Geometry | ICSE Class 7 Maths Notes

This note covers pairs of angles, parallel lines, triangle properties, verification of Pythagoras’ theorem, congruence, geometrical constructions, reflection and rotational symmetry, solid shapes, nets and simple maps.

How are angles named and related to one another?

Points, lines and angles

Geometry studies shapes and their relationships. A point marks a position. A line segment joins two endpoints; a line extends endlessly in both directions. A ray starts at a point and extends endlessly in one direction.

An angle is formed by two rays with a common starting point. The rays are its arms, and the common point is its vertex. In ∠DBE, the symbol ∠ means angle, B is the vertex, and BD and BE name its arms.

The symbol ° means degrees, the units used here to measure angles. A complete turn is 360°, a straight angle is 180°, and a right angle is 90°. An angle measures a turn, not the length of its drawn arms.

Property: Linear pairs and vertically opposite angles

A linear pair adds to 180°. Vertically opposite angles are equal. Complementary or supplementary angles need not be adjacent: those names describe their sums, whereas adjacency describes their positions.

Worked example 1. Two straight lines intersect. The four angles, named a, b, c and d in order around the intersection, include a = 120°. Find the other three angles. Here = means “is equal to”.

Answer: Angles a and b form a linear pair, so b = 180° − 120° = 60°, where − means subtraction. Angle c is vertically opposite a, so c = 120°. Angle d is vertically opposite b, so d = 60°.

To justify the equality, notice that b and d each complete the same angle a to a straight angle. Subtracting the same angle from 180° leaves the same remainder. This reasoning works without trusting the appearance of a sketch.

What happens when a transversal crosses parallel lines?

A plane is a flat surface extending in every direction. Parallel lines lie in the same plane and do not meet however far they are extended. Perpendicular lines meet at right angles. A transversal cuts two lines at distinct points.

The two intersections produce eight angles. Interior angles lie between the two lines; exterior angles lie outside them. Corresponding angles occupy matching positions at the intersections. Alternate interior angles lie between the lines on opposite sides of the transversal.

Property: Angles formed with parallel lines

• Corresponding angles are equal.
• Alternate interior angles are equal; alternate exterior angles, outside the lines on opposite sides of the transversal, are equal too.
• Interior angles on the same side of the transversal add to 180°.

Corresponding angles can be identified even when the lines are not parallel, but their equality requires parallel lines. Conversely, equal corresponding angles show that the two lines are parallel. First identify the positions, then decide which relationship applies.

Worked example 2. A transversal crosses two parallel lines. One angle is 135°. Find its corresponding angle, its vertically opposite angle and an adjacent angle at the same intersection.

Answer: The corresponding angle is 135° because the lines are parallel. The vertically opposite angle is also 135°. The adjacent angle forms a linear pair, so it is 180° − 135° = 45°.

Worked example 3. A transversal cuts two lines. At the first intersection, angles a and b form a linear pair and a = 120°. At the second intersection, angle f corresponds to b and f = 70°. Are the lines parallel?

Answer: b = 180° − 120° = 60°. The corresponding angles b and f measure 60° and 70°, so they are unequal. Therefore, the lines are not parallel.

Note: Lines drawn with thickness and measurements made with a protractor, an instrument for measuring angles, can introduce small errors. Exact geometrical relationships follow from reasoning about ideal lines.

How does the angle sum property help with triangles?

A triangle is a closed figure made from three line segments joining three vertices. The symbol △ means triangle. Thus △ABC has vertices A, B and C, sides AB, BC and CA, and interior angles at those vertices.

In a triangle, ∠A is a short name for the angle at vertex A, and similarly for ∠B and ∠C. When several angles meet at one vertex, use three letters to identify the intended angle precisely.

Property: The angle sum of a triangle

Definition: The angle sum property states that the three interior angles of a triangle add to 180°. For △ABC, ∠A + ∠B + ∠C = 180°, where + means addition.

The property can be explained using parallel lines. Draw a line through A parallel to BC. The two angles made beside ∠A on this line equal ∠B and ∠C by the alternate angle property. Together, the three angles form a straight angle.

Triangle ABC has B and C on the lower side and A above them. The line through X, A and Y is parallel to BC. Coloured angle markings connect the angles at B and C with the angles beside A.

See Fig. 7.7 in your NCERT textbook

Worked example 4. In △ABC, ∠B = 50° and ∠C = 70°. Find ∠A.

