Question

1. if bd ⊥ ac, m∠dbe=(2x - 1)°, and m∠cbe=(5x - 42)°, find the value of x.

Explanation:

Step1: Recall perpendicular - angle property

Since $BD\perp AC$, $\angle DBA = 90^{\circ}$ and $\angle DBC=90^{\circ}$. Also, $\angle CBE+\angle DBE=\angle DBC = 90^{\circ}$.

Step2: Set up the equation

We know that $m\angle CBE=(5x - 42)^{\circ}$ and $m\angle DBE=(2x - 1)^{\circ}$. So, $(5x - 42)+(2x - 1)=90$.

Step3: Simplify the left - hand side of the equation

Combine like terms: $5x+2x-42 - 1=90$, which gives $7x-43 = 90$.

Step4: Solve for x

Add 43 to both sides of the equation: $7x=90 + 43$, so $7x=133$. Then divide both sides by 7: $x=\frac{133}{7}=19$.

Answer:

$x = 19$