Question

in triangle def, \\( \angle d=(2 x+42) \\) , \\( \angle e=(2 x-15) \\) , and \\( \angle f=3 x+48 \\) .
find the value of \\( x \\).

Explanation:

Step1: Recall triangle angle - sum property

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(\angle D+\angle E+\angle F = 180^{\circ}\).
Substitute \(\angle D=(2x + 42)^{\circ}\), \(\angle E=(2x-15)^{\circ}\), and \(\angle F=(3x + 48)^{\circ}\) into the equation:
\((2x + 42)+(2x-15)+(3x + 48)=180\)

Step2: Simplify the left - hand side of the equation

Combine like terms:

$$
LATEXBLOCK0
$$

Step3: Solve for \(x\)

Subtract \(75\) from both sides of the equation \(7x+75 = 180\):
\(7x=180 - 75\)
\(7x=105\)
Divide both sides by \(7\): \(x=\frac{105}{7}=15\)

Answer:

\(x = 15\)