Question

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

Explanation:

Step1: Identify angle - relationship

Since $BD\perp AC$, $\angle DBC = 90^{\circ}$. And $\angle CBE=\angle DBC+\angle DBE$. So, $(5x - 42)=(2x - 1)+90$.

Step2: Solve the equation for $x$

First, expand the right - hand side: $5x-42 = 2x-1 + 90$.
Combine like terms: $5x-42=2x + 89$.
Subtract $2x$ from both sides: $5x-2x-42=2x-2x + 89$, which gives $3x-42 = 89$.
Add 42 to both sides: $3x-42 + 42=89 + 42$, so $3x=131$.
Divide both sides by 3: $x=\frac{131}{3}$.

Answer:

$x=\frac{131}{3}$