Question

question
in \\(\triangle def\\), \\(\overline{df}\\) is extended through point \\(f\\) to point \\(g\\), \\(m\angle def = (2x + 13)\degree\\), \\(m\angle fde = (2x + 7)\degree\\), and \\(m\angle efg = (8x - 4)\degree\\). find \\(m\angle def\\).
answer attempt 1 out of 2
\\(m\angle def = \square\degree\\)
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Explanation:

Step1: Apply Exterior Angle Theorem

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. In $\triangle DEF$, $\angle EFG$ is an exterior angle at $F$, so $m\angle EFG=m\angle DEF + m\angle FDE$.

Substitute the given angle measures: $(8x - 4)=(2x + 13)+(2x+7)$

Step2: Simplify the equation

First, simplify the right - hand side of the equation:
$(2x + 13)+(2x + 7)=2x+2x+13 + 7=4x+20$

So our equation becomes $8x-4 = 4x + 20$

Step3: Solve for x

Subtract $4x$ from both sides: $8x-4x-4=4x-4x + 20$
$4x-4=20$

Add 4 to both sides: $4x-4 + 4=20 + 4$
$4x=24$

Divide both sides by 4: $x=\frac{24}{4}=6$

Step4: Find $m\angle DEF$

We know that $m\angle DEF=(2x + 13)^{\circ}$. Substitute $x = 6$ into the expression:

$m\angle DEF=2\times6+13=12 + 13=25$

Answer:

25