Answer: ∠A + 50° + 70° = 180°. The given angles total 120°, so ∠A = 180° − 120° = 60°. Checking gives 60° + 50° + 70° = 180°.

How are triangles classified?

An equilateral triangle has three equal sides. An isosceles triangle has two equal sides, and a scalene triangle has three different side lengths. The angles opposite equal sides are equal. Each angle of an equilateral triangle is 60°.

An acute angle is greater than 0° and less than 90°. An obtuse angle is greater than 90° and less than 180°. An acute-angled triangle has three acute angles; a right-angled triangle has one right angle; an obtuse-angled triangle has one obtuse angle.

How is an exterior angle of a triangle related to its interior angles?

Extend one side of a triangle beyond a vertex. The angle between this extension and the other side meeting there is an exterior angle of the triangle. Its neighbouring interior angle forms a linear pair with it.

The other two interior angles are called the opposite interior angles for this exterior angle. They are at the other vertices and are not adjacent to the exterior angle. Distinguishing these from the neighbouring interior angle is essential.

Property: Exterior angle of a triangle

An exterior angle equals the sum of its two opposite interior angles. In △ABC, extend BC beyond C to a point D. Then the exterior angle ∠ACD equals ∠A + ∠B.

Both the exterior angle and the sum of the opposite interior angles complete ∠ACB to 180°. Subtracting ∠ACB from both totals shows why the two quantities are equal. The exterior angle property therefore follows from the angle sum and linear pair properties.

Worked example 5. In △ABC, ∠A = 50° and ∠B = 60°. Side BC is extended beyond C to D. Find ∠ACB and the exterior angle ∠ACD.

Answer: ∠ACB = 180° − 50° − 60° = 70°. Since ∠ACB and ∠ACD form a linear pair, ∠ACD = 180° − 70° = 110°. Equivalently, ∠ACD = 50° + 60° = 110°.

Keep the extension in the description when naming an exterior angle. Without knowing which side is extended and beyond which vertex, a numerical answer can refer to a different exterior angle from the one intended.

How can Pythagoras’ theorem be verified?

Theorem: Pythagoras’ relation, verification only

In a right-angled triangle, the side opposite the right angle is the hypotenuse. Let a and b be the lengths of the two sides forming the right angle, and c the hypotenuse length. All lengths must use the same unit.

c² = a² + b². The superscript ² means “squared”: a² is a multiplied by itself. Pythagoras’ theorem states that the square of the hypotenuse length equals the sum of the squares of the other two side lengths.

The relation also compares areas, the amounts of surface covered by figures. A square constructed on the hypotenuse has an area equal to the combined areas of squares constructed on the other two sides. A square is a four-sided figure with equal sides and four right angles.

1. Draw a right-angled triangle and identify its right angle.
2. Measure the two sides forming that angle and the hypotenuse using the same unit.
3. Square each measured length and add the squares of the two shorter sides.
4. Compare this sum with the square of the hypotenuse length.

Worked example 6. Verify the relation for a right-angled triangle whose sides forming the right angle are 3 and 4 units and whose hypotenuse is 5 units.

Answer: 3² + 4² = 9 + 16 = 25, while 5² = 25. The two results agree, so these lengths verify the relation. The numbers are lengths measured in one common unit.

Verification checks the relation in particular cases. Drawing or measurement errors can prevent experimental values from agreeing exactly. Use the right-angle condition explicitly, and do not replace this verification activity with an assumption about an arbitrary triangle.

When can two figures be called congruent?

Congruent figures have exactly the same shape and size. Superimposition means placing one figure over another to test whether they fit exactly. A figure may be moved, turned or flipped before the comparison.

A circle consists of points at a fixed distance from its centre. That distance is its radius; the plural is radii. Two circles are congruent when their radii are equal: placing their centres together makes their boundaries coincide.

Corresponding parts and notation

Corresponding parts are the vertices, sides or angles that match. The symbol ≅ means “is congruent to”. Writing △ABC ≅ △XYZ specifies that A matches X, B matches Y, and C matches Z.

Consequently, AB matches XY, BC matches YZ, and AC matches XZ. The angles at matching vertices are equal too. Changing the order of the letters on one side alone can change the claimed correspondence.

