SOLUTION: Prove (by contradiction) that any perpendicular height of a scalene triangle cannot bisect any angle of the triangle.

Algebra.Com
Question 957800: Prove (by contradiction) that any perpendicular height of a scalene triangle cannot bisect any angle of the triangle.
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
To prove by contradiction, we will assume the opposite and then show that this is impossible. Here are some steps to follow:
  1. Draw an acute scalen triangle and label the vertices A, B and C

  2. From vertex B draw a perpendicular height to side AC

  3. Label the point where the height intersects AC as D

  4. Assuming that this height bisects angle ABC, angles ABD and CBD must have equal measures. Label these two angles as x degrees.

  5. Triangles ABD and CBD both have angles of 90 and x degrees. Since they both have a total of 180 degrees, angles A and C must be 180 - (90 + x) or 90 - x degrees. This means angles A and C are congruent.

  6. Angles A and C are also angles of triangle ABC. Since these angles are congruent, the sides opposite them, sides AB and CB, must also be congruent.

  7. If sides AB and CB are congruent then triangle ABC is not scalene. This is the contradiction which shows why the height of a scalene triangle cannot possibly bisect an angle of the triangle.