Question

do not put degrees 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 property

Since $\overline{CD}\perp\overline{EF}$, $\angle ECD+\angle HCD = 90^{\circ}$ (perpendicular lines form right - angles).
$(x + 57)+(3x-7)=90$

Step2: Combine like terms

$x+3x+57 - 7=90$
$4x + 50=90$

Step3: Isolate the variable term

Subtract 50 from both sides:
$4x=90 - 50$
$4x=40$

Step4: Solve for x

Divide both sides by 4:
$x = 10$

Step5: Find $\angle ECH$

$\angle ECH=x + 57$. Substitute $x = 10$:
$\angle ECH=10+57=67$

Step6: Find $\angle HCD$

$\angle HCD=3x-7$. Substitute $x = 10$:
$\angle HCD=3\times10-7=23$

Answer:

$x = 10$, $\angle ECH = 67$, $\angle HCD = 23$