Maths Hub angles.notebook. June 04, Angles. Angles. Mar 18 3:28 pm. Mar 18 3:28 pm. Mar 18 3:28 pm

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Angles Aims of the session To explore the different ways we work with angles including: the notation & language Types of angles Using ICT to explore the types of angles formed on parallel lines Proof

1 Angles Aims of the session To explore the different ways we work with angles including: the notation & language Types of angles Using ICT to explore the types of angles formed on parallel lines Proof sum of the interior angles of a triangle =180 o ; the exterior angle of a triangle equals the sum of the 2 interior opposite angles Polygons types of polygon; derive the formula for finding the sum of the interior angles of any polygon What are learners' common difficulties/misconceptions? Angles The Notation & Language of Angles. an angle is a measure of amount of turn angles are measured in degrees or radians an acute angle right angle half turn (straight angle) obtuse angle reflex angle complete turn 1

2 The Notation & Language of Angles. To identify and name an angle, different notations are used for labelling, for example, the angle opposite can be referred to as 'angle CAB', or 'angle BAC' and written BAC ^ or LBAC or referred to as the angle 'theta' and labelled θ. Alternatively, we can identify an angle by using the letter/label at the vertex but with a hat on. Thus the angle θ opposite is denoted as ^A, and referred to as 'angle A'. Note: the latter notation might lead to some ambiguity if there is more than one angle at the vertex.... for you to try! Types of angles angles at a point add up to 360 o angles on a straight line add up to 180 o. vertically opposite angles are equal. Types of angles Angles formed on parallel lines include corresponding angles (the F-angle): corresponding angles are equal alternate angles (the Z-angle): alternate angles are equal interior angles (allied/c-angles): interior angles are supplementary. 2

3 Exterior & Interior Angles of a Triangle The angles inside a triangle are called interior angles. The diagram below shows the interior and exterior angles of a triangle. exterior angle a b interior angles exterior angle exterior angle c Proofs The sum of the interior angles of a triangle = 180 o. Prove The sum of the 2 interior opposite angles of a triangle is equal to the exterior angle. Prove Proof 1 Given: ΔXYZ, prove that ^x + ^y + ^z = 180o. A Y B C X Z D Proof: produce XZ (red line) as shown on the right. Through vertex Y draw a line (blue line) parallel to XZ. Label the lines. Ð AYX = Ð YXZ = x (alternate angles are equal) Ð BYZ = Ð XZY = z (alternate angles are equal) Ð AYX + Ð XYZ + Ð BYZ = 180 o (angles on a straight line) i.e. x + y + z = 180 o (Angles in a triangle add up to 180 o ) 3

4 2 interior opposite angles Proof 2 Theorem: If the side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles: To prove: ACD ^ = ABC ^ + BAC. ^ A exterior angle B C D Proof: ACD ^ + ACB ^ = 180 o (angles on a straight line)... (1) BAC ^ + ABC ^ + ACB ^ = 180 o (angles in a triangle)... (2) Equating (1) & (2) gives ACD ^ + ACB ^ = BAC ^ + ABC ^ + ACB ^ ACD ^ = BAC ^ + ABC ^ Conclusion: The exterior angle of a triangle is equal to the sum of the two interior opposite angles. Application of Proof 2 Find the unknown angle or marked angles in the diagrams below. (a) (d) Polygons what is a polygon? Any 2-d shape that is closed and bounded by straight lines. a b A polygon can be regular or irregular. What is a regular polygon? c A polygon with sides of equal length and angles of equal size. Common examples of regular polygons 4

