Q26. - 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.

Find $\mathit{m}\mathbf{\angle }\mathbf{2}$, $\mathit{m}\mathbf{\angle }\mathbf{3}$ and $\mathit{m}\mathbf{\angle }\mathbf{4}$ when the measure of $\mathbf{\angle }\mathbf{1}$ is:

a. 90    

b. 93

Step-by-Step Solution

Answer

a. The measure of the angles is $\mathit{m}\mathbf{\angle }\mathbf{2}\mathbf{=}\mathbf{90}\mathbf{°}$, $\mathit{m}\mathbf{\angle }\mathbf{3}\mathbf{=}\mathbf{90}\mathbf{°}$ and $\mathit{m}\mathbf{\angle }\mathbf{4}\mathbf{=}\mathbf{90}\mathbf{°}$.

b. The measure of the angles is $\mathit{m}\mathbf{\angle }\mathbf{2}\mathbf{=}\mathbf{87}\mathbf{°}$, $\mathit{m}\mathbf{\angle }\mathbf{3}\mathbf{=}\mathbf{93}\mathbf{°}$ and $\mathit{m}\mathbf{\angle }\mathbf{4}\mathbf{=}\mathbf{87}\mathbf{°}$.

Part a. 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.

Part a. 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:

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

Part a. Step 3 - Calculate the angle m ∠ 2

In order to calculate $m\angle 2$, substitute $90°$ for $m\angle 1$ into the equation (1) $m\angle 1+m\angle 2=180°.$

$$\begin{gathered} 90°+m\angle 2=180° \\ m\angle 2=180°-90° \\ m\angle 2=90° \end{gathered}$$

Part a. Step 4 - Calculate the angle m ∠ 3

In order to calculate $m\angle 4$, substitute $90°$ for $m\angle 1$ into the equation (2)  $m\angle 1+m\angle 4=180°$.

$$\begin{gathered} m\angle 1+m\angle 4=180° \\ 90°+m\angle 4=180° \\ m\angle 4=180°-90° \\ m\angle 4=90° \end{gathered}$$

Part a. Step 5 - Calculate the angle m ∠ 3

In order to calculate $m\angle 3$, substitute $90°$ for $m\angle 4$ into the equation (2) $m\angle 3+m\angle 4=180°$.

$$\begin{gathered} m\angle 3+m\angle 4=180° \\ m\angle 3+90°=180° \\ m\angle 3=180°-90° \\ m\angle 3=90° \end{gathered}$$

Therefore, the measure of the angles is $m\angle 2=90°$, $m\angle 3=90°$ and $m\angle 4=90°$.

Part b. Step 1 - State the Angle addition postulate

The postulate state 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.

Part b. Step 2 - Relation between angles

From the above figure it can be observe that $\overrightarrow{a}$ and $\overrightarrow{b}$ both represents straight angles. The following equations are:

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

Part b. Step 3 - Calculate the angle m ∠ 2

In order to calculate $m\angle 2$, substitute $93°$ for $m\angle 1$ into the equation (1) $m\angle 1+m\angle 2=180°$.

$$\begin{gathered} 93°+m\angle 2=180° \\ m\angle 2=180°-93° \\ m\angle 2=87° \end{gathered}$$

Step 4 - Calculate the angle m ∠ 4

In order to calculate $m\angle 4$, substitute $93°$ for $m\angle 1$ into the equation (2) $m\angle 1+m\angle 4=180°$.

$$\begin{gathered} m\angle 1+m\angle 4=180° \\ 93°+m\angle 4=180° \\ m\angle 4=180°-93° \\ m\angle 4=87° \end{gathered}$$

Step 5 - Calculate the angle m ∠ 3

In order to calculate $m\angle 3$, substitute $87°$ for $m\angle 4$ into the equation (2) $m\angle 3+m\angle 4=180°$.

$$\begin{gathered} m\angle 3+m\angle 4=180° \\ m\angle 3+87°=180° \\ m\angle 3=180°-87° \\ m\angle 3=93° \end{gathered}$$

Therefore, the measure of the angles is $m\angle 2=87°$, $m\angle 3=93°$ and $m\angle 4=87°$.