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Math 11 2.4 – Intro to Angles and Polygons Sept 24 Purpose: to explore the properties of angles of polygons Intro: a polygon is defined as a shape on a plane (a 2D shape) that is bounded by a certain number of straight line segments that form a loop. Examples at right: We often think of polygons like rectangles, triangles, pentagons, where the internal angles at each vertex are less than 180o… These are examples of convex polygons. If a polygon has at least one internal angle that is greater than 180o, then we call these non-convex (concave) polygons. Regular polygons A special subset of convex polygons is the set of regular polygons – shapes where every internal angle is the same and every side is the same length. See the image at right for examples. Regular polygons are convex polygons. 1. a) Complete the chart at right. Here’s how: Use a scrap piece of paper and construct a quadrilateral (convex) Draw a line from a vertex to a non-adjacent vertex to make a triangle. Use the “sum of angles in a triangle is 180o” to determine the sum of internal angles in the quadrilateral. Repeat this process for the next 4 shapes b) do you see a pattern for the sums? Make a conjecture about the sum of internal angles in a polygon with n sides. A chart showing the sum of internal angles of polygons 2 2. Examine the quadrilateral, at right. Notice that the (counterclockwise) external angles are labeled w, x, y, z. a) Determine the sum of the external angles for this shape. b) on a piece of scrap paper, draw a pentagon (doesn’t have to be a regular pentagon). Determine the sum of external angles for this shape. c) Complete the chart. Notice anything? Shape Triangle Quadrilateral Pentagon Hexagon # of sides 3 4 5 6 Sum of internal angles 180o Sum of external angles 360o (remember yesterday?) Summarize your conjectures: The sum of internal angles in an n-sided polygon is: The sum of external angles in an n-sided polygon is: If the polygon is a regular polygon, the measure of each internal angle is: 3. Check your understanding with these quickies: a) A loonie (the dollar coin) is a hendecagon – an 11-sided polygon. What is the sum of internal angles? b) How many internal angles does it have? c) What is the measure of each internal angle? It is a regular hendecagon ______ ______ ______ 4. 5. Polygons can be tested as convex or non-convex (concave) by drawing lines through them. a) describe a test that involves drawing a single line through a polygon b) describe a test that involves drawing diagonals (lines from a vertex to any non-adjacent vertex). 5. Three external angles of a convex pentagon measure 60o, 70o, and 90o. The other two are external angles are congruent. Determine the measure of the internal angles of the pentagon. 6. Regular polygons - quickies a) what is the common name for a regular triangle? ____________________________________ b) what is the common name for a regular quadrilateral? ____________________________________ 7. A regular polygon has an external angle of 200. How many sides does it have? 8. Is it possible for a regular polygon to have an internal angle of 1300? Why or why not? 9. Even more regular polygons: the central angle of a regular polygon is formed by tracing from one vertex to the centre to the next vertex. Think of it like spokes on a wheel, and the central angle is at the centre of the wheel. In this case, we have a regular octagon, and the central angle is indicated. What is the relationship between the external angle and the central angle? Central angle – found by External angle taking a fraction of the circle which contains the polygon. Internal angle 10. Let’s pretend we have 40cm of string to work with, and we want to make a regular polygon. But each time we make a polygon, we use one more side. See the chart below. As the area of the polygon increases, it increased by less and less each time, as if the area is limited (or will reach a maximum value). What is this maximum area? Number of sides Length of a side (cm) Perimeter (cm) Measure of internal angle Measure of external angle Area (cm2) (approximate) 3 40 3 40 60o 120o 77 4 10 40 90o 90o 100 5 8 40 108o 72o 110 40 120o 60o 115 40 ~128.6o ~51.4o 119 135o- 45o 121 6 7 20 3 40 7 8 5 40 … … … ?? (a lot!) ?? 40 … ?? ?? ?? (maximum??) 11. Read page 96 in your textbook – it’s the proof of your conjecture for the sum of internal angles for a convex polygon. Could you follow it? Could you reproduce it?