Triangle $$ X Y Z $$ is isosceles. The measure of the vertex angle, $$ Y $$, is twice the measure of a base angle. What is true about triangle $$ X Y Z $$ ? Select three options. Angle $$ Y $$ is a right angle. The measure of angle $$ Z $$ is $$ 45^{\circ} $$. The measure of angle $$ X $$ is $$ 36^{\circ} $$. The measure of the vertex angle is $$ 72^{\circ} $$. The perpendicular bisector of $$ \overline{x Z} $$ creates two smaller isosceles triangles.

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To solve the problem, we need to analyze the properties of triangle $$ XYZ $$ based on the given conditions.

1. **Understanding the triangle**: Triangle $$ XYZ $$ is isosceles, which means two of its angles are equal. Let's denote the base angles as $$ A $$ (for angle $$ X $$) and $$ A $$ (for angle $$ Z $$), and the vertex angle $$ Y $$ as $$ B $$.

2. **Given condition**: The measure of the vertex angle $$ Y $$ is twice the measure of a base angle. Therefore, we can express this relationship mathematically:
   $$
   B = 2A
   $$

3. **Sum of angles in a triangle**: The sum of the angles in any triangle is $$ 180^{\circ} $$:
   $$
   A + A + B = 180^{\circ}
   $$
   Simplifying this gives:
   $$
   2A + B = 180^{\circ}
   $$

4. **Substituting the expression for $$ B $$**:
   $$
   2A + 2A = 180^{\circ}
   $$
   $$
   4A = 180^{\circ}
   $$
   $$
   A = 45^{\circ}
   $$

5. **Finding $$ B $$**:
   $$
   B = 2A = 2 \times 45^{\circ} = 90^{\circ}
   $$

Now we have determined the measures of the angles:
- $$ A = 45^{\circ} $$ (angles $$ X $$ and $$ Z $$)
- $$ B = 90^{\circ} $$ (angle $$ Y $$)

### Conclusion:
Now we can evaluate the options provided:

1. **Angle $$ Y $$ is a right angle.** - **True** (since $$ Y = 90^{\circ} $$)
2. **The measure of angle $$ Z $$ is $$ 45^{\circ} $$.** - **True** (since $$ Z = A = 45^{\circ} $$)
3. **The measure of angle $$ X $$ is $$ 36^{\circ} $$.** - **False** (since $$ X = A = 45^{\circ} $$)
4. **The measure of the vertex angle is $$ 72^{\circ} $$.** - **False** (since $$ Y = 90^{\circ} $$)
5. **The perpendicular bisector of $$ \overline{x Z} $$ creates two smaller isosceles triangles.** - **True** (since the perpendicular bisector of the base of an isosceles triangle creates two smaller isosceles triangles).

### Final Selection:
The three true statements about triangle $$ XYZ $$ are:
1. Angle $$ Y $$ is a right angle.
2. The measure of angle $$ Z $$ is $$ 45^{\circ} $$.
3. The perpendicular bisector of $$ \overline{x Z} $$ creates two smaller isosceles triangles.
Triangle $$ XYZ $$ has a right angle at $$ Y $$, angle $$ Z $$ is $$ 45^{\circ} $$, and the perpendicular bisector of $$ \overline{x Z} $$ creates two smaller isosceles triangles.

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