Sum of Exterior Angles of a Convex Polygon — Race Car Metaphor
2011-09-27 04:18
This GeoGebra applet uses the metaphor of a race car driving around a convex polygon track to visually demonstrate the concept of exterior angles and prove that their sum is always 360°. Through geometric constructions involving Rotate, Vector, and CircleSector commands, the applet simulates the car's turning angle at each vertex, transforming the abstract summation of exterior angles into a tangi
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Drag the green points. Check the "Show Measurement" box to see the measurements of the angles." below="1. What is a + b + c + d + e, the total angles turned?

2. a, b, c, d, e are called the exterior angles of the convex pentagon ABCDE.

 In general, what is the sum of all exterior angles of a convex polygon?

3. How would you modify the result if the pentagon ABCDE is NOT convex?



Anthony Or. Education Bureau, Hong Kong. "/>
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<layer val="1"/>
<labelMode val="0"/>
<coords x="877" y="549" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="s">
<value val="0"/>
<show object="true" label="true"/>
<condition showObject="reached ≠ 0 ∧ convex"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="1"/>
<labelOffset x="-7" y="0"/>
<labelMode val="3"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="200" x="350" y="35" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
<ggbscript onUpdate="ZoomIn[xmin, ymin, xmax, ymax]"/>
<caption val="縮小看看"/>
</element>
<command name="If">
<input a0="s ≤ 0.5" a1="(-8 * (1 - (2 * s))) - (80 * s)" a2="(-40 * (1 - (2 * (s - 0.5)))) - (1120 * (s - 0.5))"/>
<output a0="xmin"/>
</command>
<element type="numeric" label="xmin">
<value val="-8"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="xmax" exp="(-xmin)" />
<element type="numeric" label="xmax">
<value val="8"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="If">
<input a0="s ≤ 0.5" a1="(-5 * (1 - (2 * s))) - (50 * s)" a2="(-25 * (1 - (2 * (s - 0.5)))) - (700 * (s - 0.5))"/>
<output a0="ymin"/>
</command>
<element type="numeric" label="ymin">
<value val="-5"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="ymax" exp="(-ymin)" />
<element type="numeric" label="ymax">
<value val="5"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="reached" exp="false"/>
<element type="boolean" label="reached">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<command name="If">
<input a0="a + b + c + d + e ≟ 360°" a1="true" a2="false"/>
<output a0="convex"/>
</command>
<element type="boolean" label="convex">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="animate">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="button" label="button1">
<show object="true" label="true"/>
<condition showObject="(¬animate) ∧ t' < 10.5"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="1"/>
<labelOffset x="354" y="466"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[animate, !animate]
show = true
StartAnimation[t', animate]"/>
<font serif="false" sizeM="2" size="20" style="0"/>
<file name="geogebra/main/nav_play.png"/>
</element>
<element type="button" label="button2">
<show object="true" label="true"/>
<condition showObject="animate ≠ 0 ∧ t' < 10.5"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="1"/>
<labelOffset x="354" y="466"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[animate, !animate]
show = true
StartAnimation[t', animate]"/>
<font serif="false" sizeM="2" size="20" style="0"/>
<file name="geogebra/main/nav_pause.png"/>
</element>
<element type="button" label="button3">
<show object="true" label="true"/>
<condition showObject="t' > 0.5"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="1"/>
<labelOffset x="409" y="466"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="t'=0.5
animate=false"/>
<font serif="false" sizeM="2" size="20" style="0"/>
<file name="geogebra/gui/images/nav_skipback.png"/>
</element>
<expression label="text6" exp=""" + (If[convex, "a + b + c + d + e = ?", "a、b、c、d、e 有甚麼關係？"]) + """ />
<element type="text" label="text6">
<show object="true" label="false" ev="8"/>
<condition showObject="reached ≠ 0"/>
<objColor r="220" g="20" b="60" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<algebra labelVisible="false"/> <fixed val="true"/>
<font serif="false" sizeM="1.4" size="8" style="1"/>
<absoluteScreenLocation x="18" y="44"/>
</element>
</construction>
</geogebra>