Question 15

We will draw different shapes based on their symmetry rules.

Step 1 — Triangle with many symmetries

We need a triangle with at least two lines of symmetry. We also need at least two angles of symmetry. An equilateral triangle has all sides equal. It has 3 lines of symmetry. It looks the same if we turn it by 120 degrees. It also looks the same if we turn it by 240 degrees. And it looks the same if we turn it by 360 degrees. So, it has 3 angles of symmetry. This triangle fits all the rules.

$\boxed{\text{Equilateral Triangle}}$

Step 2 — Triangle with one line of symmetry

We need a triangle with only one line of symmetry. It must not have rotational symmetry (other than turning it a full 360 degrees). An isosceles triangle has two sides equal. It has exactly 1 line of symmetry. This line goes from the top corner to the middle of the base. If we turn it less than a full circle, it does not look the same. So, it has no rotational symmetry (other than 360 degrees). This triangle fits all the rules.

$\boxed{\text{Isosceles Triangle}}$

Step 3 — Quadrilateral with rotational symmetry only

We need a four-sided shape with rotational symmetry. But it must have no reflection symmetry (no lines of symmetry). A parallelogram has opposite sides parallel. If it is not a rectangle or a rhombus, it has no lines of symmetry. But we can turn it by 180 degrees, and it looks the same. This means it has rotational symmetry of order 2. This shape fits all the rules.

$\boxed{\text{Parallelogram (not a rectangle or rhombus)}}$

Step 4 — Quadrilateral with reflection symmetry only

We need a four-sided shape with reflection symmetry. But it must not have rotational symmetry (other than turning it a full 360 degrees). An isosceles trapezoid has one pair of parallel sides. Its non-parallel sides are equal. It has exactly 1 line of symmetry. This line goes through the middle of the parallel sides. If we turn it less than a full circle, it does not look the same. A kite has two pairs of equal-length sides that are next to each other. It also has exactly 1 line of symmetry. It also has no rotational symmetry (other than 360 degrees). Both these shapes fit all the rules.

$\boxed{\text{Isosceles Trapezoid or Kite}}$

Answer

(a) An equilateral triangle has 3 lines of symmetry and 3 angles of symmetry (which is at least two). (b) An isosceles triangle has exactly 1 line of symmetry but no rotational symmetry (other than 360°). (c) A parallelogram (which is not a rectangle or rhombus) has rotational symmetry of order 2 but no lines of symmetry. (d) An isosceles trapezoid or a kite has reflection symmetry but no rotational symmetry (other than 360°).