Questions: Find the range of values for x .

△ Determine the range of values for \( x \). ○ Recognize isosceles triangles ☼ The larger triangle's markings indicate that the two triangles sharing the top side are isosceles, meaning two of their sides are equal in length. ○ Equal sides, equal angles ☼ In isosceles triangles, the angles opposite the equal sides are equal. ○ Set up equations for angles ☼ In the left triangle, the base angles are both \( 42^\circ \). In the right triangle, the base angles are both \( (3x + 15)^\circ \). ○ Apply triangle angle sum ☼ The sum of angles in any triangle is \( 180^\circ \). ○ Solve for \( y \) in the right triangle ☼ Using the equation \( y + y + (3x + 15) = 180 \), we get \( 2y + 3x + 15 = 180 \), leading to \( 2y + 3x = 165 \) and \( y = \frac{165 - 3x}{2} \). ○ Solve for \( x \) ☼ Since the two smaller triangles are congruent by SSS, the angles \( 42^\circ \) and \( (3x + 15)^\circ \) must be equal. Solving \( 42 = 3x + 15 \) gives \( 27 = 3x \) and \( x = 9 \). ✧ The value of \( x \) is \( 9 \). ☺ x = 9