the boundary before counting sides so that diagonals are not accidentally counted as sides.

How are angles named, measured and classified?

An angle is formed by two rays with a common starting point. That point is the vertex, and the rays are its arms. Its size measures the turn needed to move one arm onto the other about the vertex.

The symbol ∠ means angle. In ∠DBE, D and E are points on the arms and B is the vertex. The middle letter names the vertex. Rays BD and BE are the arms; ∠EBD names the same opening.

Two rays start at B. The upper ray passes through D and the lower ray through E. B is labelled vertex, and both rays are labelled arm.

See Fig. 2.8 in your NCERT textbook

For an angle smaller than a straight angle, its interior is the region between its arms. Its exterior lies outside that region. Points on the arms are on the boundary. Mark the intended opening clearly when the same rays also bound a larger angle.

What does a degree measure?

A degree is one of 360 equal parts of a complete turn. The symbol ° means degrees. A complete angle measures 360°. A straight angle is half a turn, measuring 180°. A right angle is a quarter-turn, measuring 90°.

Worked example 1. Classify angles measuring 40°, 110° and 195°.

Answer: 40° is acute because it lies between 0° and 90°. The 110° angle is obtuse because it lies between 90° and 180°. The 195° angle is reflex because it lies between 180° and 360°.

How is a protractor used?

A protractor is an instrument marked in degrees for measuring and drawing angles. Place its centre at the vertex and its baseline along one arm. Read the scale that begins at zero on that arm, then locate the other arm.

Check whether the reading agrees with the opening: an acute angle needs a reading below 90°. Extending the drawn arms can help them reach the scale. Their drawn lengths do not change the angle because the amount of turn remains unchanged.

Worked example 2. What angle is made by a quarter of a complete turn?

Answer: A complete turn is 360°. Divide 360° by 4 to obtain 90°. The resulting angle is a right angle. Here ÷, used in 360° ÷ 4 = 90°, means division; = means equals.

How do intersecting, perpendicular and parallel lines differ?

Intersecting lines meet at a point, called their point of intersection. Perpendicular lines intersect at a right angle. Parallel lines lie in the same plane and do not meet, however far they are extended in either direction.

Segments that appear separate may belong to lines that meet when extended. Two lines in different planes are not classified as parallel merely because they do not meet.

Property: Perpendicular lines form right angles

Two perpendicular lines form four right angles around their intersection. Each angle measures 90°. The right angle can be shown with a small square at the vertex. Sloping lines can also be perpendicular; their angle matters.

Opposite sides of a rectangle provide pairs of parallel segments. Neighbouring sides meet perpendicularly. Adjacent means next to one another or sharing a common endpoint in this setting. Adjacent angles share a vertex and an arm, with interiors that do not overlap.

How do known angles help find an unknown opening?

When adjacent openings together make a straight angle, their measures add to 180°. When they together make a right angle, their measures add to 90°. Establish which whole opening is given before subtracting a known part.

Worked example 3. Points B, E and R lie on one straight line, with E between B and R. Ray ES is perpendicular to ER. Ray ET lies inside ∠SER, and ∠TER is 80°. Find ∠BET and ∠SET.

Answer: ∠BER is a straight angle. Therefore ∠BET = 180° − 80° = 100°. Also ∠SER is 90°, so ∠SET = 90° − 80° = 10°. The symbol − means subtract.

The positions of the rays are part of the information: ET must lie inside the right angle for the second subtraction. A sketch helps organise that information, but the stated positions and measures supply the reasons for the answer.

How are triangles classified and their special segments identified?

A triangle is a polygon with three sides, three vertices and three angles. The notation triangle ABC names a triangle whose vertices are A, B and C. Its sides are AB, BC and CA, and its angles are ∠ABC, ∠BCA and ∠CAB.

The sides form its boundary. The enclosed region is its interior, and the region outside is its exterior. Three points lying on a single straight line do not form a triangle. The triangle must enclose a region.

What are the two ways to classify triangles?

Keep the basis of classification clear. Side names describe equality of lengths; angle names describe the sizes of openings. Finding one acute angle does not establish that a triangle is acute-angled. All three angles must be acute for that classification.

