Chapter 2.6, Problem 23WE

To determine

To find: The information depicted in red part of the given figure.

Answer to Problem 23WE

Ray $\overrightarrow{OF}$ bisects angle $\angle COD$ , so that $\angle 3\cong \angle 4$ .

Explanation of Solution

Given information: $\angle AOB$ is given so that ray $\overrightarrow{OE}$ , bisects this angle. Now rays $\overrightarrow{OE},\overrightarrow{OA}$ and $\overrightarrow{OB}$ are drawn in their opposite directions to form angle $\angle COD$ as shown in below figure,

  

Concept used: Use the property that when two lines intersect each other at a single point, then two opposite angles formed at this points are equal to each other.

As lines $\overleftrightarrow{EF}$ and $\overleftrightarrow{BC}$ intersect each other at point O, so its vertical opposite angles $\angle 3$ and $\angle 2$ are equal.

  $\Rightarrow \angle 2=\angle 3\mathrm{......}(i)$

Similarly, As lines $\overleftrightarrow{EF}$ and $\overleftrightarrow{AD}$ intersect each other at point O, so its vertical opposite angles $\angle 4$ and $\angle 1$ are equal.

  $\Rightarrow \angle 1=\angle 4\mathrm{......}(i)$

But as $\angle AOB$ is bisected by ray $\overrightarrow{OE}$ , so, $\angle 1$ and $\angle 2$ are congruent. Thus, by transitive property,

  $\angle 3=\angle 4$

That is possible only when ray $\overrightarrow{OF}$ bisects angle $\angle COD$ .

Conclusion: So, red part of the figure shows that ray $\overrightarrow{OF}$ bisects angle $\angle COD$ as its adjacent angles $\angle 3=\angle 4$ .