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Proofs without Words: Exterior Angles of Polygons

Proofs without Words: Exterior Angles of Polygons
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2015-07-30 12:32
PolygonAngleTransformationGeometric Construction
In collections Polygons & Angles Polygons & Angles Polygons & Angles ( s n attar) Polygons & Angles Polygons & Angles Polygons & Angles Polygons & Angles Geometry: Polygons & Angles Geometry: Polygons & Angles
This interactive applet provides a visual 'proof without words' demonstrating that the sum of the exterior angles of any convex polygon is always 360 degrees. The mathematical focus is on the properties of convex polygons and the constant sum of their exterior angles. The construction employs GeoGebra commands such as Ray, Vector, and Translate to shift each exterior angle to a common vertex, utilizing Segment, Point, and Intersect to form a complete circle. This tool is ideal for middle school geometry students and educators to intuitively grasp polygon theorems through dynamic visualization.

GeoGebra Commands & Parameters

  1. Ray(A, B) → a
  2. Ray(B, C) → b
  3. Ray(C, A) → c
  4. Vector(B, D) → u
  5. Vector(C, D) → v
  6. Vector(A, D) → w
  7. Point(a) → E
  8. Point(b) → F
  9. Point(b) → G
  10. Point(c) → H
  11. Point(c) → I
  12. Point(a) → J
  13. Segment(B, E) → e
  14. Segment(B, F) → f
  15. Segment(C, G) → g
  16. Segment(C, H) → h
  17. Segment(A, I) → i
  18. Segment(A, J) → j
  19. Translate(B, u) → B'_1
  20. Distance(B, D) → distanceBD
  21. Point(a) → K
  22. Point(b) → L
  23. Segment(B, K) → l
  24. Segment(B, L) → m
  25. Translate(B, u) → B'
  26. Translate(K, u) → K'
  27. Segment(B', K') → l'
  28. Translate(B, u) → B'_2
  29. Translate(L, u) → L'
  30. Segment(B'_2, L') → m'
  31. Translate(C, v) → C'
  32. Translate(G, v) → G'
  33. Segment(C', G') → g'
  34. Translate(C, v) → C'_1
  35. Translate(H, v) → H'
  36. Segment(C'_1, H') → h'
  37. Translate(A, w) → A'
  38. Translate(I, w) → I'
  39. Segment(A', I') → i'
  40. Translate(A, w) → A'_1
  41. Translate(J, w) → J'
  42. Segment(A'_1, J') → j'
  43. Ray(D, H') → p_1
  44. Circle(D, H') → q_1
  45. Ray(D, G') → r_1
  46. Intersect(q_1, r_1) → M,
  47. Ray(D, K') → s_1
  48. Intersect(q_1, s_1) → N,
  49. CircleSector(D, G', H') → t
  50. Circle(D, M) → d_1
  51. Circle(A, 1) → f_1
  52. Circle(B, 1) → g_1
  53. Circle(C, 1) → h_1
  54. Intersect(h_1, b, 2) → O
  55. Intersect(h_1, c, 1) → P
  56. Intersect(f_1, c, 2) → Q
  57. Intersect(f_1, a, 1) → R
  58. Intersect(g_1, a, 2) → S
  59. Intersect(g_1, b, 1) → T
  60. CircleSector(A, Q, R) → ext_3
  61. CircleSector(B, S, T) → ext_1
  62. CircleSector(C, O, P) → ext_2
  63. Translate(ext_3, w) → k'
  64. Translate(ext_1, u) → k_1
  65. Translate(ext_2, v) → ext_4
  66. Ray(U, V) → a_1
  67. Ray(V, W) → b_1
  68. Ray(W, Z) → c_1
  69. Ray(Z, U) → e_1
  70. Circle(V, 1) → t_1
  71. Circle(W, 1) → c_2
  72. Circle(Z, 1) → d_2
  73. Circle(U, 1) → e_2
  74. Intersect(t_1, a_1, 2) → A_1
  75. Intersect(t_1, b_1, 1) → B_1
  76. Intersect(c_2, b_1, 2) → C_1
  77. Intersect(c_2, c_1, 1) → D_1
  78. Intersect(d_2, c_1, 2) → E_1
  79. Intersect(d_2, e_1, 1) → F_1
  80. Intersect(e_2, e_1, 2) → G_1
  81. Intersect(e_2, a_1, 1) → H_1
  82. CircleSector(V, A_1, B_1) → f_2
  83. CircleSector(W, C_1, D_1) → g_2
