S1 Lines, angles and polygons

S1 Lines, angles and polygons
KS4 Mathematics S1 Lines, angles and polygons

S1.1 Parallel lines and angles
Contents S1 Lines, angles and polygons S1.1 Parallel lines and angles S1.2 Triangles S1.3 Quadrilaterals S1.4 Angles in polygons S1.5 Congruence S1.6 Similarity

Labelling line segments
When a line has end points we say that it has finite length. It is called a line segment. We usually label the end points with capital letters. For example, this line segment A B has end points A and B. We can call this line, line segment AB.

Lines in a plane A flat two-dimensional surface is called a plane.
Any two straight lines in a plane either intersect once … This is called the point of intersection. Ask pupils how we could draw two infinitely long lines that will never meet. The answer would be to draw them in different planes. We can imagine, for example, one plane made by one wall in the room and another plane made by the opposite wall. If we drew a line on one wall and a line on the other, they would never meet, even if the walls extended to infinity. When two lines intersect two pairs of equal angles are formed.

Lines in a plane … or they are parallel.
We use arrow heads to show that lines are parallel. Parallel lines will never meet. They stay an equal distance apart. Pupils should be able to identify parallel and perpendicular lines in 2-D and 3-D shapes and in the environment. For example: rail tracks, double yellow lines, door frame or ruled lines on a page. This means that they are always equidistant.

Vertically opposite angles
When two lines intersect, two pairs of vertically opposite angles are formed. a b c d a = c and b = d Vertically opposite angles are equal.

What is special about the angles at the point of intersection here?
Perpendicular lines What is special about the angles at the point of intersection here? a b a = b = c = d = 90 d c Pupils should be able to explain that perpendicular lines intersect at right angles. Lines that intersect at right angles are called perpendicular lines.

The distance from a point to a line
What is the shortest distance from a point to a line? O Ask pupils to point out which line they think is the shortest and ask them what they notice about it. Ask pupils if they think that the shortest line from a point to another line will always be at right angles. Reveal the rule. The shortest distance from a point to a line is always the perpendicular distance.

Angles When two lines meet at a point an angle is formed. a
An angle is a measure of rotation from one of the line segments to the other. Remind pupils that in mathematics a positive rotation is anticlockwise. We often label angles using lower-case letters or Greek letters such as , theta.

Labelling lines, angles and shapes
Sometimes, the vertices in a diagram are labelled with capital letters. For example, This is line segment AD. This is angle ABD, ABD or ABD. A D B Discuss how the letters at the vertices can be used to label angles, line segments and shapes. It is the labeling of angles that pupils often find most difficult. Explain that there are two angles at vertex B. We have to use three letters to distinguish between angle ABD and angle CBD. We could also have angle ABC which is equal to the sum of angles ABD and CBD. It is the middle letter that tells us the vertex containing the angle. Ask pupils to tell you what the quadrilateral is called using the letters at its vertices. C This is triangle BCD or BCD .

Types of angle a a a a Acute angle 0º < a < 90º Right angle
Obtuse angle 90º < a < 180º a Reflex angle 180º < a < 360º a Pupils should be able to classify angles acorrding to whether they are acute, right-angled, obtuse or reflex.

Angles on a straight line and at a point
a + b + c = 180° a b because there are 180° in a half turn. Angles on a line add up to 180 a + b + c + d = 360 because there are 360 in a full turn. Angles at a point add up to 360 a b c d This slide is a reminder of rules established at KS3.

Complementary and supplementary angles
b a + b = 90° Two complementary angles add up to 90°. Two supplementary angles add up to 180°. a b a + b = 180° Ask pupils to give examples of pairs of complementary angles. For example, 32° and 58º. Give pupils an acute angle and ask them to calculate the complement to this angle. Ask pupils to give examples of pairs of supplementary angles. For example, 113° and 67º. Give pupils an angle and ask them to calculate the supplement to this angle.

