Question

(ma912.gr.1.1)
in the diagram, $overleftrightarrow{ab}$ and $overleftrightarrow{ec}$ are perpendicular.
if $mangle heb=(9x)^{circ}$ and $mangle ceh=(19x + 6)^{circ}$, then the value of x is and $mangle heb=$

Explanation:

Step1: Recall angle - sum property

Since $\overleftrightarrow{AB}$ and $\overleftrightarrow{EC}$ are perpendicular, $\angle CEB = 90^{\circ}$. And $\angle CEB=\angle CEH+\angle HEB$. So, $(19x + 6)+9x=90$.

Step2: Combine like - terms

Combining the $x$ terms on the left - hand side of the equation, we get $19x+9x + 6=90$, which simplifies to $28x+6 = 90$.

Step3: Isolate the variable term

Subtract 6 from both sides of the equation: $28x+6 - 6=90 - 6$, resulting in $28x=84$.

Step4: Solve for x

Divide both sides of the equation by 28: $x=\frac{84}{28}=3$.

Step5: Find the measure of $\angle HEB$

Substitute $x = 3$ into the expression for $\angle HEB$. Since $m\angle HEB=(9x)^{\circ}$, then $m\angle HEB=9\times3 = 27^{\circ}$.

Answer:

$x = 3$; $m\angle HEB=27$