The included angle lies between the two specified sides. The included side joins the vertices of the two specified angles. In AAS, the angle sum property supplies the third equal angle, allowing ASA to be used.

Worked example 7. Triangles ABC and XYZ satisfy AB = XY = 6 cm, AC = XZ = 5 cm and ∠A = ∠X = 30°. Here cm means centimetres. Establish their congruence.

Answer: The 30° angles lie between the stated 6 cm and 5 cm sides. Two pairs of sides and the included angles are equal. Therefore, △ABC ≅ △XYZ by SAS, with A matching X, B matching Y and C matching Z.

Worked example 8. In triangles ABC and XYZ, ∠A = ∠X = 35°, ∠C = ∠Z = 75° and BC = YZ = 4 cm. Explain the congruence.

Answer: ∠B = ∠Y = 180° − 35° − 75° = 70°. Thus ∠B = ∠Y, BC = YZ and ∠C = ∠Z. These give ASA, so △ABC ≅ △XYZ. The original information also satisfies AAS.

AAA, meaning three equal angle pairs, does not guarantee the same size. SSA, meaning two side pairs and a non-included angle pair, does not guarantee congruence in general. The right-angled case has its separate RHS criterion.

How can a parallel line be drawn through an outside point?

A construction is a geometrical drawing made by a specified sequence of steps using suitable instruments. To draw a parallel through an outside point, the completed line must satisfy both requirements: passing through the point and being parallel to the given line.

Using a ruler and set square

A ruler guides straight lines. A set square is a triangular drawing instrument with a right angle. Call the given line l and the outside point A; these letters are labels, not numerical values.

1. Place one edge of the set square along line l.
2. Hold a ruler against another edge of the set square to guide its movement.
3. Keep the ruler fixed and slide the set square until the first edge passes through A.
4. Draw along that edge through A and call the resulting line m.

The set square keeps the same orientation as it slides. The new line therefore makes the same corresponding angle with the ruler’s direction as the original line. Equal corresponding angles establish that l and m are parallel.

Checking the idea by folding

A paper-folding method begins with a crease representing l. Make a crease through A perpendicular to l and name it t. Then make a crease m through A perpendicular to t. Both l and m form right angles with t, so they are parallel.

Check the finished construction at A as well as along the lines. A line that is parallel but misses A does not complete the task. The two conditions should remain visible in the final labelled drawing.

How are triangles constructed from given measurements?

A triangle has six basic measurements: three side lengths and three angles. Suitable combinations determine its shape and size. A compass draws circles or parts of circles called arcs; its opening fixes the radius.

Construction from three sides

For an equilateral triangle ABC of side 4 cm, all three required distances are 4 cm. The base is simply the side chosen first for the drawing. The construction locates C by satisfying its distances from both A and B together.

1. Draw the base AB of length 4 cm.
2. With A as centre and radius 4 cm, draw a sufficiently long arc.
3. With B as centre and the same radius, draw another arc to meet the first. Call an intersection C.
4. Join AC and BC. Both have length 4 cm, so △ABC is the required equilateral triangle.

The same method constructs a triangle with three unequal sides. For AB = 4 cm, AC = 5 cm and BC = 6 cm, draw AB first. Draw an arc of radius 5 cm from A and one of radius 6 cm from B. Join their intersection C to A and B.

The triangle inequality requires each side to be shorter than the sum of the other two. Three lengths do not automatically make a triangle. Lengths 3 cm, 4 cm and 8 cm fail because 3 + 4 = 7, which is less than 8.

Construction from two sides and the included angle

To construct △ABC with AB = 5 cm, AC = 4 cm and ∠A = 45°, first draw AB. At A, draw a ray making 45° with AB. Mark C on that ray, 4 cm from A, then join BC.

The specified angle is between AB and AC. Drawing the angle at B instead would change the conditions. Marking the lengths and angle on a rough sketch first helps keep each measurement attached to the correct part.

Construction from two angles and the included side

To construct △ABC with AB = 5 cm, ∠A = 45° and ∠B = 80°, draw AB. On the same side of AB, draw rays making the required angles at A and B. Their intersection is C. The given side lies between the two angles.

The two specified angles must have a sum less than 180°. If their sum reaches or exceeds 180°, no positive third interior angle remains. The angle sum property is therefore a useful check before beginning this construction.

Note: Keep construction arcs visible and label the vertices. Check the finished side lengths and angles against the original instructions, rather than judging success from the overall appearance of the triangle.

