Chapter 9.3, Problem 9WE

(a)

To determine

To Draw: a circle having points A,B, and C in a position such that $m\stackrel{⌢}{AB}+m\stackrel{⌢}{BC}$ does not equal to $m\stackrel{⌢}{AC}$ .

Answer to Problem 9WE

  ${50}^{\circ }\text{ and }{310}^{\circ }$

Explanation of Solution

Given:

Calculation:

  $m\stackrel{⌢}{AB}$ means central angle of arc AB and $m\stackrel{⌢}{BC}$ means central angle of arc BC.

In the figure shown below, it is clear that $m\stackrel{⌢}{AB}+m\stackrel{⌢}{BC}$ does not equal to $m\stackrel{⌢}{AC}$ .

  

Here, $m\stackrel{⌢}{AB}=90+110=200$ , $m\stackrel{⌢}{BC}=110$ and $m\stackrel{⌢}{AC}=90$

(b)

To determine

To Find: whether a circle having points A,B, and C in a position such that $m\stackrel{⌢}{AB}+m\stackrel{⌢}{BC}$ does not equal to $m\stackrel{⌢}{AC}$ contradict postulate which says “the measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs”.

Answer to Problem 9WE

No

Explanation of Solution

Given:

Calculation:

  $m\stackrel{⌢}{AB}$ means central angle of arc AB and $m\stackrel{⌢}{BC}$ means central angle of arc BC.

In the figure shown below, it is clear that $m\stackrel{⌢}{AB}+m\stackrel{⌢}{BC}$ does not equal to $m\stackrel{⌢}{AC}$ .

  

Here, $m\stackrel{⌢}{AB}=90+110=200$ , $m\stackrel{⌢}{BC}=110$ and $m\stackrel{⌢}{AC}=90$

According to theorem.

  $m\stackrel{⌢}{AC}+m\stackrel{⌢}{BC}=m\stackrel{⌢}{AB}$

It does not contradict the postulate ,here points are interchanged.