Question

a. Draw a circle. Place points A,B, and C on it in such positions that

\$[m\stackrel{⏜}{AB}+m\stackrel{⏜}{BC}\mathrm{\backslash ]}$does not equal

b. Does your example in part (a) contradict Postulate $16$?

Short answer

(a)The final answer is $50°$and $310°$.

(b)The final answer is $NO$.

Step-by-step solution

a.Step 1. Given information:

The given information is about the circle where $m\stackrel{⏜}{AB}$ and $m\stackrel{⏜}{BC}$ are the arcs of the circle.

Step 2. Concept used:

$m\stackrel{⏜}{AB}$ 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 $m\stackrel{⏜}{AC}.$

Step 3. Applying the concept:

$mAB$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,$\begin{array}{c}m\stackrel{⏜}{AB}=90+110\\ =200\end{array}$

$\begin{array}{l}m\stackrel{⏜}{BC}=110\\ m\stackrel{⏜}{AC}=90\end{array}$

b.Step 1. Given information:

The given information is about the circle where $m\stackrel{⏜}{AB}$ and $m\stackrel{⏜}{BC}$ are the arcs of the circle

Step 2. Concept used:

$m\stackrel{⏜}{AB}$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 $m\stackrel{⏜}{AC}.$

Step 3. Applying the concept:

$m\stackrel{⏜}{AB}$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,$\begin{array}{c}m\stackrel{⏜}{AB}=90+110\\ =200\end{array}$

$\begin{array}{l}m\stackrel{⏜}{BC}=110\\ m\stackrel{⏜}{AC}=90\end{array}$

According to theorem,

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

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