Worked example 4. Classify a triangle with side lengths 4 cm, 5 cm and 6 cm by its sides.

Answer: The lengths 4 cm, 5 cm and 6 cm are all different. The triangle is therefore scalene. This classification uses side lengths, without requiring an angle measurement.

How does a median differ from an altitude?

A midpoint divides a segment into two equal lengths. A median joins a vertex of a triangle to the midpoint of the opposite side. An altitude is a perpendicular segment from a vertex to the line containing the opposite side.

The base is the side chosen as a reference. The corresponding height is the length of the altitude to that base. For an obtuse triangle, drawing an altitude may require extending a side. A median is identified by equal lengths, while an altitude is identified by a right angle.

What properties identify the different quadrilaterals?

A quadrilateral is a polygon with four sides, four vertices and four angles. Name its vertices in order around its boundary. For quadrilateral ABCD, the boundary sides are AB, BC, CD and DA. Jumping across a diagonal gives an incorrect boundary order.

Adjacent sides share a vertex, such as AB and BC. Opposite sides do not share a vertex, such as AB and CD. Opposite vertices A and C are joined by diagonal AC; the other diagonal joins B and D.

Which side and angle properties should be checked?

Property: A square has equal sides and right angles

A square satisfies both conditions: equal side lengths and right angles. Turning a square on the page changes its orientation, meaning the direction in which it is placed, but preserves its lengths and angles. It remains a square.

A rhombus need not have right angles. A rectangle need not have all four sides equal. Use all the relevant properties when naming the most specific shape. Equal side lengths alone do not distinguish a square from another rhombus.

To bisect means to divide into two equal parts. The diagonals of a rectangle bisect each other, so their intersection is the midpoint of each diagonal. Adjacent sides of the rectangle are perpendicular, and its opposite sides are parallel.

Note: A figure's position is not its definition. Check sides, angles and parallelism before deciding whether a turned or sloping quadrilateral is a square, rectangle, rhombus or parallelogram.

What are the parts and regions of a circle?

A circle consists of all points in a plane at a fixed distance from a fixed point. The fixed point is its centre. A segment joining the centre to a point on the circle is a radius; the word also names its length.

All radii of one circle have equal length. The plural of radius is radii. A chord joins two points on the circle. A diameter is a chord passing through the centre, and its length is twice the radius.

How do arcs, sectors and segments differ?

An arc is a portion of the circle's curved boundary. A sector is a region bounded by two radii and the arc between their endpoints. A segment of a circle is a region bounded by a chord and its corresponding arc.

A semicircle is either half of a circle cut off by a diameter. The circumference is the distance around the circle. The same word can also refer to the circular boundary. Do not confuse the circumference with a straight distance across the circle.

The interior consists of points closer to the centre than the radius. Exterior points are farther from the centre than the radius. Points whose distance from the centre equals the radius lie on the circle itself.

How is a circle constructed?

Compasses are an instrument with a pointed leg and a pencil leg, used to draw circles and arcs. Set their opening using a ruler. Hold the pointed leg at the centre and turn the pencil leg without changing that opening.

Worked example 5. Construct a circle with centre P and radius 4 cm.

Answer: Mark P and set the compasses to 4 cm against a ruler. Keep the pointed leg at P and turn the pencil through a complete turn. Every point drawn is 4 cm from P, so the curve is the required circle.

One drawing shows compasses opened against a ruler. Another shows the pointed leg at the centre P of a circle, with a radius labelled between the centre and the circular boundary.

See Fig. 8.2 in your NCERT textbook

How are solid shapes and their nets recognised?

A solid is a three-dimensional shape: it has length, breadth and height. A face is a flat surface of a solid. An edge is where flat faces meet, and a vertex is a corner where edges meet.

Some solids also have curved surfaces. When describing them, separate flat faces from curved surfaces rather than treating every surface as a polygon. A flat drawing shows a solid, but the solid itself occupies space.

Which features distinguish the solids?

A prism has two equal, matching, parallel polygonal bases. In a right prism, the joining edges are perpendicular to the bases. A pyramid has one polygonal base and triangular side faces meeting at a common top vertex, called its apex.

