Q. 34 WE - Geometry

Question

$\overrightarrow{\mathit{A}\mathit{L}}$ bisects $\mathit{\angle }\mathit{K}\mathit{A}\mathit{T}$. Find the value of $\mathit{x}$.

$\mathit{m}\mathbf{\angle }\mathbf{2}\mathbf{=}\mathit{x}\mathbf{+}\mathbf{12}$, $\mathit{m}\mathbf{\angle }\mathbf{3}\mathbf{=}\mathbf{6}\mathit{x}\mathbf{-}\mathbf{20}$

Step-by-Step Solution

Answer

The value of $\mathit{x}\mathbf{=}\mathbf{22}\mathbf{°}$.

Step 1 - Line bisector and straight angle

If a line bisects angles, it divides it into two equal parts.

Measure of straight angle is $180°$.

Step 2 - Observe the figure

The given figure is

From the figure, since $\overrightarrow{AL}$ bisects $\angle KAT$ , therefore $m\angle 1=m\angle 2=x+12$,

Also $\angle SAT$ is a straight angle therefore,  $m\angle 1+m\angle 2+m\angle 3=180°$   

Step 3 - Calculate x .

In order to calculate the value of $x$, substitute $x+12$ for $m\angle 1$, $x+12$ for $m\angle 2$, $6x-2$ for $m\angle 3$ into the equation $m\angle 1+m\angle 2+m\angle 3=180°$  and simplify.

$$\begin{gathered} x+12+x+12+6x-20=180° \\ 8x+4=180° \\ 8x=176° \\ x=\frac{176°}{8} \\ x=22° \end{gathered}$$

Therefore, $x=22°$