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Hexagon Angle Theorems: Interior and Exterior Sums

Hexagon Angle Theorems: Interior and Exterior Sums
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2016-11-07 08:41
PolygonAngleTransformationGeometric ConstructionSlider Animation
In collections Polygons & Angles
This interactive GeoGebra applet allows students to discover the sum of interior angles and the sum of exterior angles of a hexagon through dynamic manipulation. The core mathematical concepts include the polygon interior angle sum formula (n-2)×180° and the invariant exterior angle sum of 360° for any convex polygon. Users can drag vertices to observe how individual angles change while the sums remain constant. The construction employs commands such as Circle, Ray, Intersect, Segment, Vector, Translate, Dilate, Polygon, Point, Midpoint, Rotate, Line, and CircleArc, along with objects like A_1, D, C, h, c, P, A, B, a, b, E, e, f, g, k, p, i, j, F, G, H, I, J, K, O, Q, R, S, T, U, V, q, r, s, c_1, d_1, W, h_1, p_1, q_1. It is suitable for middle school geometry lessons, enabling inquiry-based learning of angle theorems, and also serves as a visual aid for high school review of polygon properties.

GeoGebra Commands & Parameters

  1. Circle(C, 1.5) → h
  2. Ray(C, D) → c
  3. Intersect(h, c, 1) → P
  4. Ray(A, B) → a
  5. Ray(B, C) → b
  6. Ray(E, A) → e
  7. Circle(A, 1.5) → f
  8. Circle(B, 1.5) → g
  9. Circle(D, 1.5) → k
  10. Circle(E, 1.5) → p
  11. Segment(C, A) → i
  12. Segment(C, E) → j
  13. Intersect(f, a, 1) → F
  14. Intersect(f, i, 1) → G
  15. Intersect(f, e, 1) → H
  16. Intersect(f, e, 2) → I
  17. Intersect(p, e) → J,
  18. Intersect(p, j) → K,
  19. Intersect(k, c, 2) → O
  20. Intersect(h, j, 1) → Q
  21. Intersect(h, i, 1) → R
  22. Intersect(h, b, 1) → S
  23. Intersect(g, a, 2) → T
  24. Intersect(g, b, 1) → U
  25. Intersect(g, a, 1) → V
  26. CircleSector(A, F, G) → q
  27. CircleSector(A, G, H) → r
  28. CircleSector(B, U, V) → s
  29. CircleSector(C, Q, R) → c_1
  30. CircleSector(C, R, S) → d_1
  31. Intersect(k, c, 1) → W
  32. CircleSector(E, J, K) → h_1
  33. CircleSector(A, I, F) → p_1
  34. CircleSector(B, T, U) → q_1
  35. Intersect(h, b, 2) → Z
  36. CircleSector(C, Z, P) → r_1
  37. Vector(A, B) → u
  38. Vector(C, B) → v
  39. Vector(E, D) → w
  40. Vector(C, D) → u_1
  41. Vector(C, E) → v_1
  42. Vector(A, E) → w_1
  43. Translate(d_1, v) → s_1
  44. Dilate(C, If[1 < l ≤ 2, 2 - l, 0], E) → C'_1
  45. Dilate(C, If[2 < l ≤ 3, 3 - l, 0], D) → C'_2
  46. Dilate(E, If[2 < l ≤ 3, 3 - l, 0], D) → E'
  47. Vector(A, B) → a_1
  48. Vector(E, E') → j_1
  49. Vector(C, C'_2) → l_1
  50. Vector(C, C'_1) → m_1
  51. Translate(r, w_1) → r'_2
  52. Translate(c_1, v_1) → f_2
  53. Translate(c_1, m_1) → g_2
  54. Translate(q, a_1) → r'
  55. Vector(D, C) → u_2
  56. Vector(E, C) → v_2
  57. Vector(A, C) → w_2
  58. Vector(B, C) → l_2
  59. Dilate(B, If[0 < o < 1, 1 - o, 0], C) → B'
  60. Dilate(A, If[1 < o < 2, 2 - o, 0], C) → A'_2
  61. Dilate(E, If[2 < o < 3, 3 - o, 0], C) → E'_1
  62. Dilate(D, If[3 < o < 4, 4 - o, 0], C) → D'
  63. Vector(B, B') → m_2
  64. Vector(A, A'_2) → n_2
  65. Vector(E, E'_1) → s_2
  66. Vector(D, D') → t_2
  67. Translate(q_1, m_2) → c_3
  68. Translate(q_1, l_2) → d_3
  69. Translate(p_1, w_2) → e_3
  70. If(1 < o < 2, Translate[p_1, n_2]) → h_3
  71. Ray(D, A_1) → a_3
  72. Ray(A_1, E) → b_3
  73. Circle(A_1, 1.5) → q_3
  74. Intersect(q_3, b_3, 1) → N
  75. Intersect(q_3, a_3, 2) → B_1
  76. Intersect(k, c, 2) → C_1
  77. Intersect(k, a_3, 1) → D_1
  78. Intersect(k, c, 1) → E_1