A tetrahedron is a triangular pyramid with four triangular faces. Its name describes the number of faces. Identify the base and the way the remaining faces meet before deciding whether a model is a prism or a pyramid.

What does a net show?

A net is a flat arrangement that can be folded or rolled to form a solid's surface. A cube net contains six squares; a cuboid net contains six rectangles. The pieces must be arranged so that they close the solid without overlapping.

A cylinder net has a rectangle and two circles. A cone net has a circular sector and a circle. A tetrahedron net has four triangles. In each case, the sizes of the pieces must match where they join.

Try folding a paper net to test it. Count faces before folding, then locate the edges and vertices after folding. A collection of the correct number of pieces is not sufficient unless their arrangement and matching lengths produce the required solid.

How does reflection symmetry work?

A figure has reflection symmetry when reflection in a line makes it coincide with itself. Reflection means taking a mirror image. The line is a line of symmetry, also called an axis of symmetry; axes is the plural of axis.

Folding gives a useful test. If the two parts cover each other exactly when folded along a line, that line is a line of symmetry. Equal-sized parts alone are not enough: the positions of their corners and boundaries must match.

How can symmetry be tested?

1. Choose a possible line across the figure.
2. Fold the paper along that line, or place a mirror along it.
3. Check whether the boundary on one side matches the reflected boundary on the other.
4. Repeat with other possible lines, because a figure may have more than one line of symmetry.

A square has four lines of symmetry: the two lines through the midpoints of opposite sides and the two diagonals. Folding along each makes the halves coincide. A rectangle that is not a square does not have diagonal reflection symmetry.

Worked example 6. Find the lines of reflection symmetry of a square by folding.

Answer: Fold through the midpoints of each pair of opposite sides, giving 2 lines. Fold along each diagonal, giving 2 more. Each fold brings one half exactly onto the other, so the square has 4 lines of symmetry.

When reflecting a shape, match corresponding corners as well as the outside outline. Corresponding points are points that match under the reflection. A line through the middle of a picture is not automatically a symmetry line if the two sides fail to match.

How are line segments, triangles and quadrilaterals constructed?

A construction is a drawing made using geometrical instruments to satisfy given conditions. A ruler draws straight segments and measures lengths. A straightedge draws straight lines without measuring them. Use compasses to transfer a distance without changing its length.

To draw a segment of given length, mark its first endpoint, align the ruler's zero with it, and mark the second endpoint at the required reading. Join the endpoints. The length must be measured along the segment, with the unit written beside it.

How can equal distances locate a triangle's vertex?

A circle or arc centred at a point identifies positions at a fixed distance from that point. Intersecting two such arcs locates a point satisfying two distance conditions together. Retain the arcs so that the construction method remains visible.

Worked example 7. Construct an equilateral triangle with each side 4 cm.

1. Draw segment AB of length 4 cm.
2. With A as centre and radius 4 cm, draw an arc.
3. With B as centre and the same radius, draw an intersecting arc. Name an intersection C.
4. Join AC and BC.

Answer: AB, AC and BC each measure 4 cm. The first is the drawn base; the other two equal the radii used. The triangle is equilateral because all three lengths are equal.

How are right angles used to construct a square?

Worked example 8. Construct square PQRS with side 6 cm. The letters P, Q, R and S name its vertices in boundary order.

1. Draw PQ of length 6 cm.
2. Construct a perpendicular to PQ at P and mark S on it with PS equal to 6 cm.
3. Construct a perpendicular to PQ at Q, on the same side of PQ, and mark R with QR equal to 6 cm.
4. Join R to S and check the lengths and angles.

Answer: The completed square has four sides of 6 cm and four angles of 90°. Both conditions must be checked; equal lengths alone would not establish that the figure is a square.

For a rectangle, use the two specified neighbouring side lengths with right angles. Opposite sides must match in length. Check the completed figure against its defining properties rather than accepting a drawing because it merely looks like the intended shape.

How are perpendicular bisectors and perpendicular lines constructed?

A perpendicular bisector of a segment is a line that passes through its midpoint at a right angle. It therefore has two requirements: equal division of length and perpendicular intersection. A line satisfying only one requirement is not its perpendicular bisector.

