Chapter 15, Problem 15RP

To determine

To Prove: In the given one of its side is smaller than the other.

Explanation of Solution

Given:

  $\text{In a }\Delta AC<AD$

  

  $\overrightarrow{AB}\cong \overrightarrow{AD}$

Formula Used:

  $\begin{array}{l}\text{Angle inequality theorem}\text{.: It states that the exterior angle of any triangle is always greater}\\ \text{than the both interior angle}\text{.}\end{array}$

In a triangle angle opposite to larger side is always larger, and smallest side is smallest.

  $\text{Property of Congruence}\text{.}$

Calculation:

  $\begin{array}{l}\text{by reflexive property of congruence}\\ AC=AC\\ \text{As exterior angle must be larger than its remote angles}\text{.}\\ \angle A<\angle C\\ bySAS\text{ inequality theorem}\\ AC<AD\\ \text{Hence Proved}\text{.}\end{array}$

Conclusion:

Hence, $\text{Proved}\text{.}$