5 Naming Polygons Polygons are named according to the number of sides (See Table below). No of sides Name of polygon No of triangles Sum of Interior angles Sum of Exterior angles 3 Triangle Quadraliteral Pentagon Hendecagon/ Undecagon Dodecagon n n sided (n 2) (n 2)x A Sum of the Interior angles of a polygon = 180(n-2). Investigate Start with the vertex A and connect it to all other vertices. Notice that it is already connected to B and E. Three triangles are B 1 3 E C 2 D formed. We know that the sum of the angles in a triangle is 180 o. So for the 3 triangles we have 3 x 180 = 540. Our polygon is a pentagon which has 5 sides. The number of triangles formed by connecting the vertices is 3, i.e. the number of sides take away 2. This pattern is constant for ALL polygons. So we can make a general statement: a polygon with n-sides will contain (n-2) triangles and the sum of its interior angles is 180(n-2). Using the formula 180(n-2) There are two types of problems that arise when using the formula: 1. Questions that ask you to find the number of degrees in the sum of the interior angles of a given polygon. 2. Questions that ask you to find the number of sides of a polygon. Example 1. Find the number of degrees in the sum of interior angles of an octagon. Solution: An octagon has 8 sides so n = 8. Using the formula above, 180(n 2) = 180(8 2) = 180(6) = 1080 degrees. Example 2. How many sides does a polygon have if the sum of its interior angles is 900? Solution: Since the sum of the interior angles is given, set the formula above equal to 900 and solve for n. 180(n 2) = 900 (Divide both sides by 10 and then by 9 and then by 2) n 2 = 5 (Add 2 to both sides) n = 7, so the polygon has 7 sides. 5

Exterior angles of a polygon The sum of the exterior angles of any polygon is 360 o. For a regular polygon: each exterior angle = 360/n, where n is the number of sides.

6 Exterior angles of a polygon The sum of the exterior angles of any polygon is 360 o. For a regular polygon: each exterior angle = 360/n, where n is the number of sides. Example 1: Find the exterior angle of a regular nonagon. Solution: A nonagon has 9 sides and so n = 9. Using the formula: exterior angle = 360/n = 360/9 = 40 So each exterior angle of a regular nonagon is 40 degrees. Example 2: The exterior angle of a regular polygon is 15 degrees. How many sides has the polygon? Solution: Set the formula equal to 15 and solve for n. 360/n = 15 (Multiply both sides by n) 15n = 360 (Divide both sides by 15) n = 24, so the polygon has 24 sides. Another useful relationship: The sum of the exterior and interior angle of any polygon is 180. Learners' Common Difficulties & Misconceptions. 1. Using a protractor Learners have the following misconceptions when using a protractor A protractor must always be placed in the horizontal position, regardless of the orientation of the angle being measured. When measuring angles always start from the left/right. Learners need to be reminded that The point where the perpendicular lines intersect at the centre base of the protractor must be placed at the point at which the angle to be measured is located. One of the arms of the angle to be measured must be aligned to the right or the left with the horizontal line at the base of the protractor. The angle is measured by looking at the graduation starting from zero. Learners' Common Difficulties & Misconceptions. 2. Failure to justify answers when exams questions ask for reasons. Edexcel June 2013, Higher tier, Paper 2 Examiner's Report Teaching points 6

7 Learners' Common Difficulties & Misconceptions. 2. Failure to justify answers when exams questions ask for reasons. Helping Learners' Overcome their Difficulties The Examiner's Report is a clear evidence that there are still problems in both the teaching and learning about angles in the classroom. What follows is Edexcel's suggestions on how students should answer relevant questions on geometry in exams. Apr 28 07:27 7

8 Attachments Maths hub angles on parallel lines.zip Angle properties of parallel lines.ppt Angles 4 In a Triangle.ppt Edexcel GCSE 2010 geometrical statements.doc

Point A location in geometry. A point has no dimensions without any length, width, or depth. This is represented by a dot and is usually labelled.

Point A location in geometry. A point has no dimensions without any length, width, or depth. This is represented by a dot and is usually labelled. Test Date: November 3, 2016 Format: Scored out of 100 points. 8 Multiple Choice (40) / 8 Short Response (60) Topics: Points, Angles, Linear Objects, and Planes Recognizing the steps and procedures for