Question

If a quadrilateral has exactly 2 lines of symmetry, and both are angle bisectors, then which statement would be true?

The figure must be an isosceles trapezoid because it has 2 congruent base angles.
The figure must be a rectangle because all rectangles have exactly 2 lines of symmetry.
The figure could be a rhombus because the 2 lines of symmetry bisect the angles.
The figure could be a square because the diagonals of a square bisect the right angles.

Answer 1

To evaluate the given statements, we start by recognizing that a quadrilateral with exactly 2 lines of symmetry that are both angle bisectors typically has a certain shape.

1. **The figure must be an isosceles trapezoid because it has 2 congruent base angles.**
- This statement is true in the sense that an isosceles trapezoid has a line of symmetry along the vertical line that bisects the bases and the angles at the top and bottom. It can have 2 lines of symmetry if we consider the angle bisectors as the lines that bisect the top and bottom angles, leading to the conclusion that it could be an isosceles trapezoid; however, it is not definitive that this is the only shape.

2. **The figure must be a rectangle because all rectangles have exactly 2 lines of symmetry.**
- This statement is false; while rectangles do have lines of symmetry, they also have 4 lines of symmetry (two vertical and two horizontal). Therefore, a rectangle does not fit the criteria of having exactly 2 lines of symmetry.

3. **The figure could be a rhombus because the 2 lines of symmetry bisect the angles.**
- A rhombus does have 2 lines of symmetry that are the diagonals, and they indeed bisect the angles. Therefore, a rhombus fits this description.

4. **The figure could be a square because the diagonals of a square bisect the right angles.**
- A square is a specific type of rhombus, and it does have 4 lines of symmetry (two diagonals and two midlines). Thus, while this statement is true about a square having diagonals that bisect angles, it does not fit the criteria of having exactly 2 lines of symmetry.

From our evaluations, the best fitting statement that satisfies the criteria of having exactly 2 lines of symmetry (which are angle bisectors) is:

**The figure could be a rhombus because the 2 lines of symmetry bisect the angles.**

Thus, the true statement is that the figure could be a rhombus.