Question

find angle gcd * do not put degree symbol if cd ⊥ ef, m∠ech=(x + 5)° and m∠hcd=(3x - 7)°, find each missing value.

Explanation:

Step1: Use perpendicular - angle property

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

Step2: Combine like - terms

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

Step3: Solve for $x$

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

Step4: Find $\angle ECH$

Substitute $x = 23$ into the expression for $\angle ECH$: $\angle ECH=(x + 5)^{\circ}=(23 + 5)^{\circ}=28^{\circ}$.

Step5: Find $\angle HCD$

Substitute $x = 23$ into the expression for $\angle HCD$: $\angle HCD=(3x-7)^{\circ}=(3\times23-7)^{\circ}=(69 - 7)^{\circ}=62^{\circ}$.

Answer:

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