Question

find the value of the variables in the diagrams below:

Explanation:

Step1: Set up equation for the first diagram

Since $\angle ABC$ and $\angle CBD$ are a linear - pair, their sum is $180^{\circ}$. So, $(5x - 22)+(8x + 46)=180$.
\[

$$\begin{align*} 5x-22 + 8x+46&=180\\ 13x+24&=180 \end{align*}$$

\]

Step2: Solve for $x$ in the first diagram

Subtract 24 from both sides: $13x=180 - 24=156$. Then divide both sides by 13: $x=\frac{156}{13}=12$.

Step3: Set up equation for the second diagram

Since $\angle ABE = 90^{\circ}$ and $\angle ABC+\angle CBD = 90^{\circ}$, we have $2x+(3x - 10)=90$.
\[

$$\begin{align*} 2x+3x-10&=90\\ 5x-10&=90 \end{align*}$$

\]

Step4: Solve for $x$ in the second diagram

Add 10 to both sides: $5x=90 + 10=100$. Then divide both sides by 5: $x = 20$.

Answer:

For the first diagram, $x = 12$. For the second diagram, $x = 20$.