Question

1. the diagram shows \\(\overleftrightarrow{ea}\\) intersecting \\(\overleftrightarrow{bc}\\) and \\(\overleftrightarrow{bd}\\) at point \\(b\\).

if \\(\angle abc\\) and \\(\angle dbe\\) are complementary angles, what is the measure of \\(\angle dbe\\)?
a. \\(58^\circ\\)
b. \\(32^\circ\\)
c. \\(25.1^\circ\\)
d. \\(12.2^\circ\\)

Explanation:

Step1: Set up the equation

Since \( \angle ABC=(5x - 3)^{\circ}\), \( \angle DBE=(2x + 7.6)^{\circ}\) and \( \angle ABC+\angle DBE = 90^{\circ}\) (complementary angles), we have the equation \( (5x-3)+(2x + 7.6)=90\).
Simplify the left - hand side: \( 5x-3+2x + 7.6=7x+4.6\). So, \( 7x+4.6 = 90\).

Step2: Solve for \(x\)

Subtract \(4.6\) from both sides: \(7x=90 - 4.6=85.4\).
Divide both sides by \(7\): \(x=\frac{85.4}{7}=12.2\).

Step3: Find the measure of \( \angle DBE\)

Substitute \(x = 12.2\) into the expression for \( \angle DBE\): \( \angle DBE=(2x + 7.6)^{\circ}\).
\( \angle DBE=2\times12.2+7.6\).
First, calculate \(2\times12.2 = 24.4\), then \(24.4+7.6=32^{\circ}\).

Answer:

B. \(32^{\circ}\)