Question

1. the figure shows a triangle with an angle bisector. find the measure of ∠1 if ∠1 = 12x - 1 and ∠dfe = 23x - 1.

Explanation:

Step1: Use the angle - bisector property

Since \( \angle 1=\angle 2\) and \( \angle DFE=\angle 1+\angle 2\), then \( \angle DFE = 2\angle 1\).
Substitute \( \angle 1 = 12x - 1\) and \( \angle DFE=23x - 1\) into \( \angle DFE = 2\angle 1\).
We get \(23x - 1=2(12x - 1)\).

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(23x - 1 = 24x-2\).
Move the \(x\) terms to one side: \(24x-23x=2 - 1\).
So \(x = 1\).

Step3: Find the measure of \( \angle 1\)

Substitute \(x = 1\) into \( \angle 1=12x - 1\).
\(\angle 1=12\times1 - 1=11^{\circ}\).

Answer:

C. \(11^{\circ}\)