Grade 6 Math Circles February 29, D Geometry

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1 1 Universit of Waterloo Facult of Mathematics Centre for Education in Mathematics and Computing Grade 6 Math Circles Feruar 29, D Geometr What is Geometr? Geometr is one of the oldest ranches in mathematics. It studies the properties, measurement, and relationship etween points, lines, angles, shapes, planes, etc. The word geometr comes from Greek words meaning Earth measurement. Tpes of Angles: Angle Measurement Angle Classification Less than 90 Acute Angle 90 Right Angle Between 90 and 180 Otuse Angle 180 Straight Angle Between 180 and 360 Refle Angle Complementar Angles: Complementar angles are angles that sum to = 90 Note: Both angles and are acute angles.

2 2 Eample: 1) If ABC = 90, determine the value of. A B 29 C 29 + = 90 (Complementar Angles) = = 61 Thus, = 61. Supplementar Angles: Supplementar angles are angles that sum to = 180 Note: Angle is acute and is otuse. Eample: 1) If DEF = 180, determine the value of. D 130 E F = 180 (Supplementar Angles) = = 50 Thus, = 50. Opposite Angles: Opposite angles that are formed intersecting lines are equal. =

3 3 Wh is this true? z Let z e the supplementar angle to. This means that + z = 180. But we can also see that z and are supplementar angles. So + z = 180. Then since + z = 180 and + z = 180 we can see that + z = + z. This can onl e true if =. Eample: 1) Determine the values of and = 55 = = 35 Thus, = = 125 = = 25 Thus, = 25. (Opposite Angles) (Opposite Angles) Angle Properties of Parallel Lines and a Transversal: A transversal is a straight line that intersects two or more lines. Transversal Parallel Lines

4 4 Corresponding Angles are Equal (F-Pattern): = Interior Angles are Supplementar (C-Pattern): r + s = 180 s r Alternate Angles are Equal (Z-Pattern): a = a

5 5 z z + 20 = 120 z = z = 100 Thus, z = 100. z = = = = = 60 = 60 3 = 20 Thus, = 20. = 3 = 60 Thus, = 60. (F-Pattern) (C-Pattern) (Z-Pattern) Eercises: 1) Determine the value of. 2) Determine the value of a a 55 3) Determine the values of s and t ) Determine the value of k. 45 t s + 65 k

6 6 5) Determine the values of t and v. 6v 70 6) Determine the values of m and n n 31 t 11v m 7) Determine the values of r and w ) a) Determine the values of,, and z. 45 w + 25 z 122 5r ) Determine the value of + + z. c) What is the sum of the interior angles of a triangle? Interior Angles of Polgons: Triangles: The sum of the interior angles of a triangle is 180. Wh is this true? A a B c C From triangle ABC, if we draw a line parallel to the ase of the triangle we can see that + + = 180 (since the are supplementar angles). Then we can see that = a (from Z-Pattern). Also, = c (from Z-Pattern). The net step is to sustitute a in for and c for to get: a + + c = 180 Thus, the sum of interior angles of a triangle is 180.

7 7 What aout other polgons? To determine the sum of the interior angles of a polgon, draw a diagonal line from one verte to all the other vertices that it is not alread connected to. This divides the polgon into triangles. Since we alread know the sum of interior angles of a triangle, we can use that to determine the sum of interior angles of the polgon. Multipl the numer of triangles that ou have divided our polgon into 180 to get the sum of interior angles of the polgon. Polgon Picture Numer of Numer of Numer of Sum of Sides Angles Vertices Interior Angles Triangle Quadrilateral Pentagon Heagon Heptagon

8 8 Looking at the chart aove, do ou notice a pattern? The sum of interior angles increases 180 as the numer of sides increases one. Notice that the numer of triangles our polgon is divided into is two less than the numer of sides. The general formula for the sum of interior angles of an n-sided polgon is: sum of interior angles = (n 2) 180. Eamples: 1) a) Determine the sum of the interior angles of a regular octagon. ) What is the measure of each interior angle? a) Using the formula aove, the sum of interior angles = (n 2) 180. Since an octagon has 8 sides, n=8. Thus, the sum of interior angles = (8 2) 180 = = ) Since the octagon is a regular octagon, all of the interior angles are the same size. Let s call each angle. Then, = = 1080 = = 135 Therefore, the measure of each interior angle of a regular octagon is 135.

9 9 Eercises: 1) For each of the regular polgons in the chart aove, determine the measure of each interior angle. 2) a) Determine the measure of each interior angle of a regular decagon. ) Using our protractor draw a regular decagon with side lengths of 3cm. 3) Determine the values of the variales in the diagrams elow. a) ) 50 a c) c d) d e h f g

10 10 4)a) ) r u s i) Determine the values of the variales in the diagram aove. ii) Determine the value of + + z. z i) Determine the values of the variales in the diagram aove. ii) Determine the value of r + s + t + u. t a e c) i) Determine the values of the variales in the diagram to the left. ii) Determine the value of a + + c + d + e. c d d) i) What do ou notice aout the sum of the eterior angles of the polgons aove? ii) Is this true for all polgons?

11 11 Etra Prolems: Determine the values of the variales in each of the diagrams elow. 1) 2) v ) 4) a u c t s r 5) w a 100 z 80 c 50