  79. Intersect(p, b_3, 1) → F_1
  80. Dilate(C, If[2 < l < 3, 3 - l, 0], A_1) → C'_3
  81. Vector(E, A_1) → f_1
  82. Vector(C, A_1) → g_1
  83. Vector(C, C'_3) → b_2
  84. Intersect(q_3, a_3, 1) → I_1
  85. CircleSector(D, D_1, E_1) → k_3
  86. Vector(C, D) → j_2
  87. Vector(A_1, D) → u_3
  88. Intersect(p, b_3, 2) → J_1
  89. Polygon(A, B, C) → poly1, c_4, a_4, b_4
  90. Polygon(A, C, E) → poly2, e_4, a_5, c_5
  91. Polygon(E, C, A_1) → poly3, a_6, e_5, c_6
  92. Polygon(A_1, C, D) → poly4, d_4, a_7, c_7
  93. Polygon(A, B, C, D, A_1, E) → poly5, a_8, b_5, c_8, d_5, a_9, e_6
  94. Intersect(e, e_6) → K_1
  95. Intersect(p, e, 1) → L_1
  96. CircleSector(E, J_1, L_1) → f_4
  97. CircleSector(A_1, B_1, N) → g_4
  98. CircleSector(D, C_1, D_1) → h_4
  99. Vector(E, C) → j_3
  100. Vector(A_1, C) → l_3
  101. Vector(D, C) → m_3
  102. Translate(f_4, j_3) → k_4
  103. Translate(g_4, l_3) → p_4
  104. Translate(h_4, m_3) → q_4
  105. Dilate(E, If[2 < o < 3, 3 - o, 0], C) → E'_3
  106. Dilate(A_1, If[3 < o < 4, 4 - o, 0], C) → M_7
  107. Dilate(D, If[4 < o < 5, 5 - o, 0], C) → D'_1
  108. Vector(E, E'_3) → n_3
  109. Vector(A_1, M_7) → u_4
  110. Vector(D, D'_1) → v_4
  111. If(2 < o < 3, Translate[f_4, n_3]) → r_4
  112. If(3 < o < 4, Translate[g_4, u_4]) → s_4
  113. If(4 < o < 5, Translate[h_4, v_4]) → t_4
  114. Ray(C, A_1) → l_4
  115. Ray(C, E) → m_4
  116. Point(l_4) → N_1
  117. Point(m_4) → O_1
  118. Segment(D, C) → n_4
  119. Segment(C, B) → f_5
  120. Segment(B, A) → g_5
  121. Segment(A, E) → h_5
  122. Segment(E, A_1) → i_5
  123. Segment(A_1, D) → j_5
  124. Midpoint(f_5) → M_2
  125. Midpoint(g_5) → M_1
  126. Rotate(d_1, If[0 < l ≤ 1, (((-l) * 180))°, -180°], M_2) → d
  127. Rotate(C, If[0 < l ≤ 1, (((-l) * 180))°, -180°], M_2) → C'
  128. Segment(M_2, C') → t
  129. Rotate(q, If[0 < l ≤ 1, ((l * 180))°, 180°], M_1) → q'
  130. Rotate(A, If[0 < l ≤ 1, ((l * 180))°, 180°], M_1) → A'
  131. Segment(M_1, A') → e_1
  132. Rotate(R, 180°, M_2) → R'
  133. Line(B, R') → k_1
  134. Ray(E, O_1) → i_1
  135. Midpoint(h_5) → M_6
  136. Rotate(r, If[1 < l ≤ 2, (((-(l - 1)) * 180))°, -180°], M_6) → r'_1
  137. Rotate(A, If[1 < l ≤ 2, (((-(l - 1)) * 180))°, -180°], M_6) → A'_1
  138. Segment(M_6, A'_1) → n_1
  139. CircleSector(E, K, F_1) → c_2
  140. Midpoint(i_5) → M_5
  141. Rotate(E, If[2 < l ≤ 3, (((-(l - 2)) * 180))°, -180°], M_5) → E'_2
  142. Segment(M_5, E'_2) → a_2
  143. Rotate(c_2, If[2 < l ≤ 3, (((-(l - 2)) * 180))°, -180°], M_5) → d_2
  144. Segment(A_1, C) → e_2
  145. Intersect(h, e_2, 1) → U_1
  146. CircleSector(C, U_1, Q) → h_2
  147. CircleSector(C, P, U_1) → k_2
  148. Vector(C, C'_3) → t_1
  149. Translate(h_2, t_1) → p_2
  150. Intersect(q_3, e_2, 1) → R_1
  151. CircleSector(A_1, N, R_1) → q_2
  152. CircleSector(A_1, R_1, I_1) → r_2
  153. Vector(C, A_1) → i_2
  154. Translate(U_1, i_2) → S_1
  155. Ray(A_1, S_1) → f_3
  156. Midpoint(j_5) → M_4
  157. Midpoint(n_4) → M_3
  158. Rotate(k_2, If[3 < l ≤ 4, (((l - 3) * 180))°, 180°], M_3) → g_3
  159. Rotate(C, If[3 < l ≤ 4, (((l - 3) * 180))°, 180°], M_3) → C'_4
  160. Rotate(U_1, 180°, M_3) → W_1
  161. Segment(M_3, C'_4) → i_3
  162. Rotate(r_2, If[3 < l ≤ 4, (((-(l - 3)) * 180))°, -180°], M_4) → p_3
  163. Rotate(A_1, If[3 < l ≤ 4, (((-(l - 3)) * 180))°, -180°], M_4) → Z_1
  164. Segment(M_4, Z_1) → r_3
  165. Line(D, W_1) → s_3
  166. CircleArc(C, R, S) → t_3
  167. CircleArc(A, F, G) → k_5
  168. CircleArc(C, Q, R) → p_5
  169. CircleArc(A, G, H) → q_5
  170. CircleArc(C, U_1, Q) → r_5
  171. CircleArc(E, K, F_1) → s_5
  172. CircleArc(C, P, U_1) → t_5
  173. CircleArc(A_1, R_1, I_1) → d_6