Angles made with parallel lines
When a straight line crosses two parallel lines eight angles are formed. a b d c e f h This line is called a traversal. g Ask pupils to give any pairs of angles that they think are equal and to explain their choices. Which angles are equal to each other?

Corresponding, alternate and interior angles
Corresponding angles are equal a b a = b Alternate angles are equal a b a = b Interior angles add up to 180° a b a + b = 180° Tell pupils that these are called corresponding angles because they are in the same position on different parallel lines. Look for an F-shape Look for a Z-shape Look for a C- or U-shape

Angles and parallel lines

Hint: Add another parallel line.
Calculating angles Calculate the size of angle a. 28º a Hint: Add another parallel line. 45º Ask pupils to explain how we can calculate the size of angle a using what we have learnt about angles formed when lines cross parallel lines. If pupils are unsure reveal the hint. When a third parallel line is added we can deduce that a = 28º + 45º = 73º using the equality of alternate angles. a = 28º + 45º = 73º

Naming triangles Three equal sides and three equal angles.
Equilateral triangle Isosceles triangle Two equal sides and two equal angles. No equal sides and no equal angles. Scalene triangle Review the names of these three types of triangle. State that an equilateral triangle is a special name for a regular triangle. Any regular polygon has equal sides and equal angles. Ask pupils to tell you the size of the angles in an equilateral triangle. Ask pupils to give the symmetry properties of each shape.

Naming triangles Triangles can also be named according to their angles. Contains a right angle. Right-angled triangle Acute-angled triangle Contains three acute angles Contains an obtuse angle. Obtuse-angled triangle Ask pupils to explain why it is impossible for a triangle to contain more than one obtuse angle or to contain a reflex angle.

Angles in a triangle For any triangle, c b a a + b + c = 180°
Remind pupils that the angles in a triangle add up to 180°. a + b + c = 180°

Interior and exterior angles in a triangle
Any exterior angle in a triangle is equal to the sum of the two opposite interior angles. c c a b b a = b + c We can prove this by constructing a line parallel to this side. These alternate angles are equal. These corresponding angles are equal.

Interior and exterior angles in a triangle
Drag the vertices of the triangle to show that the exterior angle is equal to the sum of the opposite interior angles. Hide angles by clicking on them and ask pupils to calculate their sizes. Be aware that the angles are rounded to the nearest degree and that this may cause a slight error of ±1°.

Calculating angles Calculate the size of the lettered angles in each of the following triangles. 116° b 33° a 64° 82° 31° 34° 43° c 25° d 152° 131° 127° 272°

Quadrilaterals Quadrilaterals are also named according to their properties. They can be classified according to whether they have: Equal and/or parallel sides Equal angles Right angles Diagonals that bisect each other Diagonals that are at right angles List the properties that we use to classify shapes. Some of these properties define the shape. These are the minimum requirements needed to define the shape. Other properties, such as symmetry properties, are derived properties and happen as a result of the definition. Line symmetry Rotational symmetry

Parallelogram In a parallelogram opposite sides are equal
and parallel. Draw pupils’ attention to the convention of using double dashes to distinguish between the two pairs of equal sides and the use of double arrow heads to distinguish between two pairs of parallel sides. State that when two lines bisect each other, they cut each other into two equal parts. Ask pupils for other derived properties such as the fact that the opposite angles are equal and adjacent angles add up to 180º. Stress, however that a parallelogram has no lines of symmetry. Ask pupils if they know the name of a parallelogram that has four right angles (a rectangle), a parallelogram that has four right angles and four equal sides (a square) and a parallelogram with four equal sides (a rhombus). We can think of a parallelogram as a slanted rectangle. The diagonals of a parallelogram bisect each other. A parallelogram has rotational symmetry of order 2.

Rhombus A rhombus is a parallelogram with four equal sides.
Ask pupils for other derived properties such as the fact that the opposite angles are equal. Ask pupils if they know the name of a rhombus that has four right angles (a square). We can think of a rhombus as a slanted square. The diagonals of a rhombus bisect each other at right angles. A rhombus has two lines of symmetry and it has rotational symmetry of order 2.

