Show the similarity between a triangle with angles A , B , and C , where m∠A=47° and m∠B=36° , and another triangle with angles G , H , and J , where m∠G=36° and m∠J=97°

1 answer

To determine the similarity between the two triangles, we can start by summarizing the information given for each triangle.

Triangle 1:

  • Angles: \( A, B, C \)
  • Given: \( m∠A = 47° \) and \( m∠B = 36° \)

To find angle \( C \):

\[ m∠C = 180° - m∠A - m∠B = 180° - 47° - 36° = 97° \]

Thus, we have:

  • \( m∠A = 47° \)
  • \( m∠B = 36° \)
  • \( m∠C = 97° \)

Triangle 2:

  • Angles: \( G, H, J \)
  • Given: \( m∠G = 36° \) and \( m∠J = 97° \)

To find angle \( H \):

\[ m∠H = 180° - m∠G - m∠J = 180° - 36° - 97° = 47° \]

Thus, we have:

  • \( m∠G = 36° \)
  • \( m∠H = 47° \)
  • \( m∠J = 97° \)

Comparing the Angles: Now we can list the angles of both triangles:

Triangle 1:

  • \( m∠A = 47° \)
  • \( m∠B = 36° \)
  • \( m∠C = 97° \)

Triangle 2:

  • \( m∠G = 36° \)
  • \( m∠H = 47° \)
  • \( m∠J = 97° \)

Identifying Corresponding Angles:

  • \( m∠A = 47° \) corresponds to \( m∠H = 47° \)
  • \( m∠B = 36° \) corresponds to \( m∠G = 36° \)
  • \( m∠C = 97° \) corresponds to \( m∠J = 97° \)

Since all corresponding angles are equal, we can conclude that Triangle 1 and Triangle 2 are similar by the Angle-Angle (AA) similarity postulate.

Thus, the two triangles are similar because they have the same angles, which means their shapes are identical, even if their sizes may differ.