Question

do not put degree symbol
if $overline{cd}perpoverline{ef}$, $mangle ech=(x + 5)^{circ}$ and $mangle hcd=(3x - 7)^{circ}$, find each missing value.

Explanation:

Step1: Use perpendicular - angle relationship

Since $\overline{CD}\perp\overline{EF}$, $\angle ECD = 90^{\circ}$, and $\angle ECH+\angle HCD=\angle ECD$. So $\angle ECH+\angle HCD = 90$.
$(x + 5)+(3x-7)=90$.

Step2: Combine like - terms

Combine the $x$ terms and the constant terms:
$x+3x+5 - 7=90$.
$4x-2 = 90$.

Step3: Solve for $x$

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

Step4: Find $\angle ECH$

Substitute $x = 23$ into the expression for $\angle ECH$:
$\angle ECH=x + 5=23+5=28$.

Step5: Find $\angle HCD$

Substitute $x = 23$ into the expression for $\angle HCD$:
$\angle HCD=3x-7=3\times23-7=69 - 7=62$.

Answer:

$x = 23$, $\angle ECH=28$, $\angle HCD=62$