A \(45^\circ\)–\(45^\circ\)–\(90^\circ\) set-square has _____ line/lines of symmetry.
one.
Given: A (45^circ!-!45^circ!-!90^circ) set-square (a right triangle with two equal angles).
This triangle has two equal angles and one right angle:
(angle A = 45^circ), (angle B = 45^circ), (angle C = 90^circ).
Because two angles are equal, the two sides opposite them are also equal. So it is an isosceles right triangle.
A line of symmetry is a line that folds a shape into two matching halves.
After folding on this line, every point on one side lies exactly on top of a matching point on the other side.
In an isosceles triangle, the symmetry line goes through the vertex between the two equal sides and bisects that angle.
Here, the equal sides meet at the right-angle vertex. So the symmetry line is the bisector of the right angle.
That means the line makes two equal angles of (45^circ) with the sides that form the right angle.
Imagine folding the triangle along this angle-bisector line:
(angle C = 90^circ) is split into (45^circ) and (45^circ).
The two equal legs match exactly after folding.
The right-angle vertex lies on the fold line, so it stays fixed. The other two vertices swap places.
Try any other line (for example, through the midpoint of the hypotenuse or along a side):
The hypotenuse and a leg are not equal, so they cannot match after folding.
Therefore, no other line can create two identical halves.
The triangle has exactly one line of symmetry — the bisector of the right angle.
Answer: one.