GCSE Maths - Geometry and Measures Properties of Angles PDF

GCSE Maths – Geometry and Measures Properties of Angles Notes WORKSHEET This work by PMT Education is licensed under https://bit.ly/pmt-cc https://bit.ly/pmt-edu-cc CC BY-NC-N...

GCSE Maths – Geometry and Measures Properties of Angles Notes WORKSHEET This work by PMT Education is licensed under https://bit.ly/pmt-cc https://bit.ly/pmt-edu-cc CC BY-NC-ND 4.0 https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc Properties of Angles Angles at a point The angles around a point add up to 𝟑𝟑𝟑𝟑𝟑𝟑°. Example: Calculate the value of 𝑥𝑥 in the diagram below 1. Sum the angles around the point and equate the sum to 360°: 𝑥𝑥 + 50° + 155° + 97° = 360° 2. Solve the equation to find 𝑥𝑥: 𝑥𝑥 = 360° − 50° − 155° − 97° 𝒙𝒙 = 𝟓𝟓𝟓𝟓° Angles on a straight line The angles on a straight line add up to 𝟏𝟏𝟏𝟏𝟏𝟏°. Example: Calculate the value of 𝑥𝑥 in the diagram below 1. Sum the angles on the line and equate the sum to 180o: 2𝑥𝑥 + 54° = 180° 2. Solve the equation to find 𝑥𝑥: 2𝑥𝑥 = 180° − 54° 2𝑥𝑥 = 126° 𝒙𝒙 = 𝟔𝟔𝟔𝟔° https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc Alternate and corresponding angles Parallel lines are lines which never meet and always lie the same distance apart from one another. When parallel lines are intersected by another line, it produces angles with important properties. The intersecting line is called a transversal. Corresponding angles Corresponding angles are equal. They are on the same side of the transversal line and the lines on which the angles lie on form an ‘F’ shape. Alternate angles Alternate angles are equal. They are on opposite sides of the transversal line and the lines on which they lie form a ‘Z’ shape. Vertically opposite angles Vertically opposite angles are also equal. Example: Calculate the value of 𝑥𝑥 in the diagram below 1. Find angle CDH. 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶𝐶𝐶 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐷𝐷𝐷𝐷𝐷𝐷 since they are corresponding angles. So 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶𝐶𝐶 = 𝟔𝟔𝟔𝟔°. 2. Find angle 𝑥𝑥 by using the idea that angles on a straight line add up to 180o. 𝑥𝑥 + 63° = 180° 𝑥𝑥 = 180° − 63° = 117° Note, that the properties only work on parallel lines. GJ and CF are not parallel, therefore, the properties cannot be used there. https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc Angles in regular polygons Regular polygons are polygons which have sides of equal length, and which have interior angles of equal value. Interior angles Interior angles are the angles found inside the polygon. Interior angles in a triangle The sum of the interior angles in a triangle add up to 180o. This can be proven using the idea of alternate angles: 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑎𝑎 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑑𝑑 by alternate angles. 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑐𝑐 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑒𝑒 by alternate angles. Since angles 𝑑𝑑, 𝑏𝑏 and 𝑒𝑒 all lie on a straight line, 𝑑𝑑 + 𝑏𝑏 + 𝑒𝑒 = 180°. So, 𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐 = 180°. Interior angles in any polygon The sum of the interior angles in a polygon can be found by dividing the polygon into triangles. The square can be divided into two triangles so the interior angles of the square must add up to 2 × 180° = 360°. The pentagon can be divided into three triangles so the interior angles of the pentagon must add up to 3 × 180° = 540°. The hexagon can be divided into four triangles so the interior angles of the hexagon must add up to 4 × 180° = 720°. The formula for calculating the sum of interior angles in an n-sided polygon is (𝑛𝑛 − 2) × 180°. https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  Example: Calculate the sum of interior angles in a decagon 1. Count out how many triangles the polygon can be split into. As can be seen in the diagram, the decagon can be split into 8 triangles. The triangles are constructed by connecting one vertex to each of the other possible vertices. 2. Multiply the number of triangles by 180o to obtain the sum of the interior angles. 8 × 180° = 1440° Alternatively, you could use the formula (𝑛𝑛 − 2) × 180° with 𝑛𝑛 = 10 since a decagon is a 10-sided polygon. Exterior angles The angles formed outside the polygon when the sides of the polygon are extended are the exterior angles. The sum of the exterior angles add up to 360o. This can be used to find the size of each exterior angle. For an n-sided polygon: 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑜𝑜𝑜𝑜 𝑒𝑒𝑎𝑎𝑎𝑎ℎ 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 360° ÷ 𝑛𝑛 Example: The diagram shows part of a regular polygon. Calculate how many sides the polygon has. 1. Work out the value of the exterior angle by extending the side of the polygon and using the fact that angles on a line add up to 180o: 180° − 135° = 45° 2. Use the fact that the exterior angles add up to 360o to find the number of sides, 𝑛𝑛: 45𝑜𝑜 × 𝑛𝑛 = 360𝑜𝑜 360 𝑛𝑛 = =8 45 Therefore, the polygon has 8 sides and is an octagon. https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  Properties of Angles – Practice Questions 1. Find angle 𝑥𝑥 in each of the following diagrams: a) Hh hhhb) 2. Find angle 𝑥𝑥 in each of the following diagrams: a) hhhhhhb) 3. Calculate the sum of the interior angles in a polygon which has 23 sides. 4. In the following diagrams, find angle 𝑥𝑥: a) hhb) Worked solutions for the practice questions can be found amongst the worked solutions for the corresponding worksheet file. https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc