Question

1. if $overline{ab}perpoverline{cd}$, $mangle dce=(7x + 2)^{circ}$ and $mangle ecb=(x + 8)^{circ}$, find the measure of $angle dce$.

Explanation:

Step1: Use perpendicular - angle property

Since $\overline{AB}\perp\overline{CD}$, $\angle DCB = 90^{\circ}$. And $\angle DCE+\angle ECB=\angle DCB$. So, $(7x + 2)+(x + 8)=90$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $7x+x+2 + 8=90$, which simplifies to $8x+10 = 90$.

Step3: Solve for $x$

Subtract 10 from both sides: $8x=90 - 10=80$. Then divide both sides by 8: $x=\frac{80}{8}=10$.

Step4: Find the measure of $\angle DCE$

Substitute $x = 10$ into the expression for $\angle DCE$. $\angle DCE=7x + 2=7\times10+2=72^{\circ}$.

Answer:

$72^{\circ}$