Q. 23 WE - Geometry

Question

Draw a line $\overleftrightarrow{\mathit{A}\mathit{B}}$. Choose a point O between A and B. Use a protractor to investigate the following questions. 

In the plane represented by your paper, how, many lines can you draw through O that will form a $\mathbf{30}\mathbf{°}$ angle with $\overleftrightarrow{\mathit{O}\mathit{B}}$?

Step-by-Step Solution

Answer

Maximum 2 lines can be drawn.

Step 1 - Apply the protractor postulate

$\overline{\mathit{A}\mathit{H}}$ in a given plane is considered.

Any point O between A and H is chosen.

Consider $\overline{\mathit{O}\mathit{A}}\text{\hspace{0.17em}and}\overline{\mathit{O}\mathit{H}}$ and all these rays are drawn from O on one side of $\overline{\mathit{A}\mathit{H}}$.

These rays’ results in pairing with the real numbers from 0 to 180 in such a way that:

The line $\overline{\mathit{O}\mathit{A}}$ is paired with 0, and $\overline{\mathit{O}\mathit{H}}$ with 180.

If $\overline{\mathit{O}\mathit{B}}$ is paired with $\mathit{x}$, and $\overline{\mathit{O}\mathit{C}}$ with $\mathit{y}$, then $\mathit{m}\mathit{\angle }\mathit{B}\mathit{O}\mathit{C}\mathit{=}\left|x-y\right|$.

Step 2 - Sketch the figure

Thus, the figure is shown below, such that two lines are drawn through O that will form a $30°$ angle.

Step 3 -Observation of the figure

As it can be seen from the above figure, that on the plane of the paper through point O between A and B, there are maximum two lines that will make $30°$ with $\overline{OB}$.

The line $\overline{OC}$ can be drawn using protractor on one side of the $\overline{AB}$ and $\overline{OD}$ on the other side of the.

Therefore, only 2 lines can be drawn through point O to form $30°$ with $\overline{OB}$.