Question

Soloman attempts to construct a triangle similar to triangle XYZ.

x,. y,. z,.

Soloman constructs his triangle x prime, y prime, z prime,. by making angles x prime,. and y prime,. half the measures of angles X and Y, respectively. Is his triangle x prime, y prime, z prime,. similar to triangle XYZ? If so, name the theorem that indicates similarity. If not, explain why not.

Answer 1

No — not in general.

Reason: Triangles are similar if two corresponding angles are equal (AA similarity). Soloman made x' = X/2 and y' = Y/2, so neither x' nor y' equals the original X or Y (unless an angle were 0). Also the third angle of his triangle is
z' = 180° − (x' + y') = 180° − (X+Y)/2 = 90° + Z/2,
which is not equal to Z in general. For example, if XYZ is equilateral (60°,60°,60°), then x'y'z' has angles 30°,30°,120°, which is not similar to 60°,60°,60°.

Thus his construction does not produce a triangle similar to XYZ (AA similarity would be required but does not hold).