Question

1. the diagram shows (overrightarrow{ec}) intersecting (overrightarrow{bd}) and (overrightarrow{ba}) at point (b).

if (angle dbe) and (angle abc) are complementary angles, what is the measure of (angle abc)?
a. (15.5^{circ})
b. (32^{circ})
c. (58^{circ})
d. (125.5^{circ})

Explanation:

Step1: Use the property of complementary angles

Since \(\angle DBE=(2x + 1)^{\circ}\), \(\angle ABC=(6x-35)^{\circ}\) and \(\angle DBE+\angle ABC = 90^{\circ}\) (complementary angles), we have the equation \((2x + 1)+(6x-35)=90\).

Step2: Solve the equation for \(x\)

Simplify the left - hand side: \(2x+1 + 6x-35=8x-34\).
So, \(8x-34 = 90\).
Add \(34\) to both sides: \(8x=90 + 34=124\).
Divide both sides by \(8\): \(x=\frac{124}{8}=15.5\).

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

Substitute \(x = 15.5\) into \(\angle ABC=(6x-35)^{\circ}\).
\(\angle ABC=6\times15.5-35\).
\(=93-35\).
\(=58^{\circ}\).

Answer:

C. \(58^{\circ}\)