Chapter 5.6, Problem 41HP

To determine

To explain: $\angle BDC$ is always, sometimes, or never an acute angle.

Answer to Problem 41HP

  $\angle BDC$ is always obtuse never acute.

Explanation of Solution

Given:Where $\overline{BD}$ is a median and $AB<BC$ .

$\overline{BD}$ is a median and $AB<BC$ are given and it has to explain $\angle BDC$ is always, sometimes, or never an acute angle.

  

Never; from the converse of the hinge theorem, $\angle ADB<\angle BDC$ .

  $\angle ADB$ and $\angle BDC$ form a linear pair.

So, $m\angle ADB+\angle BDC=180.$

Since, $m\angle ADB<m\angle BDC,$

  $m\angle BDC$ must be greater than 90 and $m\angle ADB$ must be smaller than 90.

So, by the definition of obtuse and acute angles, $m\angle BDC$ is always obtuse and $m\angle ADB$ is always acute.