Q27. - Geometry

Question

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

Express $\mathit{m}\mathbf{\angle }\mathbf{2}$, width="43" height="20" style="max-width: none; vertical-align: -4px;" $\mathit{m}\mathbf{\angle }\mathbf{3}$ and $\mathit{m}\mathbf{\angle }\mathbf{4}$ in terms of $\mathit{t}$ when $\mathbf{\angle }\mathbf{1}\mathbf{=}\mathit{t}$.

Step-by-Step Solution

Answer

The measure of the angles in terms of $t$ is $\mathit{m}\mathbf{\angle }\mathbf{2}\mathbf{=}\mathbf{180}\mathbf{°}\mathbf{-}\mathit{t}$, $\mathit{m}\mathbf{\angle }\mathbf{3}\mathbf{=}\mathit{t}$ and $\mathit{m}\mathbf{\angle }\mathbf{4}\mathbf{=}\mathbf{180}\mathbf{°}\mathbf{-}\mathit{t}$.

Step 1 - State the Angle addition postulate

The Angle addition postulate states that if two angles are placed side by side, then the value of the resulting angle is equal to the sum of the original angles.

Step 2 - Relation between angles

From the above figure it can be observe that $\overrightarrow{a}$ and $\overrightarrow{b}$ both represents straight angles, therefore, resulting in following equations:

$$\begin{gathered} m\angle 1+m\angle 2=180°\dots \dots \dots \left(1\right) \\ m\angle 1+m\angle 4=180°\dots \dots \dots \left(2\right) \\ m\angle 3+m\angle 4=180°\dots \dots \dots \left(3\right) \end{gathered}$$

Step 3 - Calculate the angle m ∠ 2

In order to calculate $m\angle 2$, substitute $t$ for $m\angle 1$ into the equation (1) $m\angle 1+m\angle 2=180°$.

$$\begin{gathered} t+m\angle 2=180° \\ m\angle 2=180°-t \end{gathered}$$

Step 4 - Calculate the angle m ∠ 4

In order to calculate $m\angle 4$, substitute $t$ for $m\angle 1$ into the equation (2)  $m\angle 1+m\angle 4=180°$.

$$\begin{gathered} m\angle 1+m\angle 4=180° \\ t+m\angle 4=180° \\ m\angle 4=180°-t \end{gathered}$$

Step 5 - Calculate the angle m ∠ 3

In order to calculate $m\angle 3$, substitute $180°-t$ for $m\angle 4$ into the equation (2) $m\angle 3+m\angle 4=180°$.

$$\begin{gathered} m\angle 3+m\angle 4=180° \\ m\angle 3+180°-t=180° \\ m\angle 3=180°-180°+t \\ m\angle 3=t \end{gathered}$$ 

Therefore, the measure of the angles is $m\angle 2=180°-t$, $m\angle 3=t$ and $m\angle 4=180°-t$.