  84. CircleSector(Z, E_1, F_1) → h_2
  85. CircleSector(U, G_1, H_1) → k_2
  86. Vector(U, I_1) → u_1
  87. Vector(V, I_1) → v_1
  88. Vector(W, I_1) → w_1
  89. Vector(Z, I_1) → i_1
  90. Translate(k_2, u_1) → p_2
  91. Translate(f_2, v_1) → q_2
  92. Translate(g_2, w_1) → r_2
  93. Translate(h_2, i_1) → s_2
  94. Ray(J_1, K_1) → d
  95. Ray(K_1, L_1) → a_2
  96. Ray(L_1, M_1) → b_2
  97. Ray(M_1, N_1) → i_2
  98. Ray(N_1, J_1) → j_2
  99. Circle(J_1, 1) → t_2
  100. Circle(K_1, 1) → c_3
  101. Circle(L_1, 1) → d_3
  102. Circle(M_1, 1) → e_3
  103. Circle(N_1, 1) → f_3
  104. Intersect(t_2, j_2, 2) → O_1
  105. Intersect(t_2, d, 1) → P_1
  106. Intersect(c_3, d, 2) → Q_1
  107. Intersect(c_3, a_2, 1) → R_1
  108. Intersect(d_3, a_2, 2) → S_1
  109. Intersect(d_3, b_2, 1) → T_1
  110. Intersect(e_3, b_2, 2) → U_1
  111. Intersect(e_3, i_2, 1) → V_1
  112. Intersect(f_3, i_2, 2) → W_1
  113. Intersect(f_3, j_2, 1) → Z_1
  114. CircleSector(J_1, O_1, P_1) → g_3
  115. CircleSector(K_1, Q_1, R_1) → h_3
  116. CircleSector(L_1, S_1, T_1) → k_3
  117. CircleSector(M_1, U_1, V_1) → p_3
  118. CircleSector(N_1, W_1, Z_1) → q_3
  119. Vector(J_1, A_2) → u_2
  120. Vector(K_1, A_2) → v_2
  121. Vector(L_1, A_2) → w_2
  122. Vector(M_1, A_2) → l_2
  123. Vector(N_1, A_2) → m_2
  124. Translate(g_3, u_2) → r_3
  125. Translate(h_3, v_2) → s_3
  126. Translate(k_3, w_2) → t_3
  127. Translate(q_3, m_2) → c_4
  128. Translate(p_3, l_2) → d_4
  129. Ray(B_2, C_2) → l_3
  130. Ray(C_2, D_2) → m_3
  131. Ray(D_2, E_2) → q_6
  132. Ray(E_2, F_2) → a_4
  133. Ray(F_2, G_2) → b_4
  134. Ray(G_2, B_2) → e_4
  135. Circle(B_2, 1) → f_4
  136. Circle(C_2, 1) → g_4
  137. Circle(D_2, 1) → h_4
  138. Circle(E_2, 1) → k_4
  139. Circle(F_2, 1) → p_4
  140. Circle(G_2, 1) → q_4
  141. Intersect(q_4, b_4, 2) → H_2
  142. Intersect(q_4, e_4, 1) → I_2
  143. Intersect(f_4, e_4, 2) → J_2
  144. Intersect(f_4, l_3, 1) → K_2
  145. Intersect(g_4, l_3, 2) → L_2
  146. Intersect(g_4, m_3, 1) → M_2
  147. Intersect(h_4, m_3, 2) → N_2
  148. Intersect(h_4, q_6, 1) → O_2
  149. Intersect(k_4, q_6, 2) → P_2
  150. Intersect(k_4, a_4, 1) → Q_2
  151. Intersect(p_4, a_4, 2) → R_2
  152. Intersect(p_4, b_4, 1) → S_2
  153. CircleSector(F_2, R_2, S_2) → r_4
  154. CircleSector(G_2, H_2, I_2) → s_4
  155. CircleSector(B_2, J_2, K_2) → t_4
  156. CircleSector(C_2, L_2, M_2) → c_5
  157. CircleSector(D_2, N_2, O_2) → d_5
  158. CircleSector(E_2, P_2, Q_2) → e_5
  159. Vector(B_2, T_2) → u_3
  160. Vector(C_2, T_2) → v_3
  161. Vector(D_2, T_2) → w_3
  162. Vector(E_2, T_2) → n_3
  163. Vector(F_2, T_2) → u_4
  164. Vector(G_2, T_2) → v_4
  165. Translate(s_4, v_4) → f_5
  166. Translate(t_4, u_3) → g_5
  167. Translate(c_5, v_3) → h_5
  168. Translate(d_5, w_3) → k_5
  169. Translate(e_5, n_3) → p_5
  170. Translate(r_4, u_4) → q_5
  171. Ray(U_2, V_2) → i_4
  172. Ray(V_2, W_2) → j_4
  173. Ray(W_2, Z_2) → l_4
  174. Ray(Z_2, A_3) → m_4
  175. Ray(A_3, B_3) → n_4
  176. Ray(B_3, C_3) → a_5
  177. Ray(C_3, U_2) → b_5
  178. Circle(B_3, 1) → r_5
  179. Circle(C_3, 1) → s_5
  180. Circle(U_2, 1) → t_5
  181. Circle(V_2, 1) → c_6
  182. Circle(W_2, 1) → d_6
  183. Circle(Z_2, 1) → e_6
  184. Circle(A_3, 1) → f_6
  185. Intersect(r_5, n_4, 2) → D_3
  186. Intersect(r_5, a_5, 1) → E_3
  187. Intersect(s_5, a_5, 2) → F_3
  188. Intersect(s_5, b_5, 1) → G_3
  189. Intersect(t_5, b_5, 2) → H_3
