Question

c.6 angle bisectors if m∠dfe = m∠dfg = 40°, de = 2v, and dg = v + 27, what is the value of v? v = □

Explanation:

Step1: Identify the Angle Bisector Theorem

Since \( \angle DFE = \angle DFG = 40^\circ \), \( DF \) is the angle bisector of \( \angle EFG \). Also, \( DE \perp FE \) and \( DG \perp FG \) (right angles as shown), so by the Angle Bisector Theorem, the distances from a point on the angle bisector to the sides of the angle are equal. Thus, \( DE = DG \).

Step2: Set Up the Equation

Given \( DE = 2v \) and \( DG = v + 27 \), set them equal:
\( 2v = v + 27 \)

Step3: Solve for \( v \)

Subtract \( v \) from both sides:
\( 2v - v = v + 27 - v \)
\( v = 27 \)

Answer:

\( 27 \)