Chapter 5.3, Problem 47HP

To determine

To find: The numbered angles in the order from smallest to largest given that $m\angle 2=m\angle 5$ and then explain the reasoning.

Answer to Problem 47HP

The required order of the angles is $m\angle 1<m\angle 2=m\angle 5<m\angle 4<m\angle 6<m\angle 3$ .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider a triangle that has one side longer than the other side only when the angle that is only that is opposite to the longer side greater measure than the angle opposite to the shorter side.

By the angle-side relations $m\angle 1<m\angle 2<m\angle 6$ as $BC<AB<AC$ . Also, $m\angle 5<m\angle 4<m\angle 6$ as $Bd<CD<CB$ .

It is given that $m\angle 2=m\angle 5$ , then, by the property of inequality,

  $\begin{array}{c}m\angle 1<m\angle 2\\ <m\angle 5\\ <m\angle 4\end{array}$

Consider the triangle sum property as,

  $\begin{array}{l}m\angle 1+m\angle 2+m\angle 3=180\\ m\angle 4+m\angle 5+m\angle 6=180\end{array}$

Then, it follows that,

  $\begin{array}{c}m\angle 1+m\angle 3=180-m\angle 2\\ =180-m\angle 5\\ =m\angle 4+m\angle 6\end{array}$

Since, $m\angle 1<m\angle 4$ and $m\angle 1+m\angle 3=m\angle 4+m\angle 6$ , then $m\angle 3>m\angle 6$ and by the transitive property of inequality,

  $m\angle 1<m\angle 2=m\angle 5<m\angle 4<m\angle 6<m\angle 3$