Rectangle A rectangle has opposite sides of equal length
and four right angles. Ask pupils for other derived properties such as the fact that the diagonals are of equal length and bisect each other. Ask pupils to explain why it is possible to describe a square as a special type of parallelogram. A rectangle has two lines of symmetry.

Square A square has four equal sides and four right angles.
Ask pupils for other derived properties such as the fact that the diagonals are of equal length and bisect each other at right angles. Ask pupils to explain why it is possible to describe a square as a special type of parallelogram, a special type of rhombus or a special type of rectangle. and rotational symmetry of order 4. It has four lines of symmetry

Trapezium A trapezium has one pair of opposite sides that are parallel. A trapezium has one line of symmetry when the pair of non-parallel opposite sides are of equal length. It can never have rotational symmetry. Can a trapezium have any lines of symmetry? Can a trapezium have rotational symmetry?

Isosceles trapezium In an isosceles trapezium the two opposite non-parallel sides are the same length. Ask pupils for other derived properties such as the fact that there are two pairs of equal adjacent angles. The diagonals of an isosceles trapezium are the same length. It has one line of symmetry.

Kite A kite has two pairs of adjacent sides of equal length.
Ask pupils for other derived properties such as the fact that there is one pair of opposite angles that are equal. Ask pupils if a kite can ever have parallel sides. Conclude that this could only happen if the four sides were of equal length, in which case it would no longer be a kite, but a rhombus. The diagonals of a kite cross at right angles. A kite has one line of symmetry.

Arrowhead An arrowhead or delta has two pairs of adjacent sides of equal length and one interior angle that is more than 180°. Ask pupils for other derived properties such as the fact that one pair of angles is equal. Ask pupils if a kite can ever have parallel sides. The answer is no. Its diagonals cross at right angles outside the shape. An arrowhead has one line of symmetry.

Quadrilaterals on a 3 by 3 pegboard
Challenge pupils to find the 16 distinct quadrilaterals (not including reflections, rotations and translations) that can be made on a 3 by 3 pegboard. Classify them according to their side, angle and symmetry properties.

Polygons A polygon is a 2-D shape made when line segments enclose a region. A The end points are called vertices. One of these is called a vertex. B The line segments are called sides. E C D These two dimensions are length and width. A polygon has no height. 2-D stands for two-dimensional.

Polygons A regular polygon has equal sides and equal angles.
In a convex polygon all of the interior angles are less than 180°. All regular polygons are convex. Define these key terms. Explain that in a concave polygon some of the interior angles are reflex angles. In a concave polygon some of the interior angles are more than 180°.

Polygons are named according to their number of sides.
Naming polygons Polygons are named according to their number of sides. Number of sides Name of polygon 3 4 5 6 7 8 9 10 Triangle Quadrilateral Pentagon Hexagon Heptagon Pupils should learn any of the names that they do not know already. Octagon Nonagon Decagon

Interior angles in polygons
The angles inside a polygon are called interior angles. c a b The sum of the interior angles of a triangle is 180°.

Exterior angles in polygons
When we extend the sides of a polygon outside the shape exterior angles are formed. e d f

Sum of the interior angles in a quadrilateral
What is the sum of the interior angles in a quadrilateral? d c f a e b We can work this out by dividing the quadrilateral into two triangles. a + b + c = 180° and d + e + f = 180° So, (a + b + c) + (d + e + f ) = 360° The sum of the interior angles in a quadrilateral is 360°.

Sum of interior angles in a polygon
We already know that the sum of the interior angles in any triangle is 180°. c a + b + c = 180 ° a b We have just shown that the sum of the interior angles in any quadrilateral is 360°. a b c d Pupils may remember the rule for finding the sum of the interior angles in a polygon from Key Stage 3 work. a + b + c + d = 360 ° Do you know the sum of the interior angles for any other polygons?

