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Geometry 3 Dec 2012 1) Empty folders and place papers in correct section in your binder. 2) Place book and binder on your desk. 3) DO Flow Chart Proof Handout, # 3 Finished? Do 4.6 CPCTC handout, #12. Finished? Choose one proof from handout and rewrite in a 2nd “style”. objective Students will apply polygon interior and exterior sum conjectures to find angle measures. Students will take notes, work collaboratively and present to the class. Homework due Dec 4 5.1, pg 259: 3 – 8 5.2, pg. 264: 8 – 13, 16, 17 Inductive Reasoning the process of OBSERVING FINDING PATTERNS MAKING GENERALIZATIONS… GEOMETRY the generalization is called A CONJECTURE!!! Conjecture is an IF…THEN… statement Exterior angle Vertex Side Diagonal Polygons Interior angle • Equiangular: all angles have equal measure • Equilateral: all sides are the same length Polygons • Regular Polygon: a polygon where all the angles are equal and all of the sides are the same length. They are both equilateral and equiangular Polygons Examples of Regular Polygons Polygons Exterior Angles of a Polygon Is there a pattern for the sum for exterior angles of polygons? DEMONSTRATION What about a octagon? What is the measure of ONE exterior angle for a regular quadralateral? regular pentagon? http://math.kendallhunt.com/x19427.html No matter what type of polygon we have, the sum of the exterior angles is ALWAYS equal to 360º. Sum of exterior angles = 360º Polygons 180 180o 180o 180 o o 180o 4 sides Quadrilateral 5 sides 2 x 180o = 360o 2 Pentagon o 1 diagonal o 3 x 180 = 540 3 2 diagonals 180o 180o 180o 180o 180o 180o 180o 180o 180o 6 sides 4 Hexagon 4 x 180o = 720o 3 diagonals Polygons 7 sides 5 Heptagon/Septagon 5 x 180o = 900o 4 diagonals Rule? Can you find a rule for the interior angle sum for an “N”-gon? What if you have a regular polygon? Regular Polygonall sides and angles have the same measure Polygons Term Polygon Sum Conjecture Definition The sum of the measures of the interior angles of an n-gon is 180 n 2 0 Exterior angle sum conjecture For any polygon, the sum of the measures of a set of external angles is 3600 Equiangular Polygon Conjecture Each interior angle of an equiangular n-gon 1800 n 2 n Example Sum of interior angles 1800 n 2 1800 n 2 n Find the unknown angles below. w 100o 2 x 180o = 360o 360 – 245 = 115o 140o 125o 100o 121o 117o Polygons x 75o 75o 70o y 4 x 180o = 720o 720 – 603 = 117o Diagrams not drawn accurately. 95o 115o 110o 3 x 180o = 540o 540 – 395 = 145o 125o z 138o 138o 133o 105o 137o 5 x 180o = 900o 900 – 776 = 124o Practice– Homework Problems CW Focus: In progress: 5.1: 3, 4 5.2: 8 Partial: 5.1: 5, 6 5.2: 9 Complete: 5.1: 7, 8 5.2: 10 Advanced: 5.1: 7 5.2: 10, 13 Debrief What is difficult about finding angle measures in an “angle chase”? What strategies can you use? What is easy?