Question

applying the vertex angle of an isosceles triangle theorem
\\( \overline { zx } \\) bisects \\( \angle wzy \\). if the measure of \\( \angle yxz \\) is \\( ( 6 m - 12 ) ^ { \circ } \\), what is
the value of \\( m \\)?

Explanation:

Step1: Recall the Vertex Angle of an Isosceles Triangle Theorem

In an isosceles triangle \( \triangle WZY\) (since \( WZ = YZ\)), if \( \overline{ZX}\) bisects \( \angle WZY\), then \( \overline{ZX}\perp\overline{WY}\) (by the Vertex - Angle Bisector Theorem of an isosceles triangle: the bisector of the vertex angle of an isosceles triangle is also the perpendicular bisector of the base). So, \( \angle YXZ = 90^{\circ}\).

Step2: Set up the equation

We know that \( \angle YXZ=(6m - 12)^{\circ}\), and since \( \angle YXZ = 90^{\circ}\), we set up the equation \(6m-12 = 90\).

Step3: Solve the equation for \(m\)

Add \(12\) to both sides of the equation: \(6m-12 + 12=90 + 12\), which simplifies to \(6m=102\).
Divide both sides by \(6\): \(m=\frac{102}{6}=17\).

Answer:

\(17\)