What does reflection symmetry mean?

A figure has reflection symmetry when reflection in a line leaves it unchanged. A line of symmetry divides the figure so that folding along the line makes the two parts overlap exactly. Reflection produces a mirror image across that line.

Testing by folding

Trace the whole outline, fold along a proposed line, and compare the two parts. The edges must match throughout. A line passing through the centre is not sufficient by itself; the complete overlap is the test.

A square has four lines of symmetry. Two join the midpoints of opposite sides, and two are its diagonals. A midpoint divides a segment into equal parts. A diagonal joins vertices that are not next to each other.

A rectangle is a four-sided figure with four right angles. Every square is a rectangle, but a rectangle need not have four equal sides. A rectangle that is not a square does not have its diagonals as lines of symmetry. Folding along a diagonal fails to make the two parts overlap exactly.

Symmetry of a circle

A diameter is a line segment through a circle’s centre with both endpoints on the circle. Every diameter gives a line of reflection symmetry. Folding the circular outline across it makes the two semicircles, or half-circles, coincide.

Compare the whole figure, including any markings that form part of it. Checking only the outline can miss a mismatch in an internal pattern. Folding and tracing make the required correspondence easier to see than an unaided guess.

How is rotational symmetry different from reflection symmetry?

Rotational symmetry means that a figure matches itself after a turn of less than a full revolution. The fixed point about which it turns is the centre of rotation. An angle of rotational symmetry is a turn that brings the figure onto itself.

Every figure returns to its original position after 360°. If this is its only matching turn, it does not have rotational symmetry in the usual sense. A smaller matching turn is needed.

Order and angles of symmetry

The order of rotational symmetry counts the matching positions during one complete turn, including the return at 360° and excluding the starting position at 0°. For a square, the matching turns are 90°, 180°, 270° and 360°, giving order 4.

To test a figure, trace and cut out a copy. Keep the original fixed, place the copy over it, and turn the copy about the chosen centre. Record a match only when the whole figure coincides.

Most figures have a smallest angle of rotational symmetry, but a circle is an exception. Rotating a circle about its centre through any angle brings it onto itself. There is no smallest positive matching turn for a circle.

Reflection uses a line; rotation uses a fixed point. A figure can show rotational symmetry without having a line of reflection symmetry. The two tests should therefore be performed separately.

How do faces, edges and nets describe solid shapes?

A two-dimensional figure lies in a plane. A three-dimensional solid occupies space and has length, breadth and height. A face, in the flat-face convention used here, is a plane part of a solid’s boundary.

An edge of a cube or cuboid is a line segment where faces meet, and a vertex is a corner where edges meet. A cube has square faces; a cuboid has rectangular faces. Count the whole solid, including parts hidden in a drawing.

A cylinder has two equal circular flat faces and one curved surface, with two circular rims and no vertices. A cone has one circular flat base, one curved surface, one circular rim and one pointed vertex, called its apex.

The circular rims are curved boundaries, not straight edges. If curved surfaces are counted together with flat faces as “surfaces”, a cylinder has three and a cone has two. State the convention instead of silently changing the meaning of face or edge.

Unfolding solids into nets

A net is a flat arrangement that folds into the surface of a solid. A cube’s net contains six squares. A cuboid’s net contains six rectangles. Extra tabs used for sticking a paper model together are not additional faces.

Six equal squares form a cross-like arrangement. Four squares run in a vertical strip, with one square attached on either side of the third square from the top.

See Fig. 4.1 in your NCERT textbook

The cylinder’s net contains a rectangle for the curved surface and two equal circles for its flat faces. For a cone, the curved surface opens into a sector, a portion of a circle bounded by two radii and an arc. Its base is a circle.

Test a proposed net by imagining the folds, then by folding a paper cut-out if needed. Having the correct number of pieces is not enough if they overlap or fail to close the solid.

How can simple maps represent the space around us?

A map represents the arrangement of places on a flat surface. A simple map can show a house, a village or a route from home to school. Its purpose is to communicate positions and connections clearly.

Making an approximate map

Visual estimation means judging position or distance by looking, without claiming an exact measurement. An approximate map uses these estimates to represent the space around us. It should not be treated as an exact record of every distance.

1. Choose the space or route to represent and observe its main features.
2. Identify which places connect and where the route changes direction.
3. Sketch their relative positions on the page, using visual estimates.
4. Label the places and check that the route can be followed from the drawing.