Sum of the interior angles in a polygon
We’ve seen that a quadrilateral can be divided into two triangles … … a pentagon can be divided into three triangles … Explain that to find the smallest number of triangles that a polygon can be divided into, we draw straight lines from one vertex to each of the other vertices. We cannot produce a triangle by joining a vertex to either of the two verticies adjacent to it. Ask pupils how many triangles a heptagon can be divided into. What about a dodecagon, a shape with 12 sides? … and a hexagon can be divided into four triangles. How many triangles can a hexagon be divided into?

Sum of the interior angles in a polygon
The number of triangles that a polygon can be divided into is always two less than the number of sides. We can say that: A polygon with n sides can be divided into (n – 2) triangles. The sum of the interior angles in a triangle is 180°. So, Pupils should learn this rule. The sum of the interior angles in an n-sided polygon is (n – 2) × 180°.

Interior angles in regular polygons
A regular polygon has equal sides and equal angles. We can work out the size of the interior angles in a regular polygon as follows: Name of regular polygon Sum of the interior angles Size of each interior angle Equilateral triangle 180° 180° ÷ 3 = 60° Square 2 × 180° = 360° 360° ÷ 4 = 90° Ask pupils to complete the table for regular polygons with up to 10 sides. Regular pentagon 3 × 180° = 540° 540° ÷ 5 = 108° Regular hexagon 4 × 180° = 720° 720° ÷ 6 = 120°

The sum of exterior angles in a polygon
For any polygon, the sum of the interior and exterior angles at each vertex is 180°. For n vertices, the sum of n interior and n exterior angles is n × 180° or 180n°. The sum of the interior angles is (n – 2) × 180°. We can write this algebraically as 180(n – 2)° = 180n° – 360°.

The sum of exterior angles in a polygon
If the sum of both the interior and the exterior angles is 180n° and the sum of the interior angles is 180n° – 360°, the sum of the exterior angles is the difference between these two. The sum of the exterior angles = 180n° – (180n° – 360°) = 180n° – 180n° + 360° Discuss this algebraic proof that the sum of the exterior angles in a polygon is always 360°. = 360° The sum of the exterior angles in a polygon is 360°.

Find the number of sides
Challenge pupils to find the number of sides in a regular polygon given the size of one of its interior or exterior angles. Establish that if we are given the size of the exterior angle we have to divide this number into 360° to find the number of sides. This is because the sum of the exterior angles in a polygon is always 360° and each exterior angle is equal. Establish that if we are given the size of the interior angle we have to divide 360° by (180° – the size of the interior angle) to find the number of sides. This is because the interior angles in a regular polygon can be found by subtracting 360° divided by the number of sides from 180°.

Congruence If shapes are identical in shape and size then we say they are congruent. Congruent shapes can be mapped onto each other using translations, rotations and reflections. These triangles are congruent because AB = PQ, BC = QR, A B C R P Q Stress that if two shapes are congruent their corresponding lengths and angles are the same. and AC = PR. A = P, B = Q, and C = R.

Congruent triangles Two triangles are congruent if the satisfy the following conditions: Side, side side (SSS) 1) The three sides of one triangle are equal to the three sides of the other. Stress that the conditions outlined on the following slides are the minimum conditions required to be sure that two triangles are congruent. If two triangles satisfy these minimum conditions, we can deduce that all the sides and all the angles are equal. We call this first rule side, side, side or SSS for short.

Congruent triangles Two triangles are congruent if the satisfy the following conditions: 2) Two sides and the included angle in one triangle are equal to two sides and the included angle in the other. Side, angle, side (SAS) We call this rule side, angle, side or SAS for short.

Congruent triangles Two triangles are congruent if the satisfy the following conditions: Angle, angle, side (AAS) 3) Two angles and one side of one triangle are equal to the corresponding two angles and side in the other. We call this rule angle, angle, side or AAS for short. This can be written as ASA if the corresponding side is between the two angles.

