Final answer:
In an isosceles triangle like triangle XYZ, two angles are equal, the sum of all angles is 180 degrees, and a perpendicular bisector of the base creates two smaller isosceles triangles. Without additional information, we cannot definitively choose options A, B, C, or D, but we can confirm the truth of option E.
Step-by-step explanation:
Triangle XYZ is isosceles, which means two sides are of equal length and the angles opposite these sides are also equal. Without more specific information about triangle XYZ, we cannot determine whether angle Y is a right angle (option A), or whether angle Z is specifically 45° (option B), or whether angle X is 36° (option C). We are also not given information that confirms the vertex angle is 72° (option D); however, it is possible to deduce certain truths about isosceles triangles in general:
If the vertex angle is known and is 72°, then the base angles would also be equal and sum to 108° (since the sum of all angles in any triangle is 180°), so the base angles would each be 54°, not 36° or 45°.
If a perpendicular bisector of the base of an isosceles triangle is drawn, it will indeed bisect the vertex angle, creating two smaller isosceles triangles, with the line of symmetry being the bisector (option E).
To answer the student's question directly, we can confirm that the following is true:
The sum of the angles in the triangle is 180 degrees.
If the perpendicular bisector of the base (XZ) is drawn, it creates two smaller isosceles triangles (option E).
Since it's an isosceles triangle, two angles have to be equal, which are the angles opposite the equal sides.