Question

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).
So, $(x + 5)+(3x-7)=90$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $x+3x + 5-7=90$, which gives $4x-2 = 90$.

Step3: Solve for $x$

Add 2 to both sides: $4x-2 + 2=90 + 2$, so $4x=92$.
Then 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:

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