Congruent triangles Two triangles are congruent if the satisfy the following conditions: 4) The hypotenuse and one side of one right-angled triangle is equal to the hypotenuse and one side of another right-angled triangle. Stress that both triangles must have a right angle. We call this rule right angle, hypotenuse, side or RHS for short. Right angle, hypotenuse, side (RHS)

Congruent triangles

Similar shapes If one shape is an enlargement of the other then we say the shapes are similar. The angle sizes in two similar shapes are the same and their corresponding side lengths are in the same ratio. A similar shape can be a reflection or a rotation of the original. These triangles are similar because Give an example of what is meant by the corresponding side lengths being in the same ratio. For example, if the lengths of all the sides in the second shape are double the lengths of the corresponding sides in the first shape then the ratios of the corresponding side lengths will all simplify to 2. This number is the scale factor for the enlargement. A B C R P Q A B C R P Q A B C R P Q A B C R P Q A B C R P Q A B C R P Q A B C R P Q A =  P, B = Q, and C = R. PQ AB = QR BC = PR AC

Similar shapes Which of the following shapes are always similar?
Any two rectangles? Any two squares? Any two isosceles triangles? Any two circles? Any two equilateral triangles? Establish that any two regular polygons with the same number of sides are mathematically similar as are any two circles. Any two regular polyhedra (of the same number of sides) are also similar. Any two cubes? Any two cylinders?

Finding the scale factor of an enlargement
We can find the scale factor for an enlargement by finding the ratio between any two corresponding lengths. Scale factor = length on enlargement corresponding length on original If a shape and its enlargement are drawn to scale, the the two corresponding lengths can be found using a ruler. Always make sure that the two lengths are written using the same units before dividing them.

Finding the scale factor of an enlargement
The following rectangles are similar. What is the scale factor for the enlargement? 6 cm 9 cm The scale factor for the enlargement is 9/6 = 1.5

Finding the lengths of missing sides
The following shapes are similar. What is the size of each missing side and angle? 3.6 cm c 3 cm 37° 6 cm 37° a 4.8 cm 5 cm 4 cm e 53° b Mare sure that pupils realize that all corresponding angles in similar shapes are equal. Point out that to scale from the left-hand shape to the right-hand shape we multiply by 6/5 (or 1.2). In other words, we multiply by 6 and divide by 5. To scale from right-hand shape to the left-hand shape we multiply by the reciprocal 5/6 (or divide by 1.2). in other words we multiply by 5 and divide by 6. 53° 6 cm 7.2 cm d The scale factor for the enlargement is 6/5 = 1.2

Finding lengths in two similar shapes
Start by working out the scale factor for the enlargement. Use the scale factor to calculate the lengths of the missing sides before revealing them.

Similar triangles 1 Discuss why, if BC and DE are parallel, the interior angels of triangle ABC and triangle ADE are the same. Conclude that triangles ABC and ADE are similar. Drag points D and E to modify the angles and lengths. BC and DE will remain parallel. Show that the ratio of the side lengths remains constant. Hide some of the side lengths and modify the shape. Ask pupils to calculate the missing side lengths. Change the ratio between the two similar triangles and repeat the activity.

Similar triangles 2 Discuss why, if AB and DE are parallel, the interior angels of triangle ABC and triangle CDE are the same. Conclude that triangles ABC and CDE are similar. Drag points A, B, D and E to modify the angles and lengths. AB and DE will remain parallel. Calculate the ratio of the corresponding side lengths to find the scale factor for the enlargement. Hide some of the side lengths and modify the shape. Ask pupils to calculate the missing side lengths.

Using shadows to measure height
In ancient times, surveyors measured the height of tall objects by using a stick and comparing the length of its shadow to the length of the shadow of the tall object.

Using shadows to measure height
Stress that if the stick and the tree are close to each other the ratio between the stick’s height and its shadow is the same as the ratio between the tree’s height and its shadow. The length of the shadows depends on the position of the sun. Discuss how we find the height of the tree given the height of the stick and the lengths of the two shadows. Change the heights and shadow lengths to modify the problem.

S1 Lines, angles and polygons

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