How is a perpendicular bisector drawn with compasses?

1. Draw the given segment and label its endpoints X and Y.
2. Choose a compass opening greater than half the segment's length. From X, draw arcs above and below the segment.
3. Keep the opening unchanged. From Y, draw arcs cutting the first arcs. Label the upper intersection A and the lower intersection B.
4. Draw line AB. Label its intersection with XY as O.

Line AB is the perpendicular bisector. O is the midpoint, so OX and OY have equal lengths. The line crosses XY at 90°. A compass opening that is too small will not produce the required pair of arc intersections.

How is a perpendicular drawn at a point on a line?

Let O be the given point on the line. Mark points X and Y on opposite sides of O at equal distances from it. O is then the midpoint of XY. Construct the perpendicular bisector of XY; it is the required perpendicular through O.

Only one further point on the perpendicular is needed because O is already known. Equal-radius arcs centred at X and Y can meet at that point. Joining it to O gives the required line.

How is a perpendicular drawn from a point outside a line?

Let P be the given point outside the line. With P as centre, draw an arc large enough to cut the line at two distinct points, X and Y. Construct the perpendicular bisector of XY. It passes through P because PX and PY are equal radii.

Keep the part of this perpendicular between P and the given line when a perpendicular line segment is required. Check its right angle with the given line. In all these constructions, equal compass settings establish the equal distances on which the method depends.

How are angles constructed, bisected and copied?

To construct an angle with a protractor, draw a starting ray, centre the protractor at its endpoint, and align its zero line with the ray. Mark the required degree reading. Draw the second ray from the same endpoint through the mark.

How are 60° and 120° constructed with compasses?

1. Draw a ray with starting point O. With O as centre, draw an arc cutting the ray at A.
2. Without changing the compass opening, use A as centre to cut the arc at B above the ray.
3. Draw ray OB. The angle between OA and OB measures 60°.
4. With the same opening and B as centre, cut the original arc farther along at C. Draw ray OC. The angle from OA through OB to OC measures 120°.

The letters O, A, B and C name construction points. Keeping the opening unchanged is essential. Equal steps around the circle produce successive 60° openings at the centre. Select the new intersection farther around the arc when making the second step.

How does angle bisection produce other angles?

An angle bisector is a ray dividing an angle into two equal angles. To construct it, draw an arc centred at the vertex that meets both arms. From those meeting points, draw equal-radius arcs that intersect inside the angle. Join the vertex to their intersection.

Worked example 9. Construct a 45° angle by bisecting a right angle.

Answer: First construct a 90° angle. Draw an arc from its vertex to mark equal distances along its arms. Use these marks as centres for equal intersecting arcs inside the angle. Draw the ray through their intersection from the vertex. Each half measures 90° ÷ 2 = 45°.

Bisecting 60° gives 30°. Bisecting 30° gives 15°. Bisecting the opening between rays at 60° and 90°, measured from the same starting ray on the same side, gives a ray at 75°. These operations use equal division of an established angle.

How is an angle copied without measuring it?

1. Let A be the vertex of the given angle. Draw an arc centred at A to meet its arms at B and C.
2. Draw a new ray starting at X. With the same radius, draw an arc centred at X, meeting the ray at Z.
3. Set the compasses to the straight distance BC. From Z, mark Y on the new arc so that YZ equals BC.
4. Draw ray XY. The angle between XY and XZ copies the given angle at A.

The letters in this procedure name points, and BC and YZ name segments and their lengths. Copying transfers the opening between the arms. It does not require knowing the numerical degree measure of the original angle.

