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Hexagon Angle Sum Theorems Demo

Hexagon Angle Sum Theorems Demo
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2015-10-26 02:32
PolygonAngleGeometric Construction
In collections Polygons & Angles Polygons & Angles Polygons & Angles Geometry: Polygons & Angles Geometry: Polygons & Angles
This applet informally illustrates to students that the sum of the interior angles of any hexagon is 720 degrees and the sum of the exterior angles is 360 degrees. It involves core math concepts like polygon angle sum theorems, exterior angle properties, and angle measurement. The construction uses commands such as Segment, Circle, Intersect, Ray, CircleSector, Vector, Translate, Mirror, Dilate, If, Rotate, Polygon, and Point. Suitable for middle school and above, it helps students understand and verify polygon angle properties and enhances spatial reasoning skills.

GeoGebra Commands & Parameters

  1. Segment(C, A_1) → i_3
  2. Circle(C, 1.5) → h
  3. Intersect(h, i_3) → L,
  4. Ray(C, D) → c
  5. Intersect(h, c, 1) → P
  6. Ray(A, B) → a
  7. Ray(B, C) → b
  8. Ray(E, A) → e
  9. Circle(A, 1.5) → f
  10. Circle(B, 1.5) → g
  11. Circle(D, 1.5) → k
  12. Circle(E, 1.5) → p
  13. Segment(C, A) → i
  14. Segment(C, E) → j
  15. Intersect(f, a, 1) → F
  16. Intersect(f, i, 1) → G
  17. Intersect(f, e, 1) → H
  18. Intersect(f, e, 2) → I
  19. Intersect(p, e) → J,
  20. Intersect(p, j) → K,
  21. Intersect(k, c, 2) → O
  22. Intersect(h, j, 1) → Q
  23. Intersect(h, i, 1) → R
  24. Intersect(h, b, 1) → S
  25. Intersect(g, a, 2) → T
  26. Intersect(g, b, 1) → U
  27. Intersect(g, a, 1) → V
  28. CircleSector(A, F, G) → q
  29. CircleSector(A, G, H) → r
  30. CircleSector(B, U, V) → s
  31. CircleSector(C, Q, R) → c_1
  32. CircleSector(C, R, S) → d_1
  33. Intersect(k, c, 1) → W
  34. CircleSector(E, J, K) → h_1
  35. CircleSector(A, I, F) → p_1
  36. CircleSector(B, T, U) → q_1
  37. Intersect(h, b, 2) → Z
  38. CircleSector(C, Z, P) → r_1
  39. Vector(A, B) → u
  40. Vector(C, B) → v
  41. Vector(E, D) → w
  42. Vector(C, D) → u_1
  43. Vector(C, E) → v_1
  44. Vector(A, E) → w_1
  45. Translate(q, u) → q'
  46. Translate(d_1, v) → s_1
  47. Mirror(s_1, B) → t_1
  48. Dilate(A, If[0 < l ≤ 1, 1 - l, 0], B) → A'
  49. Dilate(C, If[0 < l ≤ 1, 1 - l, 0], B) → C'
  50. Dilate(A, If[1 < l ≤ 2, 2 - l, 0], E) → A'_1
  51. Dilate(C, If[1 < l ≤ 2, 2 - l, 0], E) → C'_1
  52. Dilate(C, If[2 < l ≤ 3, 3 - l, 0], D) → C'_2
  53. Dilate(E, If[2 < l ≤ 3, 3 - l, 0], D) → E'
  54. Vector(A, B) → a_1
  55. Vector(C, C') → b_1
  56. Vector(A, A'_1) → i_1
  57. Vector(E, E') → j_1
  58. Vector(C, C'_2) → l_1
  59. Vector(C, C'_1) → m_1
  60. Translate(d_1, b_1) → c_2
  61. Translate(r, i_1) → r'_1
  62. Translate(r, w_1) → r'_2
  63. Translate(c_1, v_1) → f_2
  64. Translate(c_1, m_1) → g_2
  65. CircleSector(C, P, L) → t
  66. Translate(t, u_1) → t'
  67. Translate(t, l_1) → t'_1
  68. Translate(q, a_1) → r'
  69. Translate(d_1, b_1) → h_2
  70. If(0 < l < 1, Rotate[h_2, ((l * 180))°, C']) → k_2
  71. Vector(A, A') → n_1
  72. Translate(q, n_1) → q'_1
  73. Mirror(r'_2, E) → p_2
  74. If(1 < l < 2, Rotate[r'_1, (((l - 1) * 180))°, A'_1]) → q_2
