Question

in the diagram, $overleftrightarrow{ab}$ and $overrightarrow{ec}$ are perpendicular. if $mangle heb=(9x)^{circ}$ and $mangle ceh=(13x + 2)^{circ}$, then the value of $x$ is select choice and $mangle heb=$ select choice$^{circ}$.

Explanation:

Step1: Recall perpendicular - angle property

Since $\overrightarrow{AB}$ and $\overrightarrow{EC}$ are perpendicular, $\angle CEB = 90^{\circ}$. Also, $\angle CEB=\angle CEH+\angle HEB$. So, $(13x + 2)+9x=90$.

Step2: Combine like - terms

Combining the $x$ terms on the left - hand side of the equation, we get $13x+9x + 2=90$, which simplifies to $22x+2 = 90$.

Step3: Isolate the variable term

Subtract 2 from both sides of the equation: $22x+2 - 2=90 - 2$, resulting in $22x=88$.

Step4: Solve for $x$

Divide both sides of the equation by 22: $x=\frac{88}{22}=4$.

Step5: Find $m\angle HEB$

Substitute $x = 4$ into the expression for $m\angle HEB$. Since $m\angle HEB=(9x)^{\circ}$, then $m\angle HEB=9\times4 = 36^{\circ}$.

Answer:

$x = 4$; $m\angle HEB=36$