Question

1. if $overline{bd}perpoverline{ac}$, $mangle dbe=(2x - 1)^{circ}$, and $mangle cbe=(5x - 35)^{circ}$, find the value of $x$.

Explanation:

Step1: Identify angle - relationship

Since $\overline{BD}\perp\overline{AC}$, $\angle DBC = 90^{\circ}$. And $\angle CBE=\angle DBC+\angle DBE$. So $\angle CBE-\angle DBE = 90^{\circ}$.

Step2: Substitute angle - measures

Substitute $m\angle DBE=(2x - 1)^{\circ}$ and $m\angle CBE=(5x - 35)^{\circ}$ into the equation: $(5x - 35)-(2x - 1)=90$.

Step3: Simplify the equation

Expand the left - hand side: $5x-35 - 2x + 1=90$. Combine like terms: $3x-34 = 90$.

Step4: Solve for x

Add 34 to both sides of the equation: $3x=90 + 34$, so $3x=124$. Then divide both sides by 3: $x=\frac{124}{3}= 41\frac{1}{3}$.

Answer:

$x = 41\frac{1}{3}$