Question 957800
To prove by contradiction, we will assume the opposite and then show that this is impossible. Here are some steps to follow:
<ol><li>Draw an acute scalen triangle and label the vertices A, B and C</li>
<li>From vertex B draw a perpendicular height to side AC</li>
<li>Label the point where the height intersects AC as D</li>
<li>Assuming that this height bisects angle ABC, angles ABD and CBD must have equal measures. Label these two angles as x degrees.</li>
<li>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.</li>
<li>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.</li>
<li>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.</li><ul>