  190. Intersect(t_5, i_4, 1) → I_3
  191. Intersect(c_6, i_4, 2) → J_3
  192. Intersect(c_6, j_4, 1) → K_3
  193. Intersect(d_6, j_4, 2) → L_3
  194. Intersect(d_6, l_4, 1) → M_3
  195. Intersect(e_6, l_4, 2) → N_3
  196. Intersect(e_6, m_4, 1) → O_3
  197. Intersect(f_6, m_4, 2) → P_3
  198. Intersect(f_6, n_4, 1) → Q_3
  199. CircleSector(B_3, D_3, E_3) → g_6
  200. CircleSector(C_3, F_3, G_3) → h_6
  201. CircleSector(U_2, H_3, I_3) → k_6
  202. CircleSector(V_2, J_3, K_3) → p_6
  203. CircleSector(W_2, L_3, M_3) → r_6
  204. CircleSector(Z_2, N_3, O_3) → s_6
  205. CircleSector(A_3, P_3, Q_3) → t_6
  206. Vector(U_2, R_3) → w_4
  207. Vector(V_2, R_3) → u_5
  208. Vector(W_2, R_3) → v_5
  209. Vector(Z_2, R_3) → w_5
  210. Vector(A_3, R_3) → i_5
  211. Vector(B_3, R_3) → j_5
  212. Vector(C_3, R_3) → l_5
  213. Translate(g_6, j_5) → c_7
  214. Translate(h_6, l_5) → d_7
  215. Translate(k_6, w_4) → e_7
  216. Translate(p_6, u_5) → f_7
  217. Translate(r_6, v_5) → g_7
  218. Translate(s_6, w_5) → h_7
  219. Translate(t_6, i_5) → k_7
  220. Ray(S_3, T_3) → m_5
  221. Ray(T_3, U_3) → n_5
  222. Ray(U_3, V_3) → a_6
  223. Ray(V_3, W_3) → b_6
  224. Ray(W_3, Z_3) → i_6
  225. Ray(Z_3, A_4) → j_6
  226. Ray(A_4, B_4) → l_6
  227. Ray(B_4, S_3) → m_6
  228. Circle(A_4, 1) → p_7
  229. Circle(B_4, 1) → q_7
  230. Circle(S_3, 1) → r_7
  231. Circle(T_3, 1) → s_7
  232. Circle(U_3, 1) → t_7
  233. Circle(V_3, 1) → c_8
  234. Circle(W_3, 1) → d_8
  235. Circle(Z_3, 1) → e_8
  236. Intersect(q_7, l_6, 2) → C_4
  237. Intersect(q_7, m_6, 1) → D_4
  238. Intersect(r_7, m_6, 2) → E_4
  239. Intersect(r_7, m_5, 1) → F_4
  240. Intersect(s_7, m_5, 2) → G_4
  241. Intersect(s_7, n_5, 1) → H_4
  242. Intersect(t_7, n_5, 2) → I_4
  243. Intersect(t_7, a_6, 1) → J_4
  244. Intersect(c_8, a_6, 2) → K_4
  245. Intersect(c_8, b_6, 1) → L_4
  246. Intersect(d_8, b_6, 2) → M_4
  247. Intersect(d_8, i_6, 1) → N_4
  248. Intersect(e_8, i_6, 2) → O_4
  249. Intersect(e_8, j_6, 1) → P_4
  250. Intersect(p_7, j_6, 2) → Q_4
  251. Intersect(p_7, l_6, 1) → R_4
  252. CircleSector(A_4, Q_4, R_4) → f_8
  253. CircleSector(B_4, C_4, D_4) → g_8
  254. CircleSector(S_3, E_4, F_4) → h_8
  255. CircleSector(T_3, G_4, H_4) → k_8
  256. CircleSector(U_3, I_4, J_4) → p_8
  257. CircleSector(V_3, K_4, L_4) → q_8
  258. CircleSector(W_3, M_4, N_4) → r_8
  259. CircleSector(Z_3, O_4, P_4) → s_8
  260. Vector(S_3, S_4) → u_6
  261. Vector(T_3, S_4) → v_6
  262. Vector(U_3, S_4) → w_6
  263. Vector(V_3, S_4) → n_6
  264. Vector(W_3, S_4) → u_7
  265. Vector(Z_3, S_4) → v_7
  266. Vector(A_4, S_4) → w_7
  267. Vector(B_4, S_4) → a_7
  268. Translate(k_8, v_6) → t_8
  269. Translate(p_8, w_6) → c_9
  270. Translate(q_8, n_6) → d_9
  271. Translate(r_8, u_7) → e_9
  272. Translate(s_8, v_7) → f_9
  273. Translate(f_8, w_7) → g_9
  274. Translate(g_8, a_7) → h_9
  275. Translate(h_8, u_6) → k_9

AI Drawing Prompt Reference

Draw a convex polygon and show its exterior angles. Translate each exterior angle to a common vertex to form a complete circle. Use dynamic visualization to demonstrate that the sum of exterior angles is always 360 degrees. The graphic should clearly show the relationship between the polygon and the circle to help understand geometric theorems.
The prompt above was compiled from the original drawing commands and is for reference only.