Question

in the diagram, $overleftrightarrow{ab}$ and $overleftrightarrow{ec}$ are perpendicular. if $mangle heb=(7x)^{circ}$ and $mangle ceh=(16x - 2)^{circ}$, then the value of $x$ is select choice and $mangle heb=$ select choice.

Explanation:

Step1: Recall perpendicular - angle property

Since $\overrightarrow{AB}$ and $\overrightarrow{EC}$ are perpendicular, $\angle CEB = 90^{\circ}$. And $\angle CEB=\angle CEH+\angle HEB$.

Step2: Set up the equation

We know that $\angle HEB=(7x)^{\circ}$ and $\angle CEH=(16x - 2)^{\circ}$, so $(7x)+(16x - 2)=90$.

Step3: Simplify the left - hand side of the equation

Combining like terms, we get $7x+16x-2 = 23x-2$. So the equation becomes $23x-2 = 90$.

Step4: Solve for x

Add 2 to both sides of the equation: $23x-2 + 2=90 + 2$, which gives $23x=92$. Then divide both sides by 23: $x=\frac{92}{23}=4$.

Step5: Find the measure of $\angle HEB$

Substitute $x = 4$ into the expression for $\angle HEB$. $\angle HEB=(7x)^{\circ}$, so $\angle HEB=7\times4^{\circ}=28^{\circ}$.

Answer:

The value of $x$ is 4 and $m\angle HEB = 28^{\circ}$