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? select three options. angle y is a right angle. the measure of angle z is 45°. the measure of angle x is 36°. the measure of the vertex angle is 72°. the perpendicular bisector of $overline{xz}$ creates two smaller isosceles triangles.

Explanation:

Step1: Let base - angle measure be $x$.

Let the measure of base angles $\angle X=\angle Z = x$. The measure of vertex angle $\angle Y = 2x$.

Step2: Use angle - sum property of a triangle.

In $\triangle XYZ$, $\angle X+\angle Y+\angle Z=180^{\circ}$. Substituting the values, we get $x + 2x+x=180^{\circ}$, which simplifies to $4x = 180^{\circ}$.

Step3: Solve for $x$.

Dividing both sides of $4x = 180^{\circ}$ by 4, we have $x=\frac{180^{\circ}}{4}=45^{\circ}$.
So, $\angle X=\angle Z = 45^{\circ}$ and $\angle Y=2x = 90^{\circ}$.
For the perpendicular bisector of $\overline{XZ}$:
Let $M$ be the mid - point of $\overline{XZ}$. In $\triangle XYZ$, since $\triangle XYZ$ is isosceles with $\angle Y = 90^{\circ}$, the perpendicular bisector of the non - congruent side $\overline{XZ}$ (from the right - angle vertex $Y$ to the mid - point $M$ of $\overline{XZ}$) divides $\triangle XYZ$ into two smaller isosceles right - triangles $\triangle XYM$ and $\triangle ZYM$.

Answer:

Angle Y is a right angle.
The measure of angle Z is 45°.
The perpendicular bisector of $\overline{XZ}$ creates two smaller isosceles triangles.