A landmark is a recognisable feature used to locate a place or follow a route. Select features that help explain the actual arrangement. Include the start and destination when drawing a route, so that the sketch communicates a complete journey.

Read before tracing: identify the labels, locate the starting place and follow each connection in order. When checking your own sketch, compare it with the observed space. Correct misplaced features or missing connections while retaining its approximate character.

Glossary

• Complementary angles — Two angles whose measures add to a right angle of 90°.
• Supplementary angles — Two angles whose measures add to a straight angle of 180°.
• Adjacent angles — Angles sharing a vertex and an arm, with non-overlapping interiors on opposite sides of that arm.
• Linear pair — Two adjacent angles whose non-common arms form a straight line.
• Vertically opposite angles — Opposite angles formed when two straight lines intersect at a point.
• Transversal — A line that intersects two other lines at distinct points.
• Corresponding angles — Angles occupying matching positions where a transversal intersects two lines.
• Hypotenuse — The side opposite the right angle in a right-angled triangle.
• Congruence — Equality of shape and size, allowing figures to fit exactly through superimposition.
• Included angle — The angle lying between the two specified sides of a triangle.
• Line of symmetry — A line along which folding makes the two parts of a figure overlap exactly.
• Rotational symmetry — Matching of a figure with itself after a turn smaller than a complete revolution.
• Net — A flat arrangement of surface pieces that folds into a solid.
• Visual estimation — Judging positions or distances by looking without claiming exact measurement.

Common errors and misconceptions

• Misconception: Supplementary angles must be next to one another. Correct: Their measures must total 180°; adjacency is a separate condition. A linear pair has both features.
• Misconception: Corresponding angles are equal whenever a transversal is drawn. Correct: Their equality depends on the two lines being parallel; check that condition before using the rule.
• Misconception: An exterior angle equals the neighbouring interior angle. Correct: It equals the sum of the two opposite interior angles and forms a linear pair with its neighbour.
• Misconception: Pythagoras’ relation applies to every triangle. Correct: It relates the sides of a right-angled triangle, with the hypotenuse opposite the right angle.
• Misconception: Equal angles alone guarantee congruent triangles. Correct: Triangles can have equal corresponding angles but different sizes. Use a valid congruence criterion with the required side information.
• Misconception: Any three side lengths can make a triangle. Correct: Each length must be less than the sum of the other two; equality does not produce a triangle.
• Misconception: A matching full turn alone establishes rotational symmetry. Correct: Every figure matches after 360°; rotational symmetry requires a matching turn smaller than a full revolution.
• Misconception: Curved surfaces and circular rims are flat faces and straight edges. Correct: Name these separately when describing a cylinder or cone, and make your counting convention clear.

Exam-style questions with model answers

Q1. Angles a and b form a linear pair. If a = 120°, find b and state the reason. [2 marks]

1. A linear pair adds to 180° because its non-common arms form a straight line.
2. Therefore, b = 180° − 120° = 60°.

Q2. A transversal cuts two parallel lines. One angle is 135°. Find its corresponding angle, its vertically opposite angle and an adjacent angle at the same intersection. Explain each answer. [3 marks]

1. The corresponding angle is 135°. Corresponding angles occupy matching positions at the two intersections, and they are equal because the given lines are parallel.
2. The vertically opposite angle is 135°, since opposite angles at the intersection of two straight lines are equal.
3. The adjacent angle is 180° − 135° = 45°, because it forms a linear pair with the given angle.

Q3. In triangle ABC, ∠A = 50° and ∠B = 60°. Side BC is extended beyond C to D. Find ∠ACB and ∠ACD, stating the properties used. [4 marks]

1. The angle sum property gives ∠A + ∠B + ∠ACB = 180° for the three interior angles of triangle ABC.
2. Substituting the given measures gives ∠ACB = 180° − 50° − 60° = 70°.
3. Since B, C and D lie on the extended side, ∠ACB and ∠ACD form a linear pair.
4. Hence ∠ACD = 180° − 70° = 110°, also equal to the sum of the two opposite interior angles.