  75. Segment(E, C) → i_2
  76. Vector(D, C) → u_2
  77. Vector(E, C) → v_2
  78. Vector(A, C) → w_2
  79. Vector(B, C) → l_2
  80. Dilate(B, If[0 < o < 1, 1 - o, 0], C) → B'
  81. Dilate(A, If[1 < o < 2, 2 - o, 0], C) → A'_2
  82. Dilate(E, If[2 < o < 3, 3 - o, 0], C) → E'_1
  83. Dilate(D, If[3 < o < 4, 4 - o, 0], C) → D'
  84. Vector(B, B') → m_2
  85. Vector(A, A'_2) → n_2
  86. Vector(E, E'_1) → s_2
  87. Vector(D, D') → t_2
  88. Translate(q_1, m_2) → c_3
  89. Translate(q_1, l_2) → d_3
  90. Translate(p_1, w_2) → e_3
  91. If(1 < o < 2, Translate[p_1, n_2]) → h_3
  92. Ray(D, A_1) → a_3
  93. Ray(A_1, E) → b_3
  94. Circle(A_1, 1.5) → q_3
  95. Intersect(q_3, i_3) → M,
  96. Intersect(q_3, b_3, 1) → N
  97. Intersect(q_3, a_3, 2) → B_1
  98. Intersect(k, c, 2) → C_1
  99. Intersect(k, a_3, 1) → D_1
  100. Intersect(k, c, 1) → E_1
  101. CircleSector(C, L, Q) → d
  102. Intersect(p, b_3, 1) → F_1
  103. Intersect(p, i_2) → G_1,
  104. CircleSector(E, G_1, N) → e_1
  105. Dilate(E, If[2 < l < 3, 3 - l, 0], A_1) → E'_2
  106. Dilate(C, If[2 < l < 3, 3 - l, 0], A_1) → C'_3
  107. Dilate(C, If[3 < l < 4, 4 - l, 0], D) → C'_4
  108. Dilate(A_1, If[3 < l < 4, 4 - l, 0], D) → H_1
  109. Vector(E, A_1) → f_1
  110. Vector(C, A_1) → g_1
  111. Translate(e_1, f_1) → k_1
  112. Translate(d, g_1) → d'
  113. Vector(E, E'_2) → a_2
  114. Vector(C, C'_3) → b_2
  115. Translate(e_1, a_2) → d_2
  116. Translate(d, b_2) → d'_1
  117. Rotate(d_2, If[2 < l < 3, (((l - 2) * 180))°, 0], E'_2) → e_2
  118. Mirror(k_1, A_1) → r_2
  119. CircleSector(A_1, N, M) → f_3
  120. Intersect(q_3, a_3, 1) → I_1
  121. CircleSector(A_1, M, I_1) → g_3
  122. CircleSector(D, D_1, E_1) → k_3
  123. Vector(C, D) → j_2
  124. Vector(A_1, D) → u_3
  125. Translate(g_3, u_3) → p_3
  126. Translate(t, j_2) → t'_2
  127. Mirror(p_3, D) → r_3
  128. Vector(A_1, H_1) → v_3
  129. Vector(C, C'_4) → w_3
  130. Translate(g_3, v_3) → s_3
  131. Translate(t, w_3) → t'_3
  132. Rotate(s_3, If[3 < l < 4, (((l - 3) * 180))°, 0], H_1) → t_3
  133. Intersect(p, b_3, 2) → J_1
  134. Polygon(A, B, C) → poly1, c_4, a_4, b_4
  135. Polygon(A, C, E) → poly2, e_4, a_5, c_5
  136. Polygon(E, C, A_1) → poly3, a_6, e_5, c_6
  137. Polygon(A_1, C, D) → poly4, d_4, a_7, c_7
  138. Polygon(A, B, C, D, A_1, E) → poly5, a_8, b_5, c_8, d_5, a_9, e_6
  139. Intersect(e, e_6) → K_1
  140. Intersect(p, e, 1) → L_1
  141. CircleSector(E, J_1, L_1) → f_4
  142. CircleSector(A_1, B_1, N) → g_4
  143. CircleSector(D, C_1, D_1) → h_4
  144. Vector(E, C) → j_3
  145. Vector(A_1, C) → l_3
  146. Vector(D, C) → m_3
  147. Translate(f_4, j_3) → k_4
  148. Translate(g_4, l_3) → p_4
  149. Translate(h_4, m_3) → q_4
  150. Dilate(E, If[2 < o < 3, 3 - o, 0], C) → E'_3
  151. Dilate(A_1, If[3 < o < 4, 4 - o, 0], C) → M_1
  152. Dilate(D, If[4 < o < 5, 5 - o, 0], C) → D'_1
  153. Vector(E, E'_3) → n_3
  154. Vector(A_1, M_1) → u_4
  155. Vector(D, D'_1) → v_4
  156. If(2 < o < 3, Translate[f_4, n_3]) → r_4
  157. If(3 < o < 4, Translate[g_4, u_4]) → s_4
  158. If(4 < o < 5, Translate[h_4, v_4]) → t_4
  159. Ray(C, A_1) → l_4
  160. Ray(C, E) → m_4
  161. Point(l_4) → N_1
  162. Point(m_4) → O_1
  163. Ray(A_1, N_1) → i_4
  164. Ray(E, O_1) → j_4