10:05 5G 80 Assignment Overview 10 DeltaMath Interior and Exterior Triangle Angles © This is the only question in this section. Question Video Examples In \( \triangle \mathrm{DEF}, \overline{D F} \) is extended through point F to point \( \mathrm{G}, \mathrm{m} \angle F D E=(x+2)^{\circ} \), \( \mathrm{m} \angle D E F=(2 x+3)^{\circ} \), and \( \mathrm{m} \angle E F G=(6 x-16)^{\circ} \). What is the value of \( x \) ? Answer Attempt 1 out of 2 \( \square \) Submit Answer Copyright ©2025 DeltaMath.com All Rights Reserved. Terms \& Policies deltamath.com

To determine the value of \( x \) in the given triangle configuration, let's analyze the information step by step. ### Given: - **Triangle DEF** with points \( D \), \( E \), and \( F \). - **\( \overline{DF} \)** is extended through point \( F \) to point \( G \). - **Angles:** - \( \angle FDE = (x + 2)^\circ \) - \( \angle DEF = (2x + 3)^\circ \) - \( \angle EFG = (6x - 16)^\circ \) ### Objective: Find the value of \( x \). ### Step-by-Step Solution: 1. **Sum of Interior Angles in Triangle DEF:** The sum of the interior angles in any triangle is \( 180^\circ \). \[ \angle FDE + \angle DEF + \angle DFE = 180^\circ \] Plugging in the given expressions: \[ (x + 2) + (2x + 3) + \angle DFE = 180 \] Combine like terms: \[ 3x + 5 + \angle DFE = 180 \] Solving for \( \angle DFE \): \[ \angle DFE = 180 - (3x + 5) = 175 - 3x \] 2. **Exterior Angle at Point F (Angle EFG):** When \( \overline{DF} \) is extended to \( G \), \( \angle EFG \) becomes an exterior angle at point \( F \). **Exterior Angle Theorem:** The measure of an exterior angle is equal to the sum of the measures of the two non-adjacent interior angles. Therefore: \[ \angle EFG = \angle FDE + \angle DEF = (x + 2) + (2x + 3) = 3x + 5 \] 3. **Setting Up the Equation:** According to the problem, \( \angle EFG = (6x - 16)^\circ \). Therefore: \[ 3x + 5 = 6x - 16 \] 4. **Solving for \( x \):** Subtract \( 3x \) from both sides: \[ 5 = 3x - 16 \] Add \( 16 \) to both sides: \[ 21 = 3x \] Divide both sides by \( 3 \): \[ x = 7 \] ### **Answer:** \( x = 7 \)