Question

triangle xyz is isosceles. the measure of the vertex angle, y, is twice the measure of a base angle.
what is true about triangle xyz? choose three correct answers.
the perpendicular bisector of \\( \overline{xz} \\) creates two smaller isosceles triangles.
the measure of angle x is \\( 36 ^ { \circ } \\).
the measure of the vertex angle is \\( 72 ^ { \circ } \\).
the measure of angle z is \\( 45 ^ { \circ } \\).
angle y is a right angle.

Explanation:

Step1: Let the measure of base angles \( \angle X=\angle Z = x\)

Since the vertex angle \( \angle Y = 2x\)

Step2: Use the angle - sum property of a triangle

\(x + x+2x=180^{\circ}\) (sum of angles in a triangle is \(180^{\circ}\))
\(4x = 180^{\circ}\)
\(x = 45^{\circ}\)
\(\angle X=\angle Z = 45^{\circ}\), \(\angle Y=90^{\circ}\)

Step3: Analyze the perpendicular bisector of \(\overline{XZ}\)

Let \(M\) be the mid - point of \(\overline{XZ}\). In \(\triangle XYZ\), \(XY = YZ\) (isosceles triangle). The perpendicular bisector of \(\overline{XZ}\) (from \(Y\) to \(M\)):
In \(\triangle XYM\) and \(\triangle ZYM\), \(XY = YZ\), \(XM = ZM\), \(YM = YM\) (by SSS congruence). Also, \(\angle XYM=\angle ZYM = 45^{\circ}\), \(\angle XMY=\angle ZMY = 90^{\circ}\), \(\angle X=\angle Z = 45^{\circ}\). So, \(\triangle XYM\) and \(\triangle ZYM\) are isosceles.

Answer:

The perpendicular bisector of \(\overline{XZ}\) creates two smaller isosceles triangles; The measure of angle \(Z\) is \(45^{\circ}\); Angle \(Y\) is a right angle.