triangle xyz is isosceles. the measure of the vertex angle, y, is twice the measure of a base angle. what is…

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.

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.

Answer

Explanation:

Step1: Let base - angle measure

Let the measure of each base angle ($\angle X$ and $\angle Z$) be $x$. Then the measure of the vertex angle $\angle Y$ is $2x$.

Step2: Use angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. So, $x + x+2x=180^{\circ}$. Combining like - terms, we get $4x = 180^{\circ}$.

Step3: Solve for $x$

Dividing both sides of the equation $4x = 180^{\circ}$ by 4, we have $x=\frac{180^{\circ}}{4}=45^{\circ}$. So, $\angle X=\angle Z = 45^{\circ}$ and $\angle Y=2x = 90^{\circ}$.

Step4: Analyze the perpendicular bisector property

Let $M$ be the mid - point of $\overline{XZ}$. The perpendicular bisector of $\overline{XZ}$ in right - isosceles $\triangle XYZ$ (since $\angle Y = 90^{\circ}$ and $\angle X=\angle Z = 45^{\circ}$) 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.