Glossary

• Point — An exact position with no length, breadth or height, represented by a small dot.
• Line segment — The shortest straight path between two endpoints, including both of those endpoints.
• Ray — A portion of a line starting at one endpoint and continuing endlessly in one direction.
• Angle — An opening formed by two rays sharing a common starting point called the vertex.
• Polygon — A simple closed plane figure whose entire boundary consists of straight line segments.
• Regular polygon — A polygon with all sides equal in length and all interior angles equal in measure.
• Median — A segment joining a triangle's vertex to the midpoint of its opposite side.
• Altitude — A perpendicular segment from a triangle's vertex to the line containing its opposite side.
• Diagonal — A line segment joining two vertices of a polygon that are not neighbours.
• Radius — A segment joining a circle's centre to any point on its circular boundary.
• Sector — A region of a circle bounded by two radii and an arc between them.
• Net — A flat arrangement of connected surface pieces that can form the surface of a solid.
• Reflection symmetry — The property of a figure that coincides with itself when reflected in a suitable line.
• Perpendicular bisector — A line crossing a segment at its midpoint and forming a right angle with it.
• Angle bisector — A ray beginning at an angle's vertex and dividing the angle into two equal angles.

Common errors and misconceptions

• Misconception: A line, segment and ray all have two endpoints. Correct: A segment has two endpoints, a ray has one, and a line has none. Arrowheads show directions in which the drawing continues endlessly.
• Misconception: Longer drawn arms make a larger angle. Correct: Angle size depends on the turn between the arms. Extending their drawn lengths leaves the opening unchanged.
• Misconception: Any angle greater than 90° is obtuse. Correct: An obtuse angle must also be less than 180°. Angles greater than 180° but less than 360° are reflex angles.
• Misconception: One acute angle makes a triangle acute-angled. Correct: All three angles must be acute. Classify by checking the full condition, rather than using a single angle.
• Misconception: A median and an altitude are the same segment in every triangle. Correct: A median reaches the opposite side's midpoint; an altitude is perpendicular to the line containing that side.
• Misconception: Every quadrilateral with equal sides is a square. Correct: A square also requires four right angles. A rhombus can have equal sides without having right angles.
• Misconception: A circle's sector and segment are the same region. Correct: A sector is bounded by two radii and an arc. A segment is bounded by a chord and an arc.
• Misconception: Any line dividing a shape into equal-sized parts is a symmetry line. Correct: Folding along a line of reflection symmetry must make the two parts match exactly, including their boundaries.

Exam-style questions with model answers

Q1. Give one distinguishing endpoint property of a line segment and one of a ray. [2 marks]

1. A line segment has two endpoints and a fixed length between them.
2. A ray has one endpoint and extends endlessly in one direction from it.

Q2. Classify angles of 40°, 110° and 195°, giving the defining range for each. [3 marks]

1. The 40° angle is acute. Its measure is greater than 0° and less than 90°, so it is smaller than a right angle.
2. The 110° angle is obtuse. It is greater than 90° and less than 180°, lying between a right angle and a straight angle.
3. The 195° angle is reflex. It is greater than 180° and less than 360°, lying between a straight angle and a complete angle.

Q3. B, E and R lie on one straight line, with E between B and R. Ray ES is perpendicular to ER. Ray ET lies inside ∠SER, and ∠TER = 80°. Find ∠BET and ∠SET, stating the whole angle used in each calculation. [4 marks]

1. ∠BER is a straight angle measuring 180° because EB and ER point in opposite directions along the given straight line.
2. ∠BET = 180° − 80° = 100°, since ET divides that straight opening into ∠BET and the given ∠TER.
3. ∠SER is a right angle measuring 90° because ES is given perpendicular to ER.
4. ∠SET = 90° − 80° = 10°, since ET lies inside ∠SER and leaves this opening between ES and ET.

Q4. In triangle ABC, M is the midpoint of BC. Point D lies on the line containing BC, and AD is perpendicular to that line. Name AM and AD, and explain the different conditions they satisfy. [3 marks]

1. AM is a median of triangle ABC because it joins vertex A to M, the given midpoint of the opposite side BC.
2. AD is an altitude of triangle ABC because it joins vertex A perpendicularly to the line containing the opposite side BC.
3. The median uses the equality of BM and MC. The altitude uses a right angle with the base line; the given information does not require D to be M.

