GCSE: Angles and Geometry

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Presentation transcript:

GCSE: Angles and Geometry Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 5th January 2014

x = 66° Starter ? Copy the diagram, then determine x. 55° 55° 33° 192° 136° 11° 70° x° 114° 44° 44° 22° 22° 11° ? x = 66°

Laws of Angles ? ? ? a a a a a a Opposite angles Alternate angles Corresponding angles ? Bro Tip: When asked to give a justification in an exam for how you determined an angle, write one of the above – NOT “Z angles” or “F angles”. You won’t get the mark.

Quick Exercises ? ? ? ? ? Q1 74° Q2 a = 65°, b = 103°, c = 38° c b 93° 77° a = 87°, b = 74° ? 115° b° Q3 Q4 The two squares are congruent. Q5 50° 68° y c 150 x 85 70° a c = 20° ? a = 136° ? ? x = 150°, y = 95°

Angles in Polygons NO YES NO ? ? ? An exterior angle of a polygon is an angle between the line extended from one side, and an adjacent side. Which of these are exterior angles of the polygon? ? ? NO YES ? NO

Click to Start Damonimation Angles in Polygons To defeat Kim Jon Il, Matt Damon must encircle his pentagonal palace. What angle does Matt Damon turn in total? 360° ? ! The sum of the exterior angles of any polygon is 360°. Click to Start Damonimation

Angles in Polygons ? ? Exterior angle of pentagon = 360 / 5 = 72° If the pentagon is regular, then all the exterior angles are clearly the same. Therefore: Exterior angle of pentagon = 360 / 5 = 72° Interior angle of pentagon = 180 – 72 = 108° ? ?

Angles in Polygons ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Num Sides Name of Regular Polygon Exterior Angle Interior Angle 3 Triangle 120° 60° 4 Quadrilateral 90° 5 Pentagon 72° 108° 6 Hexagon 7 Heptagon 51.4° 128.6° 8 Octagon 45° 135° 9 Nonagon 40° 140° 10 Decagon 36° 144° ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Bonus Question: What is the largest number of sides a shape can have such that its interior angle is an integer? 360 sides. The interior angle will be 179°. ?

Angles in Polygons ? Example GCSE Question Interior angle of hexagon = 180° – (360°/6) = 60° Interior angle of octagon = 180° – (360°/8) = 135° x = 360° – 120° – 135° = 105°

Angles in Polygons ? Copy the diagram and answer. Given that B has interior angle 60: Interior angle of A = (360 – 60)/2 = 150 Exterior angle of A = 30 So A has 360 / 30 = 12 sides

! Total interior angle = 180(n – 2) Sum of Interior Angles We know how to find the interior angle of a regular polygon. But can we find the total angle if the polygon is not regular? n sides We could split the shape up into n – 2 triangles. Now noticing that the interior angles of each triangle form the interior angles of the overall polygon, the total interior angle must be 180(n – 2) ! Total interior angle = 180(n – 2) ?

Sum of Interior Angles ? ? ? ? Num Sides Name Sum of Interior Angles 3 Triangle 180° 4 Quadrilateral 360° 5 Pentagon 540° 6 Hexagon 720° ? ? ? ?

Quick Exercises Q1 Q2 Q3 50 110 85 75 100 110 a b 80 80 60 80 80 c a = 110° ? b = 260° ? c = 70° ? Q4 Q5 Q6 The sum of the interior angles of a polygon is 3600°. How many sides does it have? 22 62 242 62 81 284 88 ? a a 67 b 86 69 a = 119°, b = 25° ? ? a = 100° ?