MA.8.G.2.3 Demonstrate that the sum of the angles in a triangle is 180- degrees and apply this fact to find unknown measure of angles. Block 26.

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MA.8.G.2.3 Demonstrate that the sum of the angles in a triangle is 180- degrees and apply this fact to find unknown measure of angles. Block 26

Triangle angle sum 2/7/10Teacher Quality Grant - AE - FAU2

Some Historical Proofs 2/7/10Teacher Quality Grant - AE - FAU3

Pythagoras (570 – 495 BC) Open GeoGebra file Pythagoras proof Move the vertices of the triangle and investigate the relationship of the angles Why are those relationships true? 2/7/10Teacher Quality Grant - AE - FAU4

Euclid (323–283 BC) Open the GeoGebra file Euclid proof Move the vertices of the triangles and discuss the relationships seen in the angles 2/7/10Teacher Quality Grant - AE - FAU5

Paper Activities to Show the Relationship 2/7/10Teacher Quality Grant - AE - FAU6

Hands-on Activity 2/7/10Teacher Quality Grant - AE - FAU7

Paper Folding Open the GeoGegra file paper folding 2/7/10Teacher Quality Grant - AE - FAU8

Other GeoGebra Files Sum_angle_triangle1 2/7/10Teacher Quality Grant - AE - FAU9

Other GeoGebra Files Sum_angle_triangle2 2/7/10Teacher Quality Grant - AE - FAU10

Other GeoGebra Files Sum_angle_triangle3 2/7/10Teacher Quality Grant - AE - FAU11

Pedagogical Discussion How could these files be used in the classroom? Which one conveys the information more precisely? Which one is more appropriate for what grade level? 2/7/10Teacher Quality Grant - AE - FAU12

Do all triangles have angles that add up to 180°? I have discovered things so wonderful that I was astounded... out of nothing I have created a strange new world. János Bolyai, /7/10Teacher Quality Grant - AE - FAU13

Back to Euclid’s time… Euclid stated 5 postulates on which he based all his theorems in his book the Elements His 5 th postulate was different from the rest – If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles. 2/7/10Teacher Quality Grant - AE - FAU14

Non-Euclidean Geometries For centuries mathematicians try to find a replacement for Postulate 5 Adrien-Marie Legendre (1752 – 1833) proved that the 5 th postulate is equivalent to: the sum of the angles of a triangle is equal to two right angles Carl Friedrich Gauss ( ) assumed that the sum of the angles of a triangle did not equal 180° Two most common Non-Euclidean Geometries: spherical and hyperbolic geometries 2/7/10Teacher Quality Grant - AE - FAU15

Spherical Geometry Geometry of the two-dimensional surface of a sphere Triangles can have more than 180° Shortest path between two points is a path on a great circle Used in navigation and astronomy 2/7/10Teacher Quality Grant - AE - FAU16

Hyperbolic Geometry The sum of the angles in a triangles are less than 180° 2/7/10Teacher Quality Grant - AE - FAU17

Find Unknown Measures of Angles 2/7/10Teacher Quality Grant - AE - FAU18

Finding Unknown Measures of Angles 2/7/10Teacher Quality Grant - AE - FAU19 Distribute worksheet Participants work in groups What kind of mistakes will students make? Why?