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OpenStax-CNX module: m39877 1 ∗ Geometry basics: Polygons [NCS] Free High School Science Texts Project This work is produced by OpenStax-CNX and licensed under the † Creative Commons Attribution License 3.0 1 Other polygons There are many other polygons, some of which are given in the table below. Sides 5 6 7 8 10 15 Name pentagon hexagon heptagon octagon decagon pentadecagon Table 1: Table of some polygons and their number of sides. Figure 1: Examples of other polygons. 2 Extra 2.1 Angles of Regular Polygons You can calculate the size of the interior angle of a regular polygon by using: ^ A= ∗ Version 1.1: Aug 5, 2011 1:48 am -0500 † http://creativecommons.org/licenses/by/3.0/ http://cnx.org/content/m39877/1.1/ n−2 × 180◦ n (1) OpenStax-CNX module: m39877 2 ^ where n is the number of sides and A is any angle. 2.2 Areas of Polygons 1. Area of triangle: 12 × base × perpendicular height Figure 2 2. Area of trapezium: 21 × (sum of k (parallel) sides) × perpendicular height Figure 3 3. Area of parallelogram and rhombus: base × perpendicular height Figure 4 4. Area of rectangle: length × breadth Figure 5 5. Area of square: length of side × length of side Figure 6 http://cnx.org/content/m39877/1.1/ OpenStax-CNX module: m39877 3 6. Area of circle: π x radius2 Figure 7 Khan Academy video on area and perimeter This media object is a Flash object. Please view or download it at <http://www.youtube.com/v/kqqmJiJez6o&rel=0&hl=en_US&feature=player_embedded&version=3> Figure 8 Khan Academy video on area of a circle This media object is a Flash object. Please view or download it at <http://www.youtube.com/v/tCrDyJsSFok&rel=0&hl=en_US&feature=player_embedded&version=3> Figure 9 2.2.1 Polygons 1. For each case below, say whether the statement is true or false. For false statements, give a counterexample to prove it: a. All squares are rectangles b. All rectangles are squares c. All pentagons are similar d. All equilateral triangles are similar e. All pentagons are congruent f. All equilateral triangles are congruent Click here for the solution1 2. Find the areas of each of the given gures - remember area is measured in square units (cm2 , m2 , mm2 ). Figure 10 1 http://www.fhsst.org/lxJ http://cnx.org/content/m39877/1.1/ OpenStax-CNX module: m39877 4 Click here for the solution2 3 Summary • Make sure you know what: quadrilaterals, vertices, sides, angles, parallel lines, perpendicular lines,diagonals, • • • • • • • • • • • • • bisectors and transversals mean. Similarities and dierences between quadrilaterals Properties of triangles and quadrilaterals Congruency of triangles Classication of angles into acute, right, obtuse, straight, reex or revolution Theorem of Pythagoras which is used to calculate the lengths of sides of a right-angled triangle Angles: · Acute angle: An angle 0 and 90 · Right angle: An angle measuring 90 · Obtuse angle: An angle 90 and 180 · Straight angle: An angle measuring 180◦ · Reex angle: An angle 180 and 360 · Revolution: An angle measuring 360 Angle properties and names Equilateral, isoceles, right-angled, scalene triangles Triangles angles = 180 Congruent and similar triangles Pythagoras Trapezium, parm, rectangle, square, rhombus, kite and properties Areas of particular gures 4 Exercises 1. Find all the pairs of parallel lines in the following gures, giving reasons in each case. a. Figure 11 b. Figure 12 c. Figure 13 2 http://www.fhsst.org/lxS http://cnx.org/content/m39877/1.1/ OpenStax-CNX module: m39877 5 Click here for the solution3 2. Find angles a, b, c and d in each case, giving reasons. a. Figure 14 b. Figure 15 c. Figure 16 Click here for the solution4 3. Which of the following claims are true? Give a counter-example for those that are incorrect. a. All equilateral triangles are similar. b. All regular quadrilaterals are similar. c. In any [U+25B5]ABC with ∠ABC = 90◦ we have AB 3 + BC 3 = CA3 . d. All right-angled isosceles triangles with perimeter 10 cm are congruent. e. All rectangles with the same area are similar. Click here for the solution5 4. Say which of the following pairs of triangles are congruent with reasons. a. Figure 17 b. Figure 18 3 http://www.fhsst.org/lxh 4 http://www.fhsst.org/laq 5 http://www.fhsst.org/lal http://cnx.org/content/m39877/1.1/ OpenStax-CNX module: m39877 6 c. Figure 19 d. Figure 20 Click here for the solution6 5. For each pair of gures state whether they are similar or not. Give reasons. Figure 21 Click here for the solution7 4.1 Challenge Problem 1. Using the gure below, show that the sum of the three angles in a triangle is 180◦ . Line DE is parallel to BC . Figure 22 Click here for the solution8 6 http://www.fhsst.org/lai 7 http://www.fhsst.org/la3 8 http://www.fhsst.org/laO http://cnx.org/content/m39877/1.1/