Question

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

Explanation:

Step1: Identify angle - relationship

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

Step2: Solve the equation for x

Combine like - terms: $7x+x+2 + 8=90$, which simplifies to $8x+10 = 90$. Subtract 10 from both sides: $8x=90 - 10=80$. Then divide both sides by 8: $x=\frac{80}{8}=10$.

Step3: Find the measure of $\angle DCE$

Substitute $x = 10$ into the expression for $\angle DCE$. $\angle DCE=(7x + 2)^{\circ}$. So, $\angle DCE=7\times10+2=70 + 2=72^{\circ}$.

Answer:

$72^{\circ}$