Q4. A right-angled triangle has perpendicular sides of 3 and 4 units and a hypotenuse of 5 units. Verify Pythagoras’ relation and explain the scope of this check. [3 marks]

1. The two sides forming the right angle have lengths 3 and 4 units. Their squared lengths add to 3² + 4² = 9 + 16 = 25.
2. The side opposite the right angle is the hypotenuse, whose squared length is 5² = 25. Both sides of the relation therefore agree.
3. This calculation verifies the relation for the stated lengths. It is a numerical check of this case, rather than a proof for every right-angled triangle.

Q5. Triangles ABC and XYZ satisfy AB = XY = 6 cm, AC = XZ = 5 cm and ∠A = ∠X = 30°. Establish their congruence and identify the corresponding vertices. [3 marks]

1. The corresponding side pairs AB and XY, and AC and XZ, have equal lengths of 6 cm and 5 cm respectively.
2. The equal angles at A and X are included between those side pairs. Thus the Side Angle Side, or SAS, condition applies.
3. Therefore, △ABC ≅ △XYZ. The matching vertices are A with X, B with Y, and C with Z, as recorded by the order of the letters.

Q6. Describe how to construct an equilateral triangle ABC of side 4 cm with a ruler and compass. Explain why the construction gives the required triangle. [5 marks]

1. Draw the line segment AB of length 4 cm with the ruler. This fixes the first side and the two centres needed for the arcs.
2. Set the compass opening to 4 cm. With A as centre, draw a sufficiently long arc on the chosen side of AB.
3. Keep the same compass opening and draw another arc with B as centre. Name a point where the two arcs intersect C.
4. Join AC and BC using the ruler. These two segments and AB make the closed figure triangle ABC.
5. The construction gives AC = 4 cm and BC = 4 cm because C lies on both arcs. Together with AB = 4 cm, this makes all three sides equal, as required.

Q7. A square is rotated about its centre. State all matching turns greater than 0° and up to 360°, and give its order of rotational symmetry. [2 marks]

1. The square matches itself after turns of 90°, 180°, 270° and 360° about its centre.
2. There are four matching positions during one complete turn, so its order of rotational symmetry is 4.

Q8. State the numbers of flat faces, straight edges and vertices of a cube, and describe what its net contains. [4 marks]

1. A cube has six flat faces. Each face is a square, and the count includes the faces hidden from view in a sketch.
2. It has twelve straight edges, where neighbouring square faces meet along line segments.
3. It has eight vertices, which are the corner points at which edges meet.
4. Its net contains six equal squares arranged so that they fold into the cube. Extra sticking tabs are not additional faces.

Key takeaways

• A linear pair totals 180°, while vertically opposite angles formed by two intersecting straight lines are equal.
• For parallel lines, corresponding and alternate angles are equal, and same-side interior angles total 180°.
• A triangle’s interior angles total 180°; an exterior angle equals the sum of its two opposite interior angles.
• Verify Pythagoras’ relation using a right-angled triangle, identifying the hypotenuse and measuring every length in the same unit.
• Congruence requires equal shape and size; state the criterion and preserve the order of corresponding vertices.
• Check the supplied measurements before constructing a triangle, especially the triangle inequality and the location of an included angle.
• Reflection tests a fold or mirror line; rotational symmetry tests matching turns about a fixed centre.
• Distinguish flat faces from curved surfaces, test nets by folding, and keep visually estimated maps explicitly approximate.

Test yourself

Must complementary angles share a vertex?

No. Complementary angles are defined by their total of 90°, not by being adjacent or sharing a vertex.

When can equal corresponding angles establish parallel lines?

When a transversal intersects two lines at distinct points and a pair of corresponding angles is equal, the lines are parallel.

In a triangle, two angles are 50° and 70°. What is the third?

The third angle is 180° − 50° − 70° = 60°, using the triangle angle sum property.

Which side of a right-angled triangle is its hypotenuse?

The hypotenuse is the side opposite the right angle, rather than either side forming that angle.

What extra condition makes two sides and an angle sufficient for SAS?

The given angle must be the included angle between the two corresponding sides whose lengths are known to be equal.

Can lengths 3 cm, 4 cm and 8 cm form a triangle?

No. The sum 3 + 4 = 7 is less than 8, so the triangle inequality fails.

Why is a circle exceptional when considering the smallest angle of rotational symmetry?

A circle matches itself after rotation through every angle about its centre, so it has no smallest positive matching angle.

What pieces form a cylinder’s net?

Its net has a rectangle representing the curved surface and two equal circles representing its flat faces.