Q5. Describe how to construct the perpendicular bisector of a given segment XY using a ruler and compasses. Include the compass-opening condition, the construction and the two properties of the resulting line. [5 marks]

1. Choose a compass opening greater than half the length of XY. This allows arcs from its two endpoints to cross above and below the segment.
2. With X as centre, draw arcs above and below XY. Keep the compass opening fixed while making the arcs from the other endpoint.
3. With Y as centre and the same opening, draw arcs intersecting the first pair. Name the upper intersection A and the lower intersection B.
4. Join A and B using the ruler. The resulting line meets XY at its midpoint, dividing the given segment into two equal lengths.
5. The line AB is also perpendicular to XY, so it forms a 90° angle with the segment. It therefore satisfies both requirements of a perpendicular bisector.

Q6. Explain how to construct an equilateral triangle with each side 4 cm using a ruler and compasses. Give the construction steps and explain why the three sides are equal. [5 marks]

1. Use the ruler to draw a line segment AB measuring 4 cm. This segment will be the base of the required triangle.
2. Set the compasses to 4 cm. With A as centre, draw an arc above the base, making it long enough for the next arc to cross.
3. Keep the compass opening at 4 cm. With B as centre, draw another arc to cut the first one, and name their intersection C.
4. Join A to C and B to C with straight segments. These segments complete triangle ABC, whose third vertex is the marked intersection.
5. AB is 4 cm by measurement. AC and BC are each 4 cm because they are radii of the constructed arcs. Thus all three sides are equal.

Q7. State four lines of reflection symmetry of a square and explain how folding verifies them. [5 marks]

1. The line through the midpoints of one pair of opposite sides is a symmetry line. Folding there makes the two halves of the square coincide.
2. The line through the midpoints of the other pair of opposite sides is another symmetry line. Its fold also matches the boundary on both sides.
3. One diagonal is a symmetry line. Folding along it brings the corners away from that diagonal onto each other and matches the two triangular halves.
4. The other diagonal is also a symmetry line. Folding along this second diagonal again makes the two triangular halves cover each other exactly.
5. These give four lines in total. The folding test checks complete matching of the halves, including sides and corners, rather than merely comparing their sizes.

Q8. A circle has centre P and radius 4 cm. Describe how to draw it using compasses and state the distance from P to any point on the completed circle. [3 marks]

1. Mark P as the centre and use a ruler to set the distance between the compass point and the pencil tip to 4 cm.
2. Place the pointed leg at P. Turn the pencil leg through a complete turn without changing the compass opening to draw the circular boundary.
3. Every point on the completed circle is 4 cm from P. The fixed opening maintains the same radius throughout the construction.

Key takeaways

• A point marks a position; a segment has two endpoints, a ray one, and a line none.
• A polygon is simple, closed and made of segments; a regular polygon has equal sides and equal angles.
• Angles measure turns. Use 90°, 180° and 360° to distinguish right, straight, complete, acute, obtuse and reflex angles.
• Classify triangles separately by sides and angles; distinguish a median's midpoint condition from an altitude's perpendicular condition.
• Identify quadrilaterals through equal sides, parallel sides and angle properties, even when the figure is turned.
• A radius joins the centre to the circle; distinguish chords, diameters, arcs, sectors and circular segments.
• Solid shapes have surfaces and occupy space; a suitable net folds or rolls into the required surface.
• Reflection symmetry requires exact matching, while accurate constructions depend on keeping the required distances and angles unchanged.

Test yourself

Why is a ray's starting point written first in its name?

The first letter identifies its endpoint. Reversing the letters generally changes that endpoint and therefore names a different ray.

Can a closed figure with a curved boundary be a polygon?

No. A polygon must be a simple closed figure made entirely of straight line segments.

What distinguishes a zero angle from a complete angle?

A zero angle represents no turn and measures 0°. A complete angle represents one full turn and measures 360°.

What must be checked before calling a triangle acute-angled?

All three angles must be acute. The presence of just one acute angle is insufficient.

What is the difference between a chord and a diameter?

A chord joins two points on a circle. A diameter is a chord that also passes through the centre.

What flat pieces form a cone's net?

A circular sector forms the curved surface, and a suitably sized circle forms the base.

Why is passing through a segment's midpoint insufficient for a perpendicular bisector?

The line must also meet the segment at a right angle. Equal division alone does not establish perpendicularity.

Which angle results from bisecting 60°, and which from bisecting 90°?

Bisecting 60° gives two angles of 30°. Bisecting 90° gives two angles of 45°.