Chapter 15.2, Problem 18PSB

To determine

To Prove:The given sides of the triangle $AD>CD$ .

Explanation of Solution

Given:

  

  $\overrightarrow{AC}\text{ bisects }\angle \text{BAD}$

Formula Used:

Triangle Inequality Theorem: It states that the sum of any two sides of a triangle is greater than the third.

Calculation:

  $\begin{array}{l}\angle 1\cong \angle 2\left(\text{angle bisector}\right)\\ \angle 1=\angle 2\left(\text{congruent angles}\right)\\ \angle 3>\angle 1\left(\text{exterior angle inequality}\right)\\ \text{ so by this }\\ \angle 3>\angle 2\\ \text{and by this }\\ AD>CD\\ \text{hence proved}\text{.}\end{array}$

Conclusion:

Hence, Proved.