Question

1. if ab ⊥ cd, m∠dce=(7x + 2)° and m∠ecb=(x + 8)°, find the measure of ∠dce.

Explanation:

Step1: Recall perpendicular - angle property

Since $\overline{AB}\perp\overline{CD}$, $\angle DCB = 90^{\circ}$. And $\angle DCB=\angle DCE+\angle ECB$.

Step2: Set up the equation

We know that $m\angle DCE=(7x + 2)^{\circ}$ and $m\angle ECB=(x + 8)^{\circ}$, so $(7x + 2)+(x + 8)=90$.

Step3: Simplify the left - hand side of the equation

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

Step4: Solve for $x$

Subtract 10 from both sides: $8x=90 - 10=80$. Then divide both sides by 8, so $x = 10$.

Step5: Find the measure of $\angle DCE$

